KOSZUL DUALITY PATTERNS IN REPRESENTATION THEORY

ALEXANDER BEILINSON, VICTOR GINZBURG, AND WOLFGANG SOERGEL

Abstract. The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other.

The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain ℤ-graded rings with particularily nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured in [BG86] and proved in [Soe90]. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.

Contents

1. Introduction 1 2. Koszul rings 8 3. Parabolic-singular duality and Koszul duality 24 4. Koszul rings arising from ‘mixed’ geometry. 40 References 52

1. Introduction

1.1. Parabolic-singular duality. Let 𝔤 ⊃ 𝔟 ⊃ 𝔥 be a complex semisimple Lie algebra, a Borel and a Cartan therein. Let U = U(𝔤) be the universal enveloping algebra of 𝔤. Let O be the category of all finitely generated U-modules which are locally finite over 𝔟 and semisimple over 𝔥, see for example [BG80]. For λ ∈ 𝔥^{∗}

1991 Mathematics Subject Classification. Primary 17B10; Secondary 16A03. The first author was partially supported by an NSF grant. The second author thanks Harvard University and MIT, where part of this work has been

written.

The third author thanks the MPI and DFG for financial support.

consider in O the Verma module M(λ) = U ⊗_{U(𝔟)} ℂ_{λ}, its simple quotient L(λ) and the (indecomposable) projective cover P (λ) of L(λ) in O.

Let Z ⊂ U be the center. Let L ∈ O be the direct sum of all simple highest weight modules L(λ) which have trivial infinitesimal character, i.e. which have the same annihilator in Z as the trivial representation ℂ = L(0). There are but finitely many of those, parametrized by the Weyl group. Let P ∈ O be the direct sum of their respective projective covers. The first statement of the following theorem is the main result of [Soe90].

Theorem 1.1.1. There exists an isomorphism of finite dimensional ℂ-algebras

End_{O}P ≅ Ext_{O}^{•}(L, L)

where Ext^{•} stands for the direct sum of all Ext^{i}, made into a ring via the cup product. Furthermore Ext_{O}^{•}(L, L) is a Koszul ring.

We have to explain what a Koszul ring is.

Definition 1.1.2. A Koszul ring is a positively graded ring A = ⨁_{j≥0} A_{j} such that (1) A_{0} is semisimple and (2) A_{0} considered as a graded left A-module admits a graded projective resolution

⋯ → P^{2} → P^{1} → P^{0} ↠ A_{0}

such that P^{i} is generated by its degree i component, i.e., P^{i} = AP_{i}^{i}.

Morally a Koszul ring is a graded ring that is “as close to semisimple as a ℤ-graded ring possibly can be”. This statement will be made more precise by proposition 2.1.3.

We explain next the meaning of the ring E = End_{O}P. If A is an abelian category whose objects have finite length and P ∈ A a projective generator of A (i.e. a projective object surjecting onto each simple object in A), then there is a canonical standard (see e.g., [Bas68] the exercise following theorem 1.3 from chapter I) equivalence of categories

Hom_{A}(P, ) : A → Mof-E  ,  E = End_{A}P

where Mof- stands for the category of finitely generated right modules. Now let Z^{+} = Ann_{Z}ℂ ⊂ Z be the augmentation ideal of the center. Apply the considerations above to the category O_{0} = {M ∈ O | (Z^{+})^{n}M = 0 for n ≫ 0}. Thus we get an equivalence of categories

O_{0} ≅ Mof-(End_{O}P).

So the theorem says in particular that the category O_{0} is governed by a Koszul ring, i.e. that it is in some sense “graded semisimple”. We hope to convince our readers that this is a good way to formalize the beauty of this part of representation theory, and we believe that similar results should hold for all kinds of representation theories associated to root systems.

As a first step we explain how to generalize the above to other infinitesimal characters. Let ρ ∈ 𝔥^{∗} be the half sum of positive roots, i.e. of the roots in 𝔟. Let (W, S) be the Coxeter system of 𝔤 ⊃ 𝔟 and let w_{∘} be the longest element of the Weyl group W. For w ∈ W, λ ∈ 𝔥^{∗} we put w · λ = w(λ + ρ) − ρ. For any subset S_{ι} ⊂ S—here ι is a variable index—let W_{ι} ⊂ W be the subgroup generated by S_{ι}, w_{ι} ∈ W_{ι} its longest element and W^{ι} ⊂ W the set of longest representatives of the cosets W/W_{ι}.

KOSZUL DUALITY PATTERNS… 3

For λ ∈ 𝔥^{∗} integral dominant but perhaps singular (in other words, λ is integral and λ + ρ lies in the closure of the dominant Weyl chamber) define the category O_{λ} ⊂ O to consist of all objects of O whose (generalized) infinitesimal character coincides with the one of L(λ). Set S_{λ} = {s ∈ S | s · λ = λ}. Then {L(x · λ)}_{x∈W^{λ}} represents the isomorphism classes of simple objects in O_{λ}. The P (x · λ) are the projective covers of L(x · λ) in O_{λ} as well.

For 𝔮 ⊂ 𝔤 a parabolic subalgebra containing 𝔟 define the subcategory O^{𝔮} ⊂ O_{0} to consist of all 𝔮-locally finite objects. Let S_{𝔮} ⊂ S be the simple reflections corresponding to 𝔮. For all x ∈ W^{𝔮} we set L_{x}^{𝔮} = L(x^{−1}w_{∘} · 0) ∈ O^{𝔮}. The L_{x}^{𝔮} represent the isomorphism classes of simple objects in O^{𝔮}. Any L_{x}^{𝔮} has a projective cover P_{x}^{𝔮} in O^{𝔮}.

We will prove in section 3

Theorem 1.1.3. Let λ, 𝔮 be as above and suppose S_{λ} = S_{𝔮}. Then there are isomorphisms of finite dimensional ℂ-algebras

End_{O_{λ}}(⨁ P (x · λ)) ≅ Ext_{O^{𝔮}}^{•}(⨁ L_{x}^{𝔮}, ⨁ L_{x}^{𝔮})
End_{O^{𝔮}}(⨁ P_{x}^{𝔮})  ≅ Ext_{O_{λ}}^{•}(⨁ L(x · λ), ⨁ L(x · λ)).

Both algebras on the right are Koszul rings and, moreover, these rings are Koszul dual to one another.

Remarks . 1. We have to explain when two Koszul rings are dual to one another. For every positively graded ring A we may consider A_{0} as a left A-module A_{0} = A/A_{>0} and form the positively graded ring E(A) = Ext_{A}^{•}(A_{0}, A_{0}). For a Koszul ring A satisfying a suitable finiteness condition we prove as theorem 2.10.2 that E(A) is Koszul as well and E(E(A)) = A canonically. Motivated by this equality we call E(A) the dual of A.

2. Remark that O^{𝔟} = O_{0}. So the theorem tells us in particular that the Koszul ring from theorem 1.1.1 is selfdual.

3. All sums go over x ∈ W^{𝔮} = W^{λ}. Let 1_{x} be the obvious pairwise orthogonal idempotents in our four algebras. The isomorphisms of the theorem can (and will) be choosen so that the 1_{x} correspond to each other under the isomorphisms above.

4. From what we explained before it is clear that the endomorphism algebras of projectives appearing in the theorem govern the categories O_{λ} and O^{𝔮} respectively.

5. The projective covers appearing in the theorem are unique only up to nonunique isomorphism, hence we cannot expect the isomorphisms predicted by the theorem to be canonical. We can however construct without choices a collection of possible P (x · λ) along with a canonical isomorphism as in the theorem for this special collection.

6. Morally we should rather consider categories O_{λ} and O^{𝔮} for Langlands-dual Lie algebras. In fact one might then view O_{λ} as a category of Harish-Chandra modules for a complex group and O^{𝔮} as a category of perverse sheaves on the corresponding space of Langlands parameters, more precisely its modified version defined by Adams-Barbasch-Vogan [ABV92]. Our theorem can then be understood as a refinement of Langlands’ philosophy in the case of complex groups. More details on this point of view can be found in [Soe92].

This article is the result of our efforts to join the two partially wrong and quite incomplete preprints [Soe89b] and [BG86] into one readable article.

First we approach the above results in as elementary a way as possible. The only facts relying on “mixed” geometry we really use are the computation of the intersection cohomology of Schubert varieties and lemma 3.5.4. In the last section we adopt a more powerful technique of mixed geometry.

1.2. Koszul duality. The results described here and proved in section 2 are essentially not new but rather generalizations of results of Priddy [Pri70], Löfwall [Löf86], Backelin [Bac82], Backelin-Fröberg [BF85], Bernstein-Gelfand-Gelfand [BGG78] and specializations of results from [BGS88]. We start repeating our definition of a Koszul ring.

Definition 1.2.1. A Koszul ring is a positively graded ring A = ⨁_{j≥0} A_{j} such that (1) A_{0} is semisimple and (2) A_{0} considered as a graded left A-module admits a graded projective resolution

⋯ → P^{2} → P^{1} → P^{0} ↠ A_{0}

such that P^{i} is generated by its component of degree i, P^{i} = AP_{i}^{i}.

Remarks . 1. A complex of graded A-modules

⋯ → P^{2} → P^{1} → P^{0} ↠ A_{0}

as in the definition is unique up to unique isomorphism and has a natural right A_{0}-action. This follows from the obvious fact that if Q^{•} is another resolution of A_{0} as in the definition, then any chain map P^{•} → Q^{•} which respects the grading on the P^{i}, Q^{i} and is homotopic to zero has itself to be zero.

2. Let V be a finite dimensional vector space over a field k. Then the symmetric algebra SV is Koszul, as can be seen from the Koszul complex

⋯ → SV ⊗ ⋀^{2} V → SV ⊗ V → SV ↠ k.

3. We will prove in section 2.2 that A is Koszul if and only if A^{opp} is.

Definition 1.2.2. A quadratic ring is a positively graded ring A = ⨁_{j≥0} A_{j} such that A_{0} is semisimple and A is generated over A_{0} by A_{1} with relations of degree two.

This may be spelled out as follows: Let

T_{A_{0}}A_{1} = A_{0} ⊕ A_{1} ⊕ (A_{1} ⊗_{A_{0}} A_{1}) ⊕ … = ⨁_{i≥0} A_{1}^{⊗i}

be the free tensor algebra of the A_{0}-bimodule A_{1}. Then the canonical map can : T_{A_{0}}A_{1} → A is required to be a surjection and ker(can) is required to be generated, as a two-sided ideal, by its component R = ker(can) ∩ (A_{1} ⊗_{A_{0}} A_{1}) of degree two, ker(can) = (R), so that T_{A_{0}}A_{1}/(R) = A. In section 2.3 we show:

Proposition 1.2.3. Any Koszul ring is quadratic.

Let us remark that for any semisimple ring k and any k-bimodule V the quadratic ring with no relations, T_{k}V, is Koszul. To have duality we need some finiteness conditions.

Definition 1.2.4. A graded ring A = ⊕_{i}A_{i} is called “left finite” (resp. “right finite”) if and only if all A_{i} are finitely generated as left (resp. right) A_{0}-modules.

For any positively graded ring A = ⨁_{j≥0} A_{j} consider now the left A-module A_{0} and form the graded ring E(A) = Ext_{A}^{•}(A_{0}, A_{0}) of self-extensions of A_{0}. We will prove as theorem 2.10.2

Theorem 1.2.5. Suppose A is a left finite Koszul ring. Then E(A) is a left finite Koszul ring as well, and E(E(A)) = A canonically.

In the situation of the theorem we say that the Koszul rings A and E(A) are “dual to each other”. For example, the Koszul dual of the symmetric algebra SV above is the exterior algebra ⋀ V^{∗} on the dual space of V, and for a bimodule V over a semisimple ring k that is finitely generated for the left k-action the Koszul dual of T_{k}V is (k ⊕ Hom_{k}(V, k))^{opp}. The theorem is proved by identifying E(A) with the opposed ring of the more elementary quadratic dual ring A^{!} of A, which we will however not define in this introduction.

Next we explain Koszul duality at the level of derived categories. To do so we need more notations. For a graded ring A = ⨁_{j} A_{j} let A-Mod denote the category of all left A-modules M and A-mod the category of all graded left Amodules M = ⨁_{j} M_{j}. Let Hom_{A} (resp. hom_{A}) denote homomorphisms in A-Mod (resp. A-mod). So hom_{A}(M, N) = {f ∈ Hom_{A}(M, N) | f(M_{j}) ⊂ N_{j}}. We define the grading shifts ⟨n⟩ by (M⟨n⟩)_{j} = M_{j−n}. The respective subcategories of finitely generated (graded) modules will be denoted A-Mof and A-mof. These are abelian categories provided A is left noetherian.

For any abelian category A let D^{b}(A) be the corresponding bounded derived category. It comes equipped with the structure of a triangulated category, in particular with a shift functor [1] : D^{b}(A) → D^{b}(A). It also comes with a canonical degree zero embedding A → D^{b}(A).

Let now A be a Koszul ring that satisfies the following two finiteness conditions:

(1) A is a finitely generated A_{0}-module from the left as well as from the right, in particular A_{i} = 0 for all i ≫ 0; (2) E(A) is right noetherian. Put A^{!} = E(A)^{opp}. Then A_{0}^{!} = A_{0} canonically, and we will denote this semisimple ring by k. We will also need the injective graded A-module A^{∗} = Hom_{k}(A, k). Remark that it comes with a right k-action.

Under the above assumptions (1) and (2) we will construct in section 2.12 (in particular theorem 2.12.6) an exact functor between triangulated categories

K : D^{b}(A-mof) → D^{b}(A^{!}-mof)

(the “Koszul duality functor”) and prove:

Theorem 1.2.6. The functor K is an equivalence of triangulated categories. We have K(M⟨n⟩) = (KM)[−n]⟨−n⟩ canonically for M ∈ D^{b}(A-mof). Furthermore for all p ∈ k we have K(A_{0}p) = A^{!}p, K(A^{∗}p) = A_{0}^{!}p.

Remarks . 1. The last statement is to be understood via the embeddings A-mof → D^{b}(A-mof), A^{!}-mof → D^{b}(A^{!}-mof). Note also that any simple object of A-mof (resp. A^{!}-mof) has up to a grading shift the form A_{0}p (resp. A_{0}^{!}p) for suitable p ∈ k. Furthermore A^{!}p is the projective cover of A_{0}^{!}p and A^{∗}p the injective hull of A_{0}p for any p ∈ k. So roughly speaking Koszul duality transforms simples to indecomposable projectives and indecomposable injectives to simples.

2. For a Koszul ring A one might interpret E(A) as being RHom_{A}(k, k) and then the Koszul duality functor K is just the functor RHom_{A}(k, ). We will however not pursue this perspective in our paper. More details on this point of view can be found in [Kel92].

3. In the case of the symmetric algebra the equivalence

D^{b}(⋀ V^{∗}-mof) ≅ D^{b}(SV -mof)

was established in [BGG78].

1.3. Parabolic-singular duality and Koszul duality. Let us start with an application. As explained in section 2.4, every module of finite length M has a radical filtration

M = rad^{0}M ⊃ rad^{1}M ⊃ rad^{2}M ⊃ …

with semisimple subquotient \overline{ra}d^{i} M = rad^{i}M/rad^{i+1}M called the i-th layer of the radical filtration. For x ∈ W^{𝔮} let M_{x}^{𝔮} ∈ O^{𝔮} be the corresponding parabolic Verma module, i.e. the maximal locally 𝔮-finite quotient of M(x^{−1}w_{∘} · 0).

Proposition 1.3.1. Let λ, 𝔮 be as in theorem 1.1.3 and x, y ∈ W^{𝔮} = W^{λ}. Then we have

dim_{ℂ} Ext_{O}^{i}(M(x · λ), L(y · λ)) = [\overline{ra}d^{i} M_{x}^{𝔮} : L_{y}^{𝔮}]
dim_{ℂ} Ext_{O^{𝔮}}^{i}(M_{x}^{𝔮}, L_{y}^{𝔮})  = [\overline{ra}d^{i} M(x · λ) : L(y · λ)].

This can be proved without much trouble by expressing both sides in terms of coefficients of Kazhdan-Lusztig polynomials. Our approach is different however. Not only it is independent of the combinatorics of the Kazhdan-Lusztig polynomials, but it provides a concrete meaning to the numerical formulas above.

Namely, fix λ, 𝔮 as in theorem 1.1.3 and denote the graded algebras there by A_{Q} and A^{Q} respectively. Using the ordinary duality on category O it is easy to see that both A_{Q} and A^{Q} are their own opposed algebras and even to construct isomorphisms A_{Q} = A_{Q}^{opp}, A^{Q} = (A^{Q})^{opp}. Now by theorem 1.1.3 we know that A^{Q} and A_{Q} are Koszul dual to one another, A^{Q} ≅ E(A_{Q}). So the Koszul duality functor gives us an equivalence of triangulated categories

D^{b}(A_{Q}-mof) ≅ D^{b}(A^{Q}-mof).

But we know also from theorem 1.1.3 and remark 4 following it that

\begin{aligned}O^{𝔮} & ≅ & A^{Q}-Mof, \\ O_{λ} & ≅ & A_{Q}-Mof. \end{aligned}

In view of that we consider A_{Q}-mof and A^{Q}-mof as “graded versions” of O_{λ} and O^{𝔮} respectively, and to emphasize this we denote them by

\begin{aligned}O^{Q} & = & A^{Q}-mof, \\ O_{Q} & = & A_{Q}-mof. \end{aligned}

With this change in notation our equivalence reads

D^{b}(O_{Q}) ≅ D^{b}(O^{Q}).

As any Koszul duality this transforms indecomposable injectives to simples and simples to indecomposable projectives. We prove that it also transforms (graded versions of) dual Vermas in O_{Q} to (graded versions of) parabolic Vermas in O^{Q}. In this framework the equalities of the proposition can then be understood as manifestations of Koszul duality. These ideas are developed section 3.11.

1.4. Koszul rings arising from mixed geometry. The purpose of the third chapter is to show how Koszul rings arise naturally in the framework of “mixed” geometry. The latter stands either for the theory of “mixed perverse sheaves” on an algebraic variety over a finite field, as introduced in [BBD82], or for the theory of “mixed Hodge modules” on a complex algebraic variety, as developed by Saito [Sai90]. To make this part of the Introduction accessible to a non-expert, we shall formulate our results without the language of mixed geometry first, and then explain how the mixed geometry enters the game.

Thus, we start with a complex algebraic variety X equipped with an algebraic Whitney stratification X = ∪X_{w}, w ∈ W where W is a finite set indexing the strata. In the rest of the introduction we assume that each stratum X_{w} is a locally-closed algebraic subvariety of X isomorphic to an affine linear space. Let D^{b}(X) denote the bounded derived category of algebraically constructible complexes of ℂ-sheaves on X, D^{b}(X, W) ⊂ D^{b}(X) the full triangulated subcategory whose objects are all complexes F such that the cohomology sheaves H^{•}F are constant along the strata, and P(X, W) ⊂ D^{b}(X, W) the subcategory of all perverse sheaves in D^{b}(X, W). Thus, P(X, W) is an abelian subcategory of D^{b}(X, W). The strata being affine, it follows that for each w ∈ W, the imbedding j_{w} : X_{w} ↪ X is affine. Hence, Δ_{w} = (j_{w})_{!}(ℂ_{𝕏_{≾}})[dim 𝕏_{≾}], ∇_{≾} = (ℷ_{≾})_{∗}(ℂ_{𝕏_{≾}})[dim 𝕏_{≾}], are well-defined objects of P(X, W). We first prove the following result which was obtained earlier, in [MV87], by different methods.

Proposition 1.4.1. 1. Each object of P(X, W) has a finite projective resolu-

tion, in particular, the category P(X, W) has enough projectives.

2. The category P(X, W) is equivalent to the category of finite-dimensional modules over a finite-dimensional ℂ-algebra.

3. Any projective in P(X, W) has a finite filtration with all successive subquotients being isomorphic to Δ_{w} for various w ∈ W.

Let X̄_{w} denote the closure of a stratum X_{w}, w ∈ W, and L_{⊒} = IC(ℂ_{𝕏_{≾}}) the intersection cohomology complex. The simple objects of P(X, W) are associated to the constant sheaf ℂ_{𝕏_{≾}} (supported on X̄_{w} and extended by zero to the whole of X). The intersection complexes L_{w}, w ∈ W are precisely the simple objects of the category P(X, W). For y ∈ W, the restriction j_{y}^{∗}(L_{w}) is a complex with constant cohomology sheaves H^{i}(j_{y}^{∗}L_{w}[− dim X_{y}]) = n_{y,w}^{i} · ℂ_{X_{y}} where n_{y,w}^{i} ∈ ℕ are certain multiplicities which vanish for positive i. Hence, we may introduce a ℤ[t]-valued W × W-matrix IC with entries IC_{y,w} = ∑_{i≤0} n_{y,w}^{i} · t^{−i}.

From now on we make the following additional assumption on the stratification X = ∪X_{w}:

(∗): For each w ∈ W, there exists a resolution of singularities X̂_{w} ↠ X̄_{w} (of the closure of X_{w}) such that the rational homology of X̂_{w} is spanned by algebraic cycles.

The main results of the chapter are summarized in the following theorem.

Theorem 1.4.2. If the condition (*) holds then:

1. The category P(X, W) is equivalent to A-Mof, the category of finite dimensional modules over a finite dimensional graded Koszul ℂ-algebra A = ⊕_{i≥0}A_{i} where A_{0} = ℂ ⊕ … ⊕ ℂ; (#W times).

2. There is a graded algebra isomorphism

A^{!} = Ext_{D^{b}(X)}^{•}(L, L), L := ⨁_{w∈W} L_{w}

In particular, the Ext-algebra above is Koszul.

3. There is a matrix identity: P (A^{!}, t) = IC^{T} · IC

Observe that we claim that P(X, W) = A-Mof is the category of non-graded Amodules. Now we may explain how the mixed geometry enters the game. Namely, a mixed version of the category P(X, W) is equivalent to the category A-mof of graded A-modules. This gives a “geometric” explanation of the graded setting.

2. Koszul rings

2.1. An alternative definition of Koszul rings. Recall that we denote A-mod the category of graded modules, as opposed to A-Mod for the category of ordinary modules over A. Both of these are abelian categories, and we will denote morphisms and extensions in the graded category by hom_{A}, ext_{A} as opposed to Hom_{A}, Ext_{A}.

Definition 2.1.1. A graded module M over a graded ring is called “pure of weight m” if and only if it is concentrated in degree −m, i.e. M = M_{−m}.

Suppose A is a positively graded ring such that A_{0} is semisimple. Then any simple object of A-mod is pure, and any pure object is semisimple. We will use later

Lemma 2.1.2. Let A be a positively graded ring such that A_{0} is semisimple. Let M, N ∈ A-mod be pure of weights m, n. Then ext_{A}^{i}(M, N) = 0 for i > m − n.

Proof. Without restriction we may assume m = 0. Now suppose even more generally that M lives only in positive degrees, M = ⊕_{j≥0}M_{j}. We may easily prove by induction that M admits a graded projective resolution

… → P^{2} → P^{1} → P^{0} ↠ M

such that P^{i} lives only in degrees ≥ i, P^{i} = ⊕_{j≥i}P_{j}^{i}. We may use this resolution to compute our ext-groups, and the lemma follows.

Now comes an alternative characterization of Koszul rings.

Proposition 2.1.3. Let A = ⨁_{j≥0} A_{j} be a positively graded ring and suppose A_{0} is semisimple. The following conditions are equivalent:

1. A is Koszul.

2. For any two pure A-modules M, N of weights m, n respectively we have ext_{A}^{i}(M, N) = 0 unless i = m − n.

3. ext_{A}^{i}(A_{0}, A_{0}⟨n⟩) = 0 unless i = n.

Remark . One may interpret this proposition as saying that a Koszul ring is a positively graded ring that is “as close to semisimple as it can possibly be”.

Proof. (1) ⇒ (2) : Suppose A is Koszul. Then by definition A_{0} admits a graded projective resolution

… → P^{2} → P^{1} → P^{0} ↠ A_{0}

such that P^{i} = AP_{i}^{i}.

Now let M, N be as above. By shifting the grading of M and N we may assume M to be pure of weight zero, hence a direct summand of a free A_{0}-module, so without restriction we may assume M = A_{0}. Then ext_{A}^{i}(M, N) is the i-th cohomology group of the complex hom_{A}(P^{j}, N). But since P^{j} = AP_{j}^{j} and N = N_{−n} all terms of this complex other then hom_{A}(P^{−n}, N) vanish. Thus ext_{A}^{i}(M, N) = 0 unless i = −n.

(2) ⇒ (3) is trivial.

(3) ⇒ (1) : To find the graded projective resolution of A_{0} required in the definition of a Koszul ring we proceed by induction. Certainly A will do as P^{0}. Suppose now a resolution

P^{i} → … → P^{2} → P^{1} → P^{0} ↠ A_{0}

as required has already been constructed up to step i, and put K = ker(P^{i} → P^{i−1}). Then ext_{A}^{i+1}(A_{0}, N) = hom_{A}(K, N) and (3) implies that hom_{A}(K, A_{0}⟨n⟩) = 0 unless n = i + 1. But this means precisely that K = AK_{i+1}, so we may put P^{i+1} = A ⊗_{A_{0}} K_{i+1} and thus complete the induction step.

2.2. The opposed ring of a Koszul ring is Koszul. Denote Mod-R (resp. mod-R) the categories of (resp. graded) right modules over a (graded) ring R. Morphisms and extensions in these categories will be denoted Hom_{−R}, Ext_{−R} and hom_{−R}, ext_{−R} respectively. Now let A, k be rings and X an A-k-bimodule. Then for M ∈ Mod-A, N ∈ Mod-k we have a standard isomorphism

Hom_{−A}(M, Hom_{−k}(X, N)) = Hom_{−k}(M ⊗_{A} X, N).

For V ∈ Mod-k put ^{∗}V = Hom_{−k}(V, k) ∈ k-Mod. Consider now k as a graded ring concentrated in degree zero. Then for V ∈ mod-k define ^{⊛}V ∈ k-mod by (^{⊛}V)_{i} = ^{∗}(V_{−i}).

Suppose next that A is a graded ring, k a graded ring concentrated in degree zero, M ∈ mod-A a graded A-module and X a graded A-k-bimodule. Then ^{⊛}X is a graded k-A-bimodule, and it is easy to see that our standard isomorphism above, when applied to N = k, induces an isomorphism

hom_{−A}(M,^{⊛}X) = ^{∗}((M ⊗_{A} X)_{0})

with (M ⊗_{A} X)_{0} the degree zero component of the graded space (M ⊗_{A} X). In particular, if k is semisimple and X projective as a left A-module, then ^{⊛}X is an injective object of mod-A. Now we can prove

Proposition 2.2.1. The opposed ring of a Koszul ring is Koszul.

Proof. Let A be our Koszul ring and put A_{0} = k. Consider a resolution

… → P^{2} → P^{1} → P^{0} ↠ k

of k ∈ A-mod such that P^{i} = AP_{i}^{i}. By the first remark following the definition 1.2.1 of a Koszul ring we know that this is in fact a complex of graded A-k-bimodules. The arguments above now show that

k ↪ ^{⊛}P^{0} → ^{⊛}P^{1} → ^{⊛}P^{2} → …

is an injective resolution of k in mod-A. Furthermore

hom_{−A}(k⟨−n⟩,^{⊛}P^{i}) = ^{∗}((k ⊗_{A} P^{i})_{n}) = ^{∗}((P_{i}^{i})_{n}) = 0 if i ≠ n.

This in turn shows ext_{−A}^{i}(k⟨−n⟩, k) = 0 for i ≠ n, and hence A^{opp} is Koszul.

2.3. Any Koszul ring is quadratic. Let us fix from now on a semisimple ring k and put ⊗ = ⊗_{k}. Let A = ⨁_{i≥0} A_{i} be a positively graded ring and suppose A_{0} = k.

Proposition 2.3.1. The following conditions are equivalent: (1) ext_{A}^{1}(k, k⟨n⟩) ≠ 0 only for n = 1 and (2) A is generated by A_{1} over k.

Proof. Put I = A_{>0} and decompose I = V ⊕ I^{2} in k-mod. Clearly A is generated by V over k. Consider the exact sequence A ⊗ V →^{m} A ↠ k where m is the multiplication. Since im(m) = I and ker(m) ⊂ I ⊗ V we find

ext_{A}^{1}(k, k⟨n⟩) = hom_{A}(I, k⟨n⟩)
= hom_{A}(A ⊗ A_{1}, k⟨n⟩)
= hom_{k}(V, k⟨n⟩).

So both (1) and (2) are equivalent to V = A_{1}.

Theorem 2.3.2. Suppose A is generated by A_{1} over k. If ext_{A}^{2}(k, k⟨n⟩) ≠ 0 implies n = 2, then A is quadratic.

Proof. Put W = ker(m) so that we have an exact sequence

0 → W → A ⊗ V → A → k → 0.

Certainly W ⊂ I ⊗ A_{1}. Hence

ext_{A}^{2}(k, k⟨n⟩) = hom_{A}(W, k⟨n⟩)
= hom_{k}(W/I · W, k⟨n⟩).

So our assumption implies that W/I ·W is concentrated in degree 2. Now write A as a quotient of T_{k}A_{1} and let x ∈ A_{1}^{⊗n} be a relation of A of degree > 2, not contained in the twosided ideal of T_{k}A_{1} generated by all relations of smaller degrees. Consider the natural surjections

A_{1}^{⊗n} = A_{1}^{⊗n−1} ⊗ A_{1} →^{p} A_{n−1} ⊗ A_{1} →^{m} A_{n}.

Certainly m(p(x)) = 0, thus p(x) ∈ W. We claim p(x) ∉ I · W. Let R_{n} ⊂ A_{1}^{⊗n} be the relations, i.e. the kernel of the multiplication map to A_{n}. Certainly p(x) ∈ I · W implies p(x) ∈ A_{1} · W, hence p(x) = p(y) for suitable y ∈ A_{1} ⊗ R_{n−1}. But then p(x − y) = 0, hence x − y ∈ R_{n−1} ⊗ A_{1}. So from p(x) ∈ I · W follows x ∈ A_{1} ⊗ R_{n−1} + R_{n−1} ⊗ A_{1}, contrary to our assumptions on x. Thus indeed p(x) ∉ I · W. But then p(x) gives an element of degree > 2 in W/I · W and this contradiction proves the theorem.

Corollary 2.3.3. Any Koszul ring is quadratic.

Proof. This follows from the above proposition and theorem along with the characterization of a Koszul ring given in proposition 2.1.3.

2.4. Filtrations on modules. Let M be a module over a ring. Then M has a biggest semisimple submodule socM, the socle of M. We put 0 = soc^{0}M and define the socle filtration on M inductively by soc^{i}M = π^{−1}soc(M/soc^{i−1}M), where π : M ↠ M/soc^{i−1}M is the projection. If M has finite length, then the set of submodules {N ⊂ M | M/N is semisimple} contains a smallest element, called the radical radM of M. We put M = rad^{0}M and define the radical filtration on M inductively by rad^{i}M = rad(rad^{i−1}M).

Let now A be a positively graded ring and M ∈ A-mod a graded A-module. Then M has a filtration by the submodules G_{j}M = ⊕_{i≥j}M_{i}. We call this the “grading filtration”. If A_{0} is semisimple, the sucessive subquotients of the grading filtration are semisimple.

Proposition 2.4.1. Let A be a graded ring. Suppose (1) that A_{0} is semisimple and (2) that A is generated by A_{1} over A_{0}. Let M ∈ A-mod be a graded A-module of finite length.

If socM (resp. M/radM) is simple, then the socle (resp. radical) filtration on M coincides with the grading filtration, up to shift.

Proof. Put I = A_{>0}. Certainly I^{n}M = 0 for n ≫ 0, thus IN = 0 for any simple subquotient of M. We thus find socM = {m ∈ M | Im = 0} (resp. radM = IM) and more generally soc_{i}M = {m ∈ M | I^{i}m = 0} (resp. rad^{i}M = I^{i}M). But since A is generated by A_{1} over A_{0} we have I^{i} = A_{≥i}. So soc_{i}M = {m ∈ M | A_{≥i}m = 0} (resp. rad^{i}M = A_{≥i}M).

Suppose now that M/radM is simple or, more generally, concentrated in one degree j. Then M = AM_{j} and hence rad^{i}M = A_{≥i}M = A_{≥i}M_{j} = ⊕_{n≥i+j}M_{n}. Suppose dually that socM is simple or, more generally, concentrated in one degree j. Then for any m ∈ M_{j−ν}, m ≠ 0, there exists a ∈ A_{ν} such that 0 ≠ am ∈ M_{j}. In other words, ⊕_{n≥j−i}M_{n} = {m ∈ M | A_{≥i}m = 0} = soc_{i}M.

2.5. Unicity of gradings.

Proposition 2.5.1. Let A be a positively graded ring such that A is generated by A_{1} over A_{0}. Put I = A_{>0}. Then we have a canonical isomorphism of graded rings

A ≅ ⊕_{i}I^{i}/I^{i+1}.

Proof. This is clear.

Let A be a ring. By a “Koszul grading on A” we mean a grading such that the corresponding graded ring is Koszul.

Corollary 2.5.2. Let A be an artinian ring. Then any two Koszul gradings on A give rise to isomorphic graded rings.

Proof. They both give rise to graded rings that are canonically isomorphic to the graded ring ⊕_{i}(rad^{i}A/rad^{i+1}A).

Let A be a graded ring. We have the functor forgetting the grading

v : A-mod → A-Mod.

By a “lift” of M ∈ A-Mod we mean an object M̃ ∈ A-mod together with an isomorphism vM̃ ≅ M.

Lemma 2.5.3. Suppose M ∈ A-Mod is indecomposable and of finite length. Then, if M admits a lift, this lift is unique up to grading shift and isomorphism.

Proof. Since M is indecomposable, its endomorphism ring is local, i.e. the non-au-tomorphisms form an ideal. If M̃′ is another lift of M, we have Hom_{A}(M, M) = ⊕_{n} hom_{A}(M̃′, M̃⟨n⟩) and can decompose id = ∑ id_{n}. Since id is an automorphism of M, one of the id_{n} has to induce an automorphism of M, and one sees easily that this id_{n} has to be an isomorphism M̃′ → M̃⟨n⟩.

2.6. The Koszul complex. Recall our fixed semisimple ring k. Let V be a kbimodule, R ⊂ V ⊗ V a sub-bimodule and A = T_{k}V/(R) the corresponding quadratic ring. We want to define its Koszul complex … → K^{2} → K^{1} → K^{0}. This is to be a complex in A-mod. Let us define the k-sub-bimodules

K_{i}^{i} = ⋂_{ν} V^{⊗ν} ⊗ R ⊗ V^{⊗i−ν−2} ⊂ V^{⊗i}.

In particular K_{0}^{0} = k, K_{1}^{1} = V, K_{2}^{2} = R. Then put K^{i} = A ⊗ K_{i}^{i} (with the grading K^{i} = ⨁ K_{j}^{i} defined in such a way that K_{i}^{i} = K_{i}^{i}). Define the differential d : K^{i} → K^{i−1} to be the restriction of A ⊗ V^{⊗i} → A ⊗ V^{⊗i−1}, a ⊗ v_{1} ⊗ … v_{i} ↦ av_{1} ⊗ … v_{i}. Clearly d^{2} = 0, so here is our Koszul complex. Clearly K^{0} = A.

Theorem 2.6.1. Let A = T_{k}V/(R) be a quadratic ring. Then A is Koszul if and only if its Koszul complex is a resolution of k = A/A_{>0}.

Proof. If the Koszul complex is a resolution of k, then A is Koszul by definition.

On the other hand suppose A is Koszul, so by proposition 2.1.3 ext_{A}^{i}(k, k⟨n⟩) = 0 if i ≠ n. Certainly K^{1} → K^{0} ↠ k is right exact. Thus we need only show that H^{p}(K^{•}) = 0 for all p ≥ 1. We use induction on p. Let Z^{p} = ker(d) ⊂ K^{p} be the submodules of cycles. Since the differential d induces an injection K_{p}^{p} ↪ K_{p}^{p−1} we find that Z^{p} lives only in degrees ≥ p + 1. Using induction we get

ext_{A}^{p+1}(k, k⟨n⟩) = coker{hom_{A}(K^{p}, k⟨n⟩) → hom_{A}(Z^{p}, k⟨n⟩)}.

Hence ext_{A}^{p+1}(k, k⟨n⟩) = hom_{A}(Z^{p}, k⟨n⟩), and by Koszulity this doesn’t vanish only for n = p + 1. In particular Z^{p} ∈ A-mod is generated by Z_{p+1}^{p}.

But it is easy to see directly that Z_{p+1}^{p} is precisely the image of K_{p+1}^{p+1} under the differential of the Koszul complex. Thus Z^{p} = im(d) and H^{p}(K^{•}) = 0.

Remark . Generalizing the arguments of [Bac82] (see also [BGS88]) one can prove in the above setting that A is a Koszul if and only if for all n ≥ 1 the set {V^{⊗n−i−1}⊗ R ⊗ V^{⊗i−1}}_{i=1}^{n−1} generates a distributive lattice of submodules of V^{⊗n}.

2.7. Some linear algebra. Let k be a fixed semisimple ring. All the claims below are easily checked on free modules and follow for arbitrary modules. For V ∈ k-Mod define V^{∗} = Hom_{k}(V, k) ∈ Mod-k via (fa)(v) = (f(v))a ∀f ∈ V^{∗}, v ∈ V, a ∈ k. For W ∈ Mod-k define ^{∗}W = Hom_{−k}(W, k) ∈ k-Mod via (ag)(w) = a(g(w)) ∀g ∈ ∗W, w ∈ W, a ∈ k. For finitely generated V, W the canonical maps V → ^{∗}(V^{∗}) and W → (^{∗}W)^{∗} given by evaluation are isomorphisms. If V_{1} ⊂ V (resp. W_{1} ⊂ W) is a submodule, we put V_{1}^{⊥} = {f ∈ V^{∗} | f(V_{1}) = 0} (resp. ^{⊥}W_{1} = {g ∈ ^{∗}W | g(W_{1}) = 0}). These are submodules.

For a k-bimodule V ∈ k-Mod-k both V^{∗} and^{∗}V are bimodules defined as follows: Define (af)(v) = f(va) ∀f ∈ V^{∗}, v ∈ V, a ∈ k and (ga)(v) = g(av) ∀g ∈ ^{∗}V, v ∈ V, a ∈ k. Both canonical maps V → ^{∗}(V^{∗}) and V → (^{∗}V)^{∗} are now morphisms of bimodules. If V_{1} ⊂ V is a subbimodule, both V_{1}^{⊥} ⊂ V^{∗} and ^{⊥}V_{1} ⊂ ^{∗}V are subbimodules.

Recall ⊗ = ⊗_{k} always. For V ∈ k-Mod-k, W ∈ k-Mod define a map W^{∗} ⊗ V^{∗} → (V ⊗ W)^{∗} by (g ⊗ f)(v ⊗ w) = f(vg(w)). This is an isomorphism for W = k, hence for any finitely generated W ∈ k-Mod. If W ∈ k-Mod-k this is actually a morphism of k-bimodules. More generally, if V_{1}, …, V_{n} ∈ k-Mod-k are finitely generated as left k-modules, then canonically V_{1}^{∗} ⊗ … ⊗ V_{n}^{∗} = (V_{n} ⊗ … ⊗ V_{1})^{∗} as k-bimodules, where the isomorphism is given by (f_{1} ⊗ … ⊗ f_{n})(v_{n} ⊗ … ⊗ v_{1}) = f_{n}(v_{n}f_{n−1}(v_{n−1} … f_{1}(v_{1}) …)).

Dually, for V ∈ Mod-k, W ∈ k-Mod-k define a map ^{∗}W ⊗ ^{∗}V → ^{∗}(V ⊗ W) by (g ⊗ f)(v ⊗ w) = g(f(v)w). This is an isomorphism for finitely generated V ∈ Mod-k. If V ∈ k-Mod-k the above morphism is actually a morphism of k-bimodules. Again if V_{1}, …, V_{n} ∈ k-Mod-k are finitely generated as right k-modules, then ∗V_{1}⊗…⊗^{∗}V_{n} = ^{∗}(V_{n}⊗…⊗V_{1}) as k-bimodules, where (f_{1}⊗…⊗f_{n})(v_{n}⊗…⊗v_{1}) = f_{1}(… f_{n−1}(f_{n}(v_{n})v_{n−1}) … v_{1}).

For V, W ∈ k-Mod define a map V^{∗} ⊗W → Hom_{k}(V, W) by (f ⊗w)(v) = f(v)w. This is an isomorphism if V or W is finitely generated. If V (resp. W) is a kbimodule, then our map is compatible with the resulting left (resp. right) k-actions on both spaces. For V, W ∈ Mod-k define a map W ⊗ ^{∗}V → Hom_{−k}(V, W) by (w ⊗ f)(v) = wf(v). This is an isomorphism if V or W is finitely generated. If V (resp. W) is a k-bimodule, then our map is compatible with the resulting right (resp. left) k-actions on both spaces.

Suppose V ∈ k-Mod-k. Then there are canonical bimodule homomorphisms V ⊗ V^{∗} → k and ^{∗}V ⊗ V → k given by v ⊗ f ↦ f(v) and g ⊗ v ↦ g(v) respectively. Dually there are canonical bimodule homomorphisms k → V^{∗} ⊗ V = Hom_{k}(V, V) and k → V ⊗ ^{∗}V = Hom_{−k}(V, V) where a ∈ k is mapped to right (resp. left) multiplication on V.

2.8. The quadratic dual rings and the Koszul complex. Call a graded ring A with A_{0} = k a graded ring “over k”. We begin by defining the quadratic dual rings.

Definition 2.8.1. Let A be a left finite quadratic ring over k, say A = T_{k}V/(R). Then its left dual A^{!} is the right finite quadratic ring over k given by A^{!} = T_{k}(V^{∗})/(R^{⊥}) with R^{⊥} ⊂ V^{∗} ⊗ V^{∗} = (V ⊗ V)^{∗}.

Analogously if A is a right finite quadratic ring over k, say A = T_{k}V/(R), then its right dual ^{!}A is the left finite quadratic ring over k given by ^{!}A = T_{k}(^{∗}V)/(^{⊥}R) with ^{⊥}R ⊂ ^{∗}V ⊗^{∗}V = ^{∗}(V ⊗ V).

Remark . Obviously ^{!}(A^{!}) = A and (^{!}A)^{!} = A for a left, resp. right finite quadratic ring A.

Let now A = T_{k}V/(R) be a left finite quadratic ring over k. Then its Koszul complex is (isomorphic to) the complex

… → A ⊗^{∗}(A_{2}^{!}) → A ⊗^{∗}(A_{1}^{!}) → A

where we abreviate (A^{!})_{i} = A_{i}^{!}. The differentials d : A ⊗ ^{∗}(A_{i+1}^{!}) → A ⊗ ^{∗}(A_{i}^{!}) may be described as follows: Observe that A ⊗^{∗}(A_{i}^{!}) = Hom_{−k}(A_{i}^{!}, A). Under the canonical isomorphism Hom_{k}(V, V) = V^{∗} ⊗ V let id_{V} = ∑ v̌_{α} ⊗ v_{α}. Then (df)(a) = ∑ f(av̌_{α})v_{α} for f ∈ Hom_{−k}(A_{i+1}^{!}, A), a ∈ A_{i}^{!}. We may also write this without coordinates. Namely use the obvious maps A ⊗ V → A and A_{i}^{!} = A_{i}^{!} ⊗ k → A_{i}^{!} ⊗ V^{∗} ⊗ V → A_{i+1}^{!} ⊗ V to give the differential as the composition Hom_{−k}(A_{i+1}^{!}, A) → Hom_{−k}(A_{i+1}^{!} ⊗ V, A ⊗ V) → Hom_{−k}(A_{i}^{!}, A). To see that this is really our Koszul complex from section 2.6, write

A_{i}^{!} = (V^{∗})^{⊗i}/ ∑_{ν}(V^{∗})^{⊗i−ν−2} ⊗ R^{⊥} ⊗ (V^{∗})^{⊗ν}.

We deduce

∗(A_{i}^{!}) = ⋂_{ν} V^{⊗ν} ⊗ R ⊗ V^{⊗i−ν−2} = K_{i}^{i} ⊂ V^{⊗i}.

Thus A ⊗^{∗}(A_{i}^{!}) = K^{i} and the reader may check that the differentials correspond.

2.9. The quadratic dual of a Koszul ring is Koszul.

Proposition 2.9.1. Let A be a left finite Koszul ring. Then A^{!} is Koszul as well.

Proof. The total space of the Koszul complex A⊗^{⊛}A^{!} is an A-A^{!}-bimodule and the differential commutes with the action of both A and A^{!}. In fact we have a bigraded space

A ⊗^{⊛}A^{!} = ⊕_{i,j}A_{i} ⊗^{∗}A_{−j}^{!}

concentrated in the lower right quadrant, the differential has bidegree (1, 1) and the only cohomology appears in bidegree (0, 0).

If we apply Hom_{k}( , k) to every component of this bigraded space, we obtain a bigraded space

A^{!} ⊗ A^{⊛} = ⊕_{i,j}A_{j}^{!} ⊗ A_{−i}^{∗}

concentrated in the upper left quadrant with differential of bidegree (1, 1) commuting with the left A^{!}-action, and again the only cohomology appears in bidegree (0, 0). But we may also read this as a graded projective resolution

… → A^{!} ⊗ A_{2}^{∗} → A^{!} ⊗ A_{1}^{∗} → A^{!} ⊗ A_{0}^{∗} ↠ k

of k ∈ A^{!}-mod, and this then shows that A^{!} is Koszul.

2.10. The quadratic dual and cohomology. Let R be a ring and M ∈ R-Mod. We will repeat the definition of Ext_{R}^{•}(M, M) in a form suitable for our purposes. For two complexes P^{•}, Q^{•} of R-modules consider the graded space Hom_{R}^{•}(P^{•}, Q^{•}) with

Hom_{R}^{i}(P^{•}, Q^{•}) = ⊕_{n}Hom_{R}(P^{n}, Q^{n−i}).

The differentials d of degree (−1) on P^{•} and Q^{•} give rise to a differential of degree (+1) on Hom_{R}^{•}(P^{•}, Q^{•}) by df = d ∘ f − (−1)^{|f|}f ∘ d for f ∈ Hom_{R}^{|f|}(P^{•}, Q^{•}). For f, g ∈ End_{R}^{•}(P^{•}) we have the Leibnitz rule d(fg) = (df)g + (−1)^{|f|}f(dg). Thus the cohomology H^{•}End_{R}^{•}(P^{•}) has the structure of a graded ring. If now P^{•} ↠ M is a projective resolution of M ∈ R-Mod, then more or less by definition

H^{•}End_{R}^{•}(P^{•}) = Ext_{R}^{•}(M, M).

Theorem 2.10.1. Let A be a left finite Koszul ring over k. Then Ext_{A}^{•}(k, k) = (A^{!})^{opp} canonically.

Proof. Consider the Koszul complex (K^{•}, d) and form the complex (K̃^{•}, d̃) with K̃^{i} = K^{i}, d̃ = (−1)^{i}d : K^{i} → K^{i−1}. Then K̃^{•} is also a projective resolution of k. The right action of A^{!} on K^{•} = A⊗^{⊛}A^{!} gives us an homomorphism of graded rings

(A^{!})^{opp} → End_{A}^{•}(K^{•}).

Since this right action commutes with the differential d of the Koszul complex, the image of

(A^{!})^{opp} → End_{A}^{•}(K̃^{•})

even consists of cycles. (This is the reason we work with K̃^{•} and not with K^{•}.) Hence our morphism above induces a morphism

(A^{!})^{opp} → H^{•}End_{A}^{•}(K̃^{•})

and we just have to show that this is an isomorphism. For this consider the composition

(A^{!})^{opp} → End_{A}^{•}(K̃^{•}) → Hom_{A}^{•}(K̃^{•}, k),

where k is considered as a complex concentrated in degree zero. We consider (A^{!})^{opp} as a complex with zero differential, and then both these maps are chain maps. The second map certainly induces an isomorphism on cohomology, and the composition is just the identity from A^{!} to

Hom_{A}^{•}(K̃^{•}, k) = ⊕_{i}Hom_{A}(A ⊗^{∗}(A_{i}^{!}), k) = A^{!}.

Hence the first map has to induce an isomorphism on cohomology as well.

We can now prove theorem 1.2.5 from the introduction.

Theorem 2.10.2. Suppose A is a left finite Koszul ring. Then E(A) is also a left finite Koszul ring and E(E(A)) = A canonically.

Proof. E(A) = (A^{!})^{opp} is obviously left finite and is Koszul by propositions 2.2.1 and 2.9.1. Then E(E(A)) = (((A^{!})^{opp})^{!})^{opp} = ^{!}(A^{!}) = A.

2.11. A numerical Koszulity criterion. Fix a field F and suppose that (1) the positively graded ring A is an F -algebra, i.e. there is a central embedding F ⊂ A. Suppose in addition (2) that dim_{F} A_{i} < ∞ for all i and (3) that A_{0} = k = ⨁_{x∈W} F 1_{x} for pairwise orthogonal idempotents 1_{x} indexed by a finite set W, so in particular k^{opp} = k.

Then we may form the Hilbert polynomial P (A, t) of A, a W × W-matrix with entries in power series ℤ[[t]] given by

P (A, t)_{x,y} = ∑ t^{i} dim (1_{x}A_{i}1_{y}).

To simplify notation set dim(1_{x}M1_{y}) = [M]_{x,y}, for any M ∈ k ⊗_{F} k-Mod ⊂ k-Mod-k. The following lemma motivates our criterion.

Lemma 2.11.1. Suppose a positively graded F -algebra A satisfying conditions (1)- (3) above is Koszul. Then the matrix equation P (A, t)P (A^{!}, −t)^{T} = 1 holds.

Remark . We do not know whether for a quadratic ring A as in the proposition the matrix equation P (A, t)P (A^{!}, −t)^{T} = 1 implies that A is Koszul.

Proof. The Euler-Poincaré principle for the Koszul complex yields

∑_{i,j}(−1)^{i}t^{i+j}[A_{i} ⊗^{∗}(A_{j}^{!})]_{x,y} = [A_{0}]_{x,y}.

Now the result follows from the identities [A_{0}]_{x,y} = δ_{x,y}, [^{∗}M]_{x,y} = [M]_{y,x} and [M ⊗ N]_{x,y} = ∑_{z}[M]_{x,z}[N]_{z,x}. (Remark that ⊗ = ⊗_{k} always!)

Forget now the grading on A and form the positively graded F -algebra E = E(A) = Ext_{A}^{•}(k, k). Certainly E_{0} = k^{opp} = k, and if A is left noetherian then dim E_{i} < ∞ for all i, so we can form the Hilbert polynomial P (E, t) of E.

Theorem 2.11.1. Suppose A is a left noetherian F -algebra satisfying the conditions (1)-(3) above. Form the graded F -algebra E = E(A). Then the following conditions are equivalent:

(i) A is Koszul. (ii) The matrix equation P (A, t)P (E, −t) = 1 holds.

Remark . Löfwall [Löf86] calls (ii) a “Fröberg condition”.

Proof. (i) ⇒ (ii). If A is Koszul we know by theorem 2.10.1 that E ≅ (A^{!})^{opp}. Now apply the lemma.

(ii) ⇒ (i). Certainly k is a graded A-module and a graded right k-module and these two actions commute and induce the same F -action, in other words k ∈ A ⊗_{F} k-mod. We are going to construct inductively a resolution

… → P^{2} → P^{1} → P^{0} ↠ k

of k in A ⊗_{F} k-mod as follows: Take P^{0} = A. If P^{i} is already constructed, put K^{i} = ker(P^{i} → P^{i−1}) and choose a complement V^{i} in k⊗_{F} k-mod of A_{>0}·K^{i} ⊂ K^{i}. So V^{i} is a graded k-bimodule V^{i} = ⊕V_{j}^{i} and we put P^{i+1} = A ⊗ V^{i}. Take V^{0} = k for convenience. So we get a resolution of k of the form

… → A ⊗ V^{2} → A ⊗ V^{1} → A ⊗ V^{0} ↠ k

with the V^{i} some graded k-bimodules V^{i} = ⨁ V_{j}^{i} such that all differentials of the resolution are compatible with the right k-action. By construction P^{i+1} gets mapped to A_{>0} · P^{i}, hence the differential on the complex of the Hom_{A}(P^{•}, k) vanishes. Also by construction all P^{i} are projective in A-mod, hence Ext_{A}^{i}(k, k) = (V^{i})^{∗}, ext_{A}^{i}(k, k⟨j⟩) = (V_{j}^{i})^{∗} and the reader may check that even [E_{i}]_{x,y} = [V^{i}]_{x,y}.

We form the matrix of power series P (V, t) with P (V, t)_{x,y} = ∑[V^{i}]_{x,y}t^{i} and deduce from (ii) the equation P (A, t)P (V, −t) = 1. Thus in the Grothendieck group of a suitable category of k-bimodules

∑_{i=0}^{n}(−1)^{i}A_{i} ⊗ V^{n−i} = 0

for n ≥ 1. To prove that A is Koszul we have to check that V_{j}^{n} = 0 for all n ≥ 1, j ≠ n. ¿From lemma 2.1.2 we deduce that ext_{A}^{i}(k, k⟨j⟩) = 0 for j < i. Thus V_{j}^{i} = 0 for j < i. We need V_{j}^{i} = 0 for j ≠ i.

Proceed by induction on i. For i = 0 the statement is clear. Now take the degree n component of the above resolution. Clearly (A⊗V^{i})_{n} = ⊕_{j}A_{n−j} ⊗V_{j}^{i}. By induction we know that V^{i} = V_{i}^{i} for i < n, hence (A⊗V^{i})_{n} = A_{n−i}⊗V_{i}^{i} = A_{n−i}⊗V^{i} for i < n. By lemma 2.1.2 (or by construction) we know V_{j}^{n} = 0 for j < n, hence (A ⊗ V^{n})_{n} = A_{0} ⊗ V_{n}^{n}. We find an exact sequence

0 → A_{0} ⊗ V_{n}^{n} → … → A_{n−1} ⊗ V^{1} → A_{n} ⊗ V^{0} → 0.

Thus we get an equation

∑_{i=0}^{n−1}(−1)^{i}A_{i} ⊗ V^{n−i} + (−1)^{n}A_{0} ⊗ V_{n}^{n} = 0

in the same Grothendieck group as above. Comparing the two equations we find V_{n}^{n} = V^{n} and the induction step is complete.

2.12. Koszul duality. Let B = ⨁_{j≥0} B_{j} be a positively graded ring. Write C(B) for the homotopy category of complexes in B-mod. An object M ∈ C(B) is a complex … → M^{i} →^{∂} M^{i+1} → … of graded B-modules M^{i} = ⨁_{j} M_{j}^{i}. Let C^{↑}(B)

(resp. C^{↓}(B)) be the full subcategories of C(B) whose objects M = ⨁ M_{j}^{i} satisfy the conditions

M_{j}^{i} = 0 if \{ \begin{matrix}i ≫ 0 or i + j ≪ 0 for C^{↑}(B) \\ i ≪ 0 or i + j ≫ 0 for C^{↓}(B). \end{matrix}

Thus all nonzero components M_{j}^{i} of an object of C^{↑}(B) (resp. C^{↓}(B)) are concentrated in a region as indicated in the picture.

[figure: C^{↑}(B); C^{↓}(B); i; j]

Let D^{↑}(B), D^{↓}(B) denote the localisations of C^{↑}(B), C^{↓}(B) at quasi-isomorphisms. These are also triangulated categories. We want to prove

Theorem 2.12.1. Let A be a left finite Koszul ring. Then there exists an equivalence of triangulated categories D^{↓}(A) ≅ D^{↑}(A^{!}).

Proof[Theorem. ] The existence of this equivalence could be deduced from [Kel92]. We explicitly construct such an equivalence. This will be done in several steps. Assume A_{0} = k. Step 1: Construction of a functor DF : D^{↓}(A) → D^{↑}(A^{!}). For M ∈ C(A) consider the bigraded k-module

F M = A^{!} ⊗ M = ⨁_{l,i} A_{l}^{!} ⊗ M^{i} = ⨁_{l,i} Hom_{A}(A ⊗^{∗}(A_{l}^{!}), M^{i})

with anticommuting differentials coming, up to signs, from the differentials of the Koszul complex and of the complex M respectively, and given by

\begin{aligned}d′(a ⊗ m) & = & (−1)^{i+j} ∑ av̌_{α} ⊗ v_{α}m \\ d′′(a ⊗ m) & = & a ⊗ ∂m \end{aligned}

for a ∈ A_{l}^{!}, m ∈ M_{j}^{i}. Consider on F M the total differential d = d′ + d′′. We can interpret F M as an object of C(A^{!}) letting

(F M)_{q}^{p} = ⨁_{p=i+j,q=l−j} A_{l}^{!} ⊗ M_{j}^{i}.

It is easy to check that M ∈ C^{↓}(A) implies F M ∈ C^{↑}(A^{!}). Furthermore for M ∈ C^{↓}(A) the spectral sequence with first term H^{•}(F M, d′′) associated to the bicomplex (F M; d′, d′′) converges to the cohomology H^{•}(F M, d), i.e. M ∈ C^{↓}(A) acyclic implies F M ∈ C^{↑}(A^{!}) acyclic. Hence F induces a functor DF : D^{↓}(A) → D^{↑}(A^{!}).

Step 2: Construction of a functor DG : D^{↑}(A^{!}) → D^{↓}(A). For N ∈ C(A^{!}) consider the bigraded k-module GN given by

(GN)_{l,i} = Hom_{k}(A_{−l}, N^{i})

with anticommuting differentials d′, d′′ given by

\begin{matrix}(d′f)(a) = (−1)^{i} ∑ v̌_{α}f(v_{α}a) \\ (d′′f)(a) = ∂(f(a)) \end{matrix}

for f ∈ Hom_{k}(A_{−l}, N^{i}). The reader may check himself that d′ is well defined and is a differential. Consider on GN the total differential d = d′ + d′′. We can interpret GN as an object of C(A) letting

(GN)_{q}^{p} = ⨁_{p=i+j,q=l−j} Hom_{k}(A_{−l}, N_{j}^{i}).

It is easy to check that N ∈ C^{↑}(A^{!}) implies GN ∈ C^{↓}(A). Furthermore for N ∈ C^{↑}(A^{!}) the spectral sequence with first term H^{•}(GN, d′′) associated to the bicomplex (GN; d′, d′′) converges to the cohomology H^{•}(GN, d), i.e. N ∈ C^{↑}(A^{!}) acyclic implies GN ∈ C^{↓}(A) acyclic. Hence G induces a functor DG : D^{↑}(A^{!}) → D^{↓}(A).

Step 3: Adjointness of our functors. Consider the functors F : C^{↓}(A) → C^{↑}(A^{!}) and G : C^{↑}(A^{!}) → C^{↓}(A). We define an adjointness (F, G). In general, for any M ∈ A-Mod, N ∈ A^{!}-Mod we have canonically Hom_{A^{!}}(A^{!}⊗M, N) = Hom_{k}(M, N) = Hom_{A}(M, Hom_{k}(A, N)). If for all m ∈ M there exists l such that A_{i}m = 0 for i ≥ l then even Hom_{A^{!}}(A^{!} ⊗ M, N) = Hom_{k}(M, N) = Hom_{A}(M, ⨁_{l} Hom_{k}(A_{l}, N)).

Lemma 2.12.2. This canonical isomorphism establishes an adjointness (F, G) of functors between C^{↓}(A) and C^{↑}(A^{!}).

Proof. Choose M ∈ C^{↓}(A), N ∈ C^{↑}(A^{!}). Choose corresponding homomorphisms f̃, f, f̂ in Hom_{A^{!}}(F M, N) = Hom_{k}(M, N) = Hom_{A}(M, GN). It is easy to see that f̃((F M)_{j}^{i}) ⊂ N_{j}^{i} for all i, j if and only if f̂(M_{q}^{p}) ⊂ (GN)_{q}^{p} for all p, q. Suppose from now on that f̃ satisfies this condition. We just have to show that f̃ commutes with the differentials if and only if f̂ does. But both f̃ and f̂ commute with the differentials if and only if

(−1)^{i+j} ∑ v̌_{α}f(v_{α}m) + f(∂m) − ∂(f(m)) = 0.

This proves the lemma. Let us perform the calculation for f̂. Suppose m ∈ M_{j}^{i}. Then d(f̂(m)) = f̂(∂m) if and only if they have the same value at 1 ∈ A, i.e. if and only if (d(f̂(m)))(1) = (f̂(∂m))(1). By definition (f̂(∂m))(1) = f(∂m). On the other hand, f̂(m) = ∑_{l} f̂_{l}(m) with f̂_{l}(m) ∈ Hom_{k}(A_{l}, N_{−l−j}^{i+l+j}). Then

(d(f̂(m)))(1) = ∑_{l}(d(f̂_{l}(m)))(1)
= ∂((f̂_{0}(m))(1)) − (−1)^{i+j} ∑ v̌_{α}((f̂_{1}(m))(v_{α}))
= ∂(f(m)) − (−1)^{i+j} ∑ v̌_{α}f(v_{α}m).

Thus indeed f̂ commutes with the differentials if and only if the above equation on f holds.

Step 4: We construct an equivalence DF ∘ DG → id. More precisely we prove Lemma 2.12.3. For any N ∈ C^{↑}(A^{!}) the canonical surjection (F ∘ G)(N) → N from the adjointness is a quasi-isomorphism.

Proof. Consider the splitting over k of our surjection, given by

N = k ⊗ Hom_{k}(k, N) → ⨁_{l} A^{!} ⊗ Hom_{k}(A_{l}, N).

It is sufficient to show that this is a quasi-isomorphism. Consider N as a bicomplex (N_{p,q}; d′, d′′) with N_{0,q} = ⨁_{i+j=q} N_{j}^{i}, N_{p,q} = 0 if p ≠ 0, d′ = 0, d′′ = ∂. Recall from step 1 how F (GN) is a bicomplex with (F (GN))_{p,q} = A_{p}^{!} ⊗ (GN)^{q}. In fact our splitting map is a morphism of bicomplexes. Since the spectral sequences in question converge, we need only show that our splitting induces isomorphisms

H^{•}(N, d′) → H^{•}(F (GN), d′).

But F (GN) = ⨁ Hom_{k}(A_{l} ⊗^{∗}(A_{p}^{!}), N_{j}^{i}) and the differential d′ can be described as d′f = (−1)^{i+l}(f ∘ d_{K}) where d_{K} : A_{l} ⊗ ^{∗}(A_{p+1}^{!}) → A_{l} ⊗ ^{∗}(A_{p}^{!}) is the differential of the Koszul complex. Now the fact that the Koszul complex is a resolution of k implies that indeed our splitting induces isomorphisms on H^{•}( , d′).

Step 5: We construct an equivalence id → DG ∘ DF. More precisely we prove

Lemma 2.12.4. For any M ∈ C^{↓}(A) the canonical injection M → (G ∘ F)(M) from the adjointness is a quasi-isomorphism.

Proof. Consider the splitting G(F M) → M over k of our canonical injection given by

⨁ Hom_{k}(A_{l}, A^{!} ⊗ M) → Hom_{k}(k, k ⊗ M) = M.

It is sufficient to show that this is a quasi-isomorphism. Consider M as a bicomplex (M_{p,q}; d′, d′′) with M_{0,q} = ⨁_{i+j=q} M_{j}^{i}, M_{p,q} = 0 if p ≠ 0, d′ = 0, d′′ = ∂. Recall from step 2 how G(F M) is a bicomplex with (G(F M))_{p,q} = Hom_{k}(A_{−p}, (F N)^{q}). In fact our splitting map is a morphism of bicomplexes. Since the spectral sequences in question converge, we need only show that our splitting induces isomorphisms

H^{•}(G(F M), d′) → H^{•}(M, d′).

But G(F M) = ⨁ Hom_{k}(A_{l}, A^{!} ⊗ M) = ⨁(A_{l}^{∗} ⊗ A^{!}) ⊗ M and the differential d′ can be described as being up to signs the differential of the right Koszul complex of A^{!} tensor the identity on M. Now the fact that the Koszul complex is a resolution of k implies that indeed the above maps on cohomology are isomorphisms.

The theorem is now proven.

As usual for a complex X = (X^{i}, ∂) denote by X[n] the shifted complex (X[n])^{i} = X^{i+n} with differentials (−1)^{n}∂. For a graded ring B put B-mod^{↑} = {M ∈ B-mod | M_{j} = 0 for j ≪ 0} and B-mod^{↓} = {M ∈ B-mod | M_{j} = 0 for j ≫ 0} We will consider these categories as full subcategories B-mod^{↑} ⊂ D^{↑}(B), B-mod^{↓} ⊂ D^{↓}(B) of complexes concentrated in degree zero.

Suppose B_{0} = k is semisimple. Then any simple object of B-mod is up to a grading shift isomorphic to B_{0}p for some p ∈ k. For any p ∈ k the surjection Bp ↠ B_{0}p is a projective cover in B-mod^{↑} (and B-mod, B-Mod.) Define the “continuous dual” B^{⊛} = ⨁_{l} B_{l}^{∗} ∈ B-Mod-k. Then B_{0}p ↪ B^{⊛}p is an injective hull in B-mod^{↓} for all p ∈ k.

Let us from now on denote by

K : D^{↓}(A) → D^{↑}(A^{!})

the “Koszul duality functor” K = DF constructed in step 1.

Theorem 2.12.5. Let A be a left finite Koszul ring over k.

(i): The functor K : D^{↓}(A) → D^{↑}(A^{!}) together with the obvious canonical iso-

morphism K(M[1]) = (KM)[1] is an equivalence of triangulated categories. (ii): We have K(M⟨n⟩) = (KM)[−n]⟨−n⟩ canonically. (iii): For any p ∈ k we have K(A_{0}p) = A^{!}p and K(A^{⊛}p) = A_{0}^{!}p.

Proof. (i) has been done already. (ii) is left to the reader. Just note that the isomorphism is given in essence by (−1)^{nq} : A^{!} ⊗ M^{q} → A^{!} ⊗ M^{q}. In (iii) the first equality is evident, the second follows from the fact that the Koszul complex is a resolution of k = A_{0}.

Remark . As can be seen from ii.) already, the standard t-structures on our two triangulated categories do not correspond under K. In section 2.13 we discuss the non-standard t-structure on D^{↑}(A^{!}) coming from the standard one on D^{↓}(A) via K, and in section 2.14 we will define the notion of a “Koszul module” and see that A^{!}-mod^{↑} ∩ K(A-mod^{↓}) consists exactly of the Koszul modules for A^{!}.

Recall that for a graded ring B we let B-mof be the category of finitely generated graded modules. We want to give a version of Koszul duality involving only standard derived categories.

Theorem 2.12.6. Let A be a Koszul ring over k. Suppose A is a finitely generated k-module as well from the left as well as from the right so that A_{i} = 0 for i ≫ 0. Suppose in addition that A^{!} is left noetherian. Then Koszul duality induces an equivalence of triangulated categories

K : D^{b}(A-mof) → D^{b}(A^{!}-mof).

Proof[Theorem. ] We must first make precise the meaning of “induces”. For this we will characterize D^{b}(A-mof) and D^{b}(A^{!}-mof) as certain full subcategories of D^{↓}(A) and D^{↑}(A^{!}) respectively.

Namely define full subcategories D_{e}^{↓}(A) ⊂ D^{↓}(A) (resp. C_{e}^{↓}(A) ⊂ C^{↓}(A)) to consist of all objects with only finitely many nonvanishing cohomology groups which are in addition finitely generated A-modules. We may also interpret D_{e}^{↓}(A) as the localization of C_{e}^{↓}(A) at quasi-isomorphisms.

Lemma 2.12.7. Suppose A is finitely generated as a left k-module, so in particular is left noetherian. Then the obvious functor D^{b}(A-mof) → D_{e}^{↓}(A) is an equivalence.

Proof. First note that by [Har66, exercise to prop. 4.8 of chapter I] the canonical functor D^{−}(A-mof) → D_{A-mof}^{−}(A-mod) is an equivalence. Thus an equivalence D^{b}(A-mof) → D_{A-mof}^{b}(A-mod). Now D_{A-mof}^{b}(A-mod) can be interpreted as the localization of a suitable homotopy category of complexes C_{A-mof}^{b}(A-mod) at quasiisomorphisms. Also D_{e}^{↓}(A) is the localization of C_{e}^{↓}(A) at quasi-isomorphisms.

Define now a functor C_{e}^{↓}(A) → D_{A-mof}^{b}(A-mod) by X ↦ lim τ^{≤i}X with τ^{≤i} the usual truncation functors. The inductive system stabilizes, so there is no problem. This gives a functor D_{e}^{↓}(A) → D_{A-mof}^{b}(A-mod).

In the other direction we have the functor C_{A-mof}^{b}(A-mod) → D_{e}^{↓}(A) given by X ↦ lim X/G_{j}X, using the notation G_{j} from section 2.4. The projective system stabilizes and we get a functor D_{A-mof}^{b}(A-mod) → D_{e}^{↓}(A).

One may check that these are inverse of one another and that the canonical functor from the lemma is the composition of equivalences D^{b}(A-mof) → D_{A-mof}^{b}(A-mod) → D_{e}^{↓}(A).

Define further the full subcategories D_{e}^{↑}(A^{!}) ⊂ D^{↑}(A^{!}) (resp. C_{e}^{↑}(A^{!}) ⊂ C^{↑}(A^{!})) to consist of all complexes with only finitely many nonvanishing cohomology groups which are in addition finitely generated A^{!}-modules. We may also interpret D_{e}^{↑}(A^{!}) as the localization of C_{e}^{↑}(A^{!}) at quasi-isomorphisms.

Lemma 2.12.8. Suppose A^{!} is left noetherian. Then the obvious functor from D^{b}(A^{!}-mof) to D_{e}^{↑}(A^{!}) is an equivalence.

Proof. First of all we have as before D^{b}(A^{!}-mof) = D_{A^{!}-mof}^{b}(A^{!}-mod). Now we may define a functor C_{e}^{↑}(A^{!}) → D_{A^{!}-mof}^{b}(A^{!}-mod) by X ↦ lim τ^{≥i}X with τ^{≥i} the usual truncation functors. The projective system stabilizes and we get a functor D_{e}^{↑}(A^{!}) → D_{A^{!}-mof}^{b}(A^{!}-mod).

In the other direction we have the functor C_{e}^{↑}(A^{!}) → D_{e}^{↑}(A^{!}) given by X ↦ lim W_{i}X. The injective system stabilizes and gives a functor D_{A^{!}-mof}^{b}(A^{!}-mod) → D_{e}^{↑}(A^{!}).

One may check that these are inverse of one another and that the canonical functor from the lemma is the composition of equivalences D^{b}(A^{!}-mof) → D_{A^{!}-mof}^{b}(A^{!}-mod) → D_{e}^{↑}(A^{!}).

Now we actually prove the theorem. The equivalence D^{b}(A-mof) → D_{e}^{↓}(A) makes it clear that K : D^{↓}(A) → D^{↑}(A^{!}) maps D_{e}^{↓}(A) to D_{e}^{↑}(A^{!}) and thus induces a fully faithful functor K : D^{b}(A-mof) → D^{b}(A^{!}-mof). The only problem is to show that here K is essentially surjective. Since we know that KA_{0} = A^{!} this will follow once we show that A^{!}-mof has finite homological dimension. So let us prove this.

Since KA^{∗} = A_{0}^{!} = k we have ext_{A^{!}}^{i}(k, k⟨j⟩) = hom_{D^{b}}(A^{∗}, A^{∗}[i − j]⟨−j⟩). Since A^{∗} ∈ A-mof is the injective hull of A_{0} = k, we actually find that ext_{A^{!}}^{i}(k, k⟨j⟩) vanishes for i ≠ j and ext_{A^{!}}^{i}(k, k⟨i⟩) = hom_{A}(A^{∗}, A^{∗}⟨−i⟩) = hom_{k}(A^{∗}⟨i⟩, k) = (A_{i}^{∗})^{∗}. In particular there exists d such that for all M, N ∈ A^{!}-mod which are finitely generated over k we have ext_{A^{!}}^{i}(M, N) = 0 if i ≥ d.

Let now M ∈ A^{!}-mof be arbitrary. By lemma 2.1.2 ext_{A^{!}}^{i}(G_{1}M, k) = 0 ∀i. But M/G_{1}M is finitely generated over k, so ext_{A^{!}}^{i}(M, k) = 0 if i ≥ d.

Now choose a minimal projective resolution

… → P_{2} → P_{1} → P_{0} ↠ M

of M in A^{!}-mof. This means precisely that hom_{A^{!}}(P_{i}, k⟨j⟩) = ext_{A^{!}}^{i}(M, k⟨j⟩). We deduce P_{i} = 0 for all i ≥ d. The theorem is proven.

2.13. t-structures. Let A be a left finite Koszul ring and consider the Koszul duality functor

K : D^{↓}(A) → D^{↑}(A^{!})

defined in section 2.12. Here both sides have standard t-structures with cores A-mod^{↓} and A^{!}-mod^{↑} respectively, but under K these do not correspond. So transporting our standard t-structures via K, we obtain non-standard t-structures on both D^{↓}(A) and D^{↑}(A^{!}). We will describe this new t-structure on D^{↑}(A^{!}) quite explicitely. The case of D^{↓}(A) is very similar and will be left to the reader.

First let B be any positively graded ring such that B_{0} is semisimple. In D^{↑}(B) consider the subcategories D^{↑}(B)^{≤0,g} resp. D^{↑}(B)^{≥0,g} of all objects isomorphic to complexes of graded projective modules

… → P^{i} → P^{i+1} → …

in C^{↑}(B) such that P^{i} is generated by its components of degree ≤ −i resp. ≥ −i, for all i.

Proposition 2.13.1. The pair (D^{↑}(B)^{≤0,g}, D^{↑}(B)^{≥0,g}) is a t-structure on D^{↑}(B).

Remark . An analogous definition gives a non-standard t-structure on D^{−}(B-mod). This is the “geometric” t-structure in [Soe92], hence the notation.

Proof[Proposition. ] Most of this is straightforward and left to the reader. Let us just prove that every Y ∈ D^{↑}(B) decomposes into a distinguished triangle (Y^{<0}, Y, Y^{≥0}). Using lemma 2.1.2 or more precisely its proof we see that every object of D^{↑}(B) can be represented by a complex of projective objects in C^{↑}(B), say

… → P^{i} → P^{i+1} → ….

Now we need

Lemma 2.13.2. This complex can be choosen such that the induced maps

P^{i}/B_{>0}P^{i} → P^{i+1}/B_{>0}P^{i+1}

all vanish.

Proof. Let us simplify notation and put \overline{M} = M/B_{>0}M, k = B_{0}, ⊗ = ⊗_{k}. Remark that for any projective P ∈ B-mod every splitting in k-mod of the surjection P ↠ \overline{P} defines an isomorphism P ≅ B ⊗\overline{P} . Consider now the maps ¯∂: \overline{P^{i}} → \overline{P^{i+1}}. Choose in k-mod decompositions

\overline{P^{i}} = I^{i} ⊕ H^{i} ⊕ R^{i}

such that I^{i} = ¯∂(\overline{P^{i}}) and I^{i} ⊕ H^{i} = ker(¯∂), so that ¯∂ induces isomorphisms R^{i} → I^{i+1}. Now choose splittings of graded k-modules R^{i} → P^{i}. This leads to splittings I^{i+1} → P^{i+1} and we can extend these splittings from I^{i} ⊕ R^{i} to all of \overline{P^{i}}. Thus we can decompose

P^{i} = (B ⊗ I^{i}) ⊕ (B ⊗ H^{i}) ⊕ (B ⊗ R^{i})

in such a way that our differentials induce isomorphisms from (B⊗R^{i}) to (B⊗I^{i+1}). It is then clear that our complex P^{•} is quasi-isomorphic to the complex (B ⊗ H^{•}) with the induced differentials, and that this complex satisfies the conditions of the lemma.

So let now Y ∈ D^{↑}(B) be represented by a complex of projective objects

… → P^{i} → P^{i+1} → …

in C^{↑}(B) such that if we apply k⊗_{B} all differentials become zero. Consider the submodules BP_{<−i}^{i} ⊂ P^{i} generated by the homogeneous components of degree less than −i. They are projective, and the quotients P^{i}/BP_{<−i}^{i} are projective as well. Furthermore the differential has to map BP_{<−i}^{i} to BP_{<−i−1}^{i+1}, by our assumption on the complex. Now we just let Y^{<0} be the complex of the BP_{<−i}^{i} and Y^{≥0} the complex of the P^{i}/BP_{<−i}^{i} and are done. The proposition is proven.

As a corollary of this proof, we give a more direct description of the core of our non-standard t-structure.

Corollary 2.13.3. Any object of the core of our non-standard t-structure on D^{↑}(B) is quasi-isomorphic to a complex of projectives

… → P^{i} → P^{i+1} → …

in C^{↑}(B) such that P^{i} = BP_{−i}^{i} ∀i.

Now we prove that this is the t-structure we were looking for.

Proposition 2.13.4. The Koszul duality functor K transforms the standard tstructure on D^{↓}(A) into our non-standard t-structure (D^{↑}(A^{!})^{≤0,g}, D^{↑}(A^{!})^{≥0,g}) on D^{↑}(A^{!}).

Proof. Certainly

D^{↓}(A)^{≤0} = {X ∈ D^{↓}(A) | Hom(X, A^{⊛}[i]⟨n⟩) = 0 ∀i < 0, ∀n}

and K transforms this into

{Y ∈ D^{↑}(A^{!}) | Hom(Y, A^{⊛}[i − n]⟨−n⟩) = 0 ∀i < 0, ∀n}.

It is clear that all Y ∈ D^{↑}(A^{!})^{≤0,g} satisfy this condition. It is equally clear that if Y ∈ D^{↑}(A^{!})^{≥0,g} satisfies this condition, then any complex of projectives isomorphic to it in the derived category has to be exact, i.e. Y = 0. Decomposing an arbitrary Y ∈ D^{↑}(A^{!}) into a distinguished triangle (Y^{<0}, Y, Y^{≥0}) according to our t-structure we find then that indeed K(D^{↓}(A)^{≤0}) ⊂ D^{↑}(A^{!})^{≤0,g}.

Similar arguments show the other inclusion K(D^{↓}(A)^{≥0}) ⊂ D^{↑}(A^{!})^{≥0,g}.

2.14. Koszul modules. Let A be a positively graded ring over the semisimple ring k = A_{0}.

Definition 2.14.1. A graded A-module M ∈ A-mod is called “Koszul” if and only if M admits a graded projective resolution

… → P^{2} → P^{1} → P^{0} ↠ M

such that P^{i} is generated by its component of degree i, P^{i} = AP_{i}^{i}.

Proposition 2.14.2. Let M ∈ A-mod be bounded from below, i.e. M ∈ A-mod^{↑}. Then the following are equivalent:

1. M is Koszul.

2. ext_{A}^{i}(M, N) = 0 for any pure module N whose weight is not (−i).

3. ext_{A}^{i}(M, k⟨n⟩) = 0 unless i = n.

Proof. Almost identical to the proof of proposition 2.1.3 and left to the reader.

Definition 2.14.3. A graded A-module M ∈ A-mod is called “quadratic” if and only if it is generated in degree zero with relations in degree one.

We may spell this out as follows: The multiplication m : A ⊗ M_{0} → M is a surjection, and ker(m) is generated as a left A-module by its component of degree one S = ker(A_{1} ⊗ M_{0} → M), so that M = A ⊗ M_{0}/AS.

Proposition 2.14.4. Any Koszul module is quadratic.

Proof. This is trivial.

Suppose from now on A is a quadratic ring A = T_{k}V/(R). Extending the considerations of section 2.6, we may define the Koszul complex

… → K^{2}M → K^{1}M → K^{0}M

of a quadratic A-module M by setting

K^{i}M = A ⊗ (⋂_{ν} V^{⊗ν} ⊗ R ⊗ V^{⊗i−ν−2} ⊗ M_{0} ∩ V^{⊗i−1} ⊗ S).

Proposition 2.14.5. A quadratic module M over a quadratic ring A is Koszul if and only if its Koszul complex K^{•}M is a resolution of M.

Proof. Almost the same as the proof of theorem 2.6.1 and left to the reader.

If A is a left finite quadratic ring and M a finitely generated quadratic module over A, we may form the quadratic dual M^{!} ∈ mod-A^{!} given by S^{⊥} ⊂ (A_{1} ⊗M_{0})^{∗} = M_{0}^{∗} ⊗ A_{1}^{∗} as M^{!} = M_{0}^{∗} ⊗ A^{!}/S^{⊥}A^{!}. Just as in section 2.8 we may then identify the Koszul complex of M with the complex

… → A ⊗^{∗}(M_{2}^{!}) → A ⊗^{∗}(M_{1}^{!}) → A ⊗^{∗}(M_{0}^{!})

where we leave the definition of the differential to the reader. In case A is Koszul, theorem 2.10.1 generalizes to a canonical isomorphism M^{!} = Ext_{A}^{•}(M, k) of left modules over (A^{!})^{opp} = Ext_{A}^{•}(k, k).

There also is an analog of the distributivity criterion (explained in the remark closing subsection 2.6) for Koszul modules over Koszul rings. It shows in particular that the dual M^{!} of a finitely generated Koszul module over a left finite Koszul ring is also Koszul.

3. Parabolic-singular duality and Koszul duality

3.1. Notations from algebraic geometry. Let X be a complex algebraic variety. Then we denote by D(X) = D_{const}^{b}(Sh(X^{an})) the bounded derived category with algebraically constructible cohomology of the category Sh(X^{an}) of sheaves of complex vector spaces on the analytic space X^{an} defined by X. Let P(X) ⊂ D(X) be the perverse sheaves (for the middle perversity) and H^{i} : D(X) → P(X) the perverse cohomology functors. For any F, G ∈ D(X) we may form the derived RHom-sheaf RHom(F, G) ∈ D(X). Any morphism f : X → Y gives adjoint pairs of exact functors (f^{∗}, f_{∗}) and (f_{!}, f^{!}) relating D(X) and D(Y). More classically f^{∗}, f_{∗}, f_{!} would be written f^{−1}, Rf_{∗}, and Rf_{!} respectively. Let \underline{X} ∈ Sh_{const}(X^{an}) ⊂ D(X) be the constant sheaf ℂ. Let Hom_{D} denote homomorphisms in D(X) and put Hom_{D}^{i}(F, G) = Hom_{D}(F, G[i]) for all F, G ∈ D(X). For any F ∈ D(X) its hypercohomology ℍ^{•}F = ⨁ ℍ^{i}F is a graded ℂ-vectorspace and even a graded module over the cohomology ring H^{•}(X) of X (which should really be denoted ℍ^{•}(\underline{X}) or even End_{D}^{•}(\underline{X}, \underline{X})). More details on these notions may be found in [Bor84, KS90, BBD82].

3.2. Existence of projective objects in some abelian categories. To prove that suitable categories of perverse sheaves have enough projectives, we first work in an abstract context. The results in this and the next section can also be found in papers by Cline, Parshall, and Scott [CPS91]. Let k be a field and A an abelian k-category. Suppose that

1. Every object of A has finite length.

2. There are only finitely many simple isomorphism classes in A.

To simplify the exposition, suppose furthermore

3. The endomorphisms of simple objects in A are reduced to scalars. Let {L(s)}_{s∈S} represent the simple isomorphism classes in A. Assume on S is given a partial order ≤ and S is equipped with the order topology, i.e. closed subsets T ⊂ S are characterized by s ∈ T, s′ ≤ s ⇒ s′ ∈ T. For any closed T ⊂ S let A_{T} be the full subcategory of A of objects “supported on T ”, i.e. all of whose simple subquotients have parameter in T.

Assume we are given for all s ∈ S objects Δ(s), ∇(s) and morphisms Δ(s) → L(s), L(s) → ∇(s) in A such that

4. Whenever T ⊂ S is closed and s ∈ T is maximal, Δ(s) → L(s) is an projective cover and L(s) → ∇(s) an injective hull of L(s) in A_{T} . (In particular, both Δ(s) and ∇(s) are indecomposable.)

5. ker(Δ(s) → L(s)) and coker(L(s) → ∇(s) lie in A_{<s} ∀s ∈ S. Assume furthermore

6. Ext_{A}^{2}(Δ(s), ∇(t)) = 0 ∀s, t ∈ S. We call the Δ(s) the standard modules and ∇(s) the costandard modules.

Theorem 3.2.1. Let A be an abelian category satisfying all of the above conditions (1)-(6). Then A has enough projective objects and each projective has a finite filtration with standard subquotients.

Remarks . 1. If we let P (s) ↠ L(s) be the projective cover and (P (s) : Δ(t)) the multiplicity of Δ(t)) as a subquotient in a standard filtration of P (s), then the so-called reciprocity formula (P (s) : Δ(t)) = [∇(t) : L(s)] holds. Indeed it is easy to see that both sides are just dim_{k} Hom_{A}(P (s), ∇(t)).

2. Dual results hold for injectives.

3. As Ringel explained to one of the authors, the conditions (1)–(5) are already sufficient to ensure the existence of enough projectives. His argument goes as follows. Choose s ∈ S maximal and assume by induction the statement to be known for A_{S−{s}}. For any t ∈ S we need to show that there is a bound N(t) on the possible length of an object X with unique simple quotient L(t). For t = s this is clear, so suppose t ≠ s. Consider the short exact sequence

0 → K → X → X̄ → 0

where X̄ is the maximal quotient of X which lies in A_{S−{s}}. Then X̄ is a quotient of the projective cover of L(t) in A_{S−{s}}, so its length is bounded. Furthermore we have an inclusion

Hom(K, L(s)) ↪ Ext^{1}(X̄, L(s)),

and K/radK is a direct sum of copies of L(s), hence K is a quotient of a direct sum of at most dim Ext^{1}(X̄, L(s)) copies of Δ(s). This does the job.

Proof. We start the proof by deriving further properties of A. Let us henceforth abreviate Hom_{A}, Ext_{A}^{i}, End_{A} by Hom, Ext^{i}, End.

7. End(Δ(s)) = k = End(∇(s)) follows from (3), (4) and (5).

8. For all M, N ∈ A the space Hom(M, N) is of finite dimension, as follows from (3) and (1).

9. For all M, N ∈ A the space Ext^{1}(M, N) is of finite dimension. To prove this, we may assume without restriction that M, N are simple. First assume M = L(s), N = L(t) with t ≯ s. Then there exists a closed subset T ⊂ S such that s, t ∈ T and s is maximal in T. Now the short exact sequence ker ↪ Δ(s) ↠ L(s) in A_{T} gives us a surjection Hom(ker, L(t)) ↠ Ext^{1}(L(s), L(t)), and we are done by (8). If t > s we proceed dually.

To prove theorem 3.2.1, we prove more generally for any closed subset T ⊂ S :

Claim(T) . There are enough projectives in A_{T} and each one of those has a finite filtration with standard sucessive quotients.

For this we proceed by induction on |T |. So let T ⊂ S be closed and s ∈ S such that T′ = T ∪ {s} is closed as well. We assume theorem(T) and try to construct for all t ∈ T′ a projective cover P_{T′}(t) ↠ L(t) of L(t) in A_{T′}. If t = s then Δ(s) → L(s) is the sought-for cover by (4). If t ≠ s then t ∈ T and theorem(T) gives at least a projective cover P_{T} (t) ↠ L(t) of L(t) in A_{T} .

Let us abreviate P_{T} (t) = P. Put E = Ext^{1}(P, Δ(s)). The canonical element in E^{∗} ⊗ E = E^{∗} ⊗ Ext^{1}(P, Δ(s)) = Ext^{1}(P, E^{∗} ⊗ Δ(s)) gives rise to an extension E^{∗} ⊗Δ(s) ↪ P̃ ↠ P such that, if we take the long exact sequence … → Hom(E^{∗} ⊗ Δ(s), Δ(s)) → Ext^{1}(P, Δ(s)) → Ext^{1}(P̃, Δ(s)) → Ext^{1}(E^{∗} ⊗ Δ(s), Δ(s)) → …, then the diagram

E
can ↙  ↘ id
… → Hom(E^{∗} ⊗ Δ(s), Δ(s))  →  Ext^{1}(P, Δ(s)) → …

commutes. Since can is an isomorphism by (7), the above boundary map is actually an isomorphism. Since Ext^{1}(Δ(s), Δ(s)) = 0 by (4) we get Ext^{1}(P̃, Δ(s)) = 0. We want to prove that P̃ is a projective cover of L(t) in A_{T′}. This would establish theorem(T′). To achieve this we only have to prove for r ∈ T′ = T ∪ {s} :

dim Hom(P̃, L(r)) = \{ \begin{matrix}1 & r = t \\ 0 & else, \end{matrix}
Ext^{1}(P̃, L(r)) = 0.

Suppose first r ∈ T. We consider the short exact sequence E^{∗} ⊗ Δ(s) ↪ P̃ ↠ P. Applying Ext^{i}( , L(r)) we get a long exact sequence. Since Hom(E^{∗}⊗Δ(s), L(r)) = Ext^{1}(E^{∗} ⊗ Δ(s), L(r)) = 0 by (4) this settles the case r ∈ T.

So we are left with the task to prove Hom(P̃, L(s)) = Ext^{1}(P̃, L(s)) = 0. Consider the short exact sequence ker ↪ Δ(s) ↠ L(s). It gives rise to two long exact sequences

Hom(P, ker) ↪ Hom(P, Δ(s)) → Hom(P, L(s)) → Ext^{1}(P, ker)
↓ ↓ ↓ ↓
Hom(P̃, ker) ↪ Hom(P̃, Δ(s)) → Hom(P̃, L(s)) → Ext^{1}(P̃, ker).

By the case r ∈ T the first vertical is an isomorphism and both Ext-groups vanish. By construction of P̃ the second vertical is an isomorphism. Thus the third vertical is an isomorphism and Hom(P̃, L(s)) = Hom(P, L(s)) = 0.

Next we prove Ext^{1}(P̃, L(s)) = 0. The short exact sequence ker ↪ Δ(s) ↠ L(s) gives an exact sequence Ext^{1}(P̃, Δ(s)) → Ext^{1}(P̃, L(s)) → Ext^{2}(P̃, ker). We deduced already from the definition of P̃ that Ext^{1}(P̃, Δ(s)) = 0. To get Ext^{2}(P̃, ker) = 0 it is sufficient to prove Ext^{2}(P̃, L(r)) = 0 for all r ∈ T. But this follows via the short exact sequence L(r) ↪ ∇(r) ↠ coker since we know already Ext^{1}(P̃, coker) = 0, and since Ext^{2}(P̃, ∇(r)) = 0 by (6) because P̃ has a filtration with standard subquotients. So Ext^{2}(P̃, ker) = 0 and thus Ext^{1}(P̃, L(s)) = 0.

Corollary 3.2.2. Let A be an abelian category satisfying (1)– (6). Then A has finite cohomological dimension.

Proof. We have to prove that every object admits a projective resolution of finite length. It is clear (at least from the point of view of derived categories) that if this holds for two objects in a short exact sequence, then it also holds for the third one. Hence it is enough to prove this for simple objects, and using (5) and induction over S from below we see that it is also enough to prove it for standard objects. But then we may use induction over S from above, using that the projectives P (s) we just constructed have filtrations with a single subquotient Δ(s) and all the other subquotients of the form Δ(t), t > s.

To check for condition 6) sometimes the following lemmas from folklore are helpful.

Lemma 3.2.3. Let A be an abelian category and B ⊂ A a full abelian subcategory, closed under extensions. Then for all M, N ∈ B the canonical map

Ext_{B}^{2}(M, N) → Ext_{A}^{2}(M, N)

is an injection.

Proof. Suppose given e ∈ Ext_{B}^{2}(M, N), e ≠ 0. Certainly there exists a surjection P ↠ M in B such that the pull-back Ext_{B}^{2}(M, N) → Ext_{B}^{2}(P, N) maps e to zero. The short exact sequence ker ↪ P ↠ M gives a commutative diagram with exact rows

Ext_{B}^{1}(P, N) → Ext_{B}^{1}(ker, N) → Ext_{B}^{2}(M, N) → Ext_{B}^{2}(P, N)
↓ ↓ ↓
Ext_{A}^{1}(P, N) → Ext_{A}^{1}(ker, N) → Ext_{A}^{2}(M, N)

and a diagram chase proves the lemma.

Now consider any triangulated category D with t-structure and let P ⊂ D be the core. In general one cannot extend the embedding P ⊂ D to a triangulated functor D^{b}(P) ⊂ D in a canonical way. However we have for all M, N ∈ P canonical maps Ext_{P}^{i}(M, N) → Hom_{D}^{i}(M, N). This follows from the fact for fixed M both sides are δ-functors in N from P to the category of abelian groups, and the left hand side is even a universal δ-functor.

Lemma 3.2.4. In the situation above, the canonical map

Ext_{P}^{2}(M, N) → Hom_{D}^{2}(M, N)

is an injection.

Proof. Very similar to the proof of the previous lemma.

3.3. Existence of projective perverse sheaves. We want to prove that suitable categories of perverse sheaves have enough projective objects. This is due to [MV87]. However we think that our argument is easier. In fact, it is quite similar to an argument of [CPS91].

Let X be a complex algebraic variety with an algebraic stratification by affine linear spaces X = ˙⋃_{w∈W}X_{w}. Let D(X, W) ⊂ D(X) consist of all objects which are smooth along the stratification and set P(X, W) = P(X) ∩ D(X, W). Let j_{w} : X_{w} → X be the inclusions of the strata. Put |w| = dim X_{w}, so X_{w} ≅ ℂ^{|w|}. Let C_{w} = \underline{X}_{w}[−|w|] be the constant perverse sheaf on X_{w} and consider the standard and costandard objects M_{w} = j_{w!}C_{w}, N_{w} = j_{w∗}C_{w}. By our assumptions j_{w} is affine, hence both M_{w} and N_{w} belong to P(X, W). We will need

Theorem 3.3.1. There are enough projectives in P(X, W), and each one has a filtration with standard objects as subquotients. Dually, there also are enough injectives in P(X, W), and each one has a filtration with costandard objects as subquotients.

Proof. This is a direct consequence of theorem 3.2.1.

For simplicity suppose now in addition X to be smooth. Then we may interpret P(X) as a category of D-modules and D(X) as a category of complexes of Dmodules, and get in this way an evident extension of the embedding P(X) → D(X) to a triangulated functor D^{b}(P(X)) → D(X). This in turn gives us in an obvious way a functor D^{b}(P(X, W)) → D(X, W) and we obtain under our assumptions

Corollary 3.3.2. The obvious functor D^{b}(P(X, W)) → D(X, W) is an equivalence of categories.

Proof. By corollary 3.2.2 we see that the left hand side is generated as a triangulated category by the projectives and also by the injectives in P(X, W). So we need only show that for P, I ∈ P(X, W) a projective resp. injective object we have Hom_{D^{b}(P(X,W))}^{i}(P, I) = Hom_{D}^{i}(P, I). This is clear for i = 0, and for i ≠ 0 both sides vanish: The right hand side since it is just an Ext group in P(X, W), the left hand side since P (resp. I) has a filtration with standard (resp. costandard) sucessive quotients and Hom_{D}^{i}(M_{v}, N_{w}) = 0 for i ≠ 0.

3.4. Calculation of some extensions. Consider a complex algebraic variety X filtered by closed subvarieties X = X_{0} ⊃ X_{1} ⊃ … ⊃ X_{r} = ∅ and F ∈ D(X). To compute ℍ^{•}F we have to choose an injective resolution I^{•} of F and take cohomology of the complex of global sections Γ(I^{•}). This is filtered by supports, 0 ⊂ Γ_{X_{r}}(I^{•}) ⊂ … ⊂ Γ_{X_{0}}(I^{•}) = Γ(I^{•}). So its cohomology is the limit of a spectral sequence with E_{1}-term E_{1}^{p,q} = ℍ_{X_{p}−X_{p+1}}^{p+q}(F), the local hypercohomology of F along X_{p} − X_{p+1}. For v : Y ↪ X locally closed we have ℍ_{Y}^{•} (F) = ℍ^{•}v^{!}F. So if we let i_{p} : (X_{p} − X_{p+1}) ↪ X be the inclusion, the E_{1}-term can also be written as E_{1}^{p,q} = ℍ^{p+q}i_{p}^{!}F. For L, M ∈ D(X) we then find that Hom^{•}(L, M) = ℍ^{•}RHom(L, M) is the limit of a spectral sequence with E_{1}-term E_{1}^{p,q} = ℍ^{p+q}i_{p}^{!}RHom(L, M) = ℍ^{p+q}RHom(i_{p}^{∗}L, i_{p}^{!}M).

Let now X be a complex variety with an algebraic stratification by affine linear spaces X = ˙⋃_{w∈W}X_{w}. Let j_{w} : X_{w} → X be the inclusions of the strata. Put |w| = dim X_{w}, so X_{w} ≅ ℂ^{|w|}. Consider the intersection cohomology complexes IC_{w} ∈ P(X) of the \overline{X}_{w}. For all i the perverse cohomology H^{−i}j_{v}^{∗}IC_{w} ∈ P(X_{v}) is of the form H^{−i}j_{v}^{∗}IC_{w} ≅ n_{v,w}^{i}C_{v} where C_{v} ∈ P(X_{v}) is the constant sheaf put in degree −|v| and n_{v,w}^{i} ∈ ℕ are multiplicities. We may assemble the constants n_{v,w}^{i} together in a W × W-matrix IC(t) = IC(X, t) with entries in ℤ[t] given by

IC_{v,w}(t) = ∑_{i} n_{v,w}^{i}t^{i}.

In fact, since the j_{v}^{∗} are right exact for the perverse t-structure n_{v,w}^{i} = 0 for i < 0. By definition the diagonal entries of our matrix IC(t) are all one, and for an ordering of W compatible with the closure relations between strata the whole matrix is upper triangular.

Theorem 3.4.1. Suppose we have parity vanishing, i.e. H^{i}j_{v}^{∗}IC_{w} = 0 unless i ≡ |v| + |w| (mod 2). Then

∑_{i} t^{i} dim Hom_{D}^{i}(IC_{x}, IC_{y}) = ∑_{z} IC_{z,x}(t)IC_{z,y}(t)

is just the (x, y)-entry of IC^{T} · IC.

Remark . Instead of parity vanishing we might as well assume pointwise purity of the IC_{x}. The idea to use parity vanishing comes from [CPS91].

Proof. We apply the preceding considerations to L = IC_{x}, M = IC_{y}, and the filtration by the X_{p} = ⋃_{|x|+p≤dim X} X_{x}. Thus Hom_{D}^{•}(IC_{x}, IC_{y}) is the limit of a spectral sequence with E_{1}-term

\begin{aligned}E_{1}^{p,q} & = & ℍ^{p+q}RHom(i_{p}^{∗}IC_{x}, i_{p}^{!}IC_{y}) \\ & = & ⊕_{|z|+p=dim X}ℍ^{p+q}RHom(j_{z}^{∗}IC_{x}, j_{z}^{!}IC_{y}). \end{aligned}

Our assumptions say j_{z}^{∗}IC_{x} ≅ ⨁_{μ} n_{z,x}^{μ}C_{z}[μ] and dually j_{z}^{!}IC_{y} ≅ ⨁_{ν} n_{z,y}^{ν}C_{z}[−ν] and now parity vanishing tells us that our spectral sequence vanishes “like a chessboard”. It follows that it degenerates at the E_{1}-term and we deduce that

dim Hom_{D}^{i}(IC_{x}, IC_{y}) = ∑_{p+q=i} dim E_{1}^{p+q} = ∑_{z,μ+ν=i} n_{z,x}^{μ}n_{z,y}^{ν}.

This proves our theorem.

In the same setup we can also prove

Proposition 3.4.2. Suppose we have parity vanishing. Then hypercohomology induces an injection

Hom_{D}^{•}(IC_{x}, IC_{y}) → Hom_{ℂ}(ℍ^{•}IC_{x}, ℍ^{•}IC_{y}).

Proof. By parity vanishing the spectral sequence ℍ^{p+q}i_{p}^{!}IC_{x} ⇒ ℍ^{n}IC_{x} is degenerate. So if f ∈ Hom_{D}^{•}(IC_{x}, IC_{y}) is given such that ℍ^{•}f = 0, then necessarily 0 = i_{p}^{!}f ∈ Hom_{D}^{•}(i_{p}^{!}IC_{x}, i_{p}^{!}IC_{y}) for all p. Let a_{p} : X_{p} ↪ X be the inclusion. We have a decomposition

X_{p} − X_{p+1} ↪^{u} X_{p} ↩^{i} X_{p+1}

in an open and a closed subset and a distinguished triangle (u_{∗}u^{!}, id, i_{!}i^{!})a_{p}^{!} which shows that a_{p+1}^{!}f = 0 = i_{p+1}^{!}f ⇒ a_{p}^{!}f = 0. Hence by induction i_{p}^{!}f = 0 ∀p implies f = a_{0}^{!}f = 0.

3.5. Localization of O^{𝔮}. Let 𝔤 ⊃ 𝔮 ⊃ 𝔟 be as in the introduction and let G ⊃ Q ⊃ B be corresponding affine connected algebraic groups. We put S_{Q} = S_{𝔮} and hence W^{Q} = W^{𝔮}, etc. Let P_{B}(G/Q) ⊂ P(G/Q) be all perverse sheaves which are smooth along B-orbits. So the simple objects of P_{B}(G/Q) are just the intersection cohomology complexes L_{x}^{Q} of the closures of the B-orbits BxQ/Q, for x ∈ W^{Q}. We want to prove

Proposition 3.5.1. There exists an equivalence of categories O^{𝔮} ≅ P_{B}(G/Q) such that L_{x}^{𝔮} corresponds to L_{x}^{Q}.

In fact, we will even construct an equivalence of triangulated categories from D^{b}(O^{𝔮}) to the full subcategory D_{B}(G/Q) ⊂ D(G/Q) consisting of all objects which are smooth along B-orbits. Here the case 𝔮 = 𝔟 is localization [BB81].

Proposition 3.5.2. There exists an equivalence of triangulated categories L^{B} : D^{b}(O^{𝔟}) ≅ D_{B}(G/B) such that the obvious t-structure on the left corresponds to the perverse t-structure on the right and the induced equivalence O^{𝔟} ≅ P_{B}(G/B) maps L_{x}^{𝔟} to L_{x}^{B} for all x ∈ W.

Proof. For example [BB81, Soe86] and [Soe89a, proposition 6].

We will now deduce the general case. Let π : G/B → G/Q be the projection and d = dim(Q/B) its fibre dimension.

Theorem 3.5.3. There exists an equivalence L^{Q} : D^{b}(O^{𝔮}) ≅ D_{B}(G/Q) of triangulated categories such that the obvious t-structure on the left corresponds to the perverse t-structure on the right and the induced equivalence O^{𝔮} ≅ P_{B}(G/Q) maps L_{x}^{𝔮} to L_{x}^{Q} for all x ∈ W^{Q}. Furthermore, the diagram

D^{b}(O^{𝔮}) ≅ D_{B}(G/Q)
↓  ↓ π^{!}[−d]
D^{b}(O^{𝔟}) ≅ D_{B}(G/B)

commutes up to natural equivalence and its vertical arrows are faithful functors.

Proof[Theorem. ] We procede in several steps. Step 1: We show that π^{!} : D(G/Q) → D(G/B) is faithful. More precisely, we show

Lemma 3.5.4. There exist natural isomorphisms π_{!}π^{!}F = ⨁_{x∈W_{Q}} F[2l(x)] for F ∈ D(G/Q).

Proof. By the decomposition theorem π_{!}\underline{G/B} ≅ ⨁_{x∈W_{Q}} \underline{G/Q}[−2l(x)]. Choose such an isomorphism. Then we get

π_{!}π^{!}F = π_{∗}RHom(\underline{G/B}, π^{!}F)
= RHom(π_{!}\underline{G/B}, F)
= ⨁_{x∈W_{Q}} F[2l(x)]

where the second step is Verdier duality.

Step 2: Let P_{Q^{−1}}(G/B) be the category of all perverse sheaves on G/B which are smooth along the strata BxQ/B. The simple objects of this category are the L_{x}^{B} with x ∈ W^{Q}.

Proposition 3.5.5. The functors π^{!}[−d] and H^{d}π_{!} define an equivalence of categories P_{Q^{−1}}(G/B) ≅ P_{B}(G/Q) such that L_{x}^{B} corresponds to L_{x}^{Q}.

Proof[Proposition. ] Let us first show how H^{d}π_{!} : P(G/B) → P(G/Q) is the right adjoint of π^{!}[−d] : P(G/Q) → P(G/B). Indeed for G ∈ P(G/B), F ∈ P(G/Q) we have

Hom_{D}(G, π^{!}[−d]F) = Hom_{D}(π_{!}[d]G, F)
= Hom_{D}(H^{d}π_{!}G, F),

the latter equality follows since we know that π_{!}[d]G is concentrated in perverse degrees ≤ 0. Next we show that for any F ∈ P(G/Q) the adjointness map going from (H^{d}π_{!})(π^{!}[−d])F to F is an isomorphism. Indeed this follows immediately from the preceding lemma. Now we need

Lemma 3.5.5. The functor H^{d}π_{!} : P_{Q^{−1}}(G/B) → P_{B}(G/Q) is exact.

Proof. It is sufficient to show that for all G ∈ P_{Q^{−1}}(G/B) and all i of parity different from d we have H^{i}π_{!}G = 0. We may even assume G simple, say G = L_{x}, x ∈ W^{Q}. But then G ≅ π^{!}[−d]L_{x}^{Q} and parity vanishing follows from the preceding lemma.

Finally we show that for all G ∈ P_{Q^{−1}}(G/B) the adjointness map going from G to (π^{!}[−d])(H^{d}π_{!})G is an isomorphism. This is clear for all G of the form π^{!}[−d]F, in particular for all simple G. Then it follows for all G from the five lemma. The proposition is proven.

Step 3: It is evident that the equivalence O^{𝔟} ≅ P_{B}(G/B) of proposition 3.5.2 induces an equivalence O^{𝔮} ≅ P_{Q^{−1}}(G/B). Together with the preceding proposition we get now a diagram of exact functors

O^{𝔮} ≅ P_{B}(G/Q)
↓  ↓ π^{!}[−d]
O^{𝔟} ≅ P_{B}(G/B)

that commutes up to natural equivalence. Certainly we can go over to bounded derived categories and obtain a diagram

D^{b}(O^{𝔮}) ≅ D^{b}(P_{B}(G/Q))
↓  ↓ π^{!}[−d]
D^{b}(O^{𝔟}) ≅ D^{b}(P_{B}(G/B)).

So to prove the theorem we have only to establish a commutative diagram

D^{b}(P_{B}(G/Q)) ⟶ D_{B}(G/Q)
π^{!}[−d] ↓  ↓ π^{!}[−d]
D^{b}(P_{B}(G/B)) ⟶ D_{B}(G/B)

such that the horizontal arrows are equivalences. For the definition of these horizontal arrows remark that for smooth X we may interpret P(X) as a category of D-modules and D(X) as a category of complexes of D-modules, and get in this way an evident extension of the embedding P(X) → D(X) to an exact functor D^{b}(P(X)) → D(X). From this point of view it is clear how to define the horizontal arrows, and it is equally clear that the diagram commutes. We then apply corollary 3.3.2 to see that the horizontal functors are equivalences. The theorem is proven.

We now want to discuss standard objects. Recall the parabolic Verma modules M_{x}^{𝔮} ∈ O^{𝔮} from the introduction and let M_{x}^{Q} = j_{x!}C_{x} ∈ P_{B}(G/Q) be the standard objects.

Proposition 3.5.7. Our equivalence O^{𝔮} ≅ P_{B}(G/Q) identifies M_{x}^{𝔮} and M_{x}^{Q}.

Proof. This is clear since M_{x}^{𝔮} is the projective cover of L_{x}^{𝔮} in O_{≤x}^{𝔮} and M_{x}^{Q} is the projective cover of L_{x}^{Q} in P_{B}(G/Q)_{≤x} (see section 3.2 for the notations).

Remarks . 1. The proposition allows us to avoid the use of sections 3.2 and 3.3 in the proof of the theorem. Namely the only thing we used was corollary 3.3.2 for the stratified space G/Q, and to prove this corollary we only needed that P_{B}(G/Q) has (1) finite homological dimension, (2) enough projectives and injectives and that (3) each projective (resp. injective) has a filtration with standard (resp. costandard) subquotients. But (1)–(3) are known for O^{𝔮}, hence they follow for the equivalent category P_{B}(G/Q).

2. Let us mention that the theorem allows to interpret the derived functors of the left and right adjoint of the inclusion O^{𝔮} → O^{𝔟} geometrically as the left and right adjoints of π^{!}[−d], i.e. as π_{!}[d] and π_{∗}[−d]. This in turn allows to compute the effect of these “derived parabolic truncation functors” on simple objects.

3.6. Extensions of simple objects in O^{𝔮}. To any two elements x, y of the Weyl group Kazhdan and Lusztig associate [KL80a] the so-called KL-polynomial P_{x,y}(q) ∈ ℤ[q]. Furthermore they prove [KL80b] that the intersection cohomology matrix IC(G/Q, t) of the stratified space G/Q = ˙⋃_{w∈W^{Q}}BwQ/Q can be expressed in terms of their polynomials as

IC(G/Q, t)_{x,y} = P_{x,y}(t^{−2})t^{l(y)−l(x)} ∀x, y ∈ W^{Q}.

Let us denote this matrix by P^{Q} = P^{Q}(t). Remember now our algebra

A_{Q} = Ext_{O^{𝔮}}^{•}(⊕L_{x}^{𝔮}, ⊕L_{x}^{𝔮})

from section 1.3.

Proposition 3.6.1. The Hilbert polynomial of A_{Q} is P (A_{Q}, t) = (P^{Q})^{T} P^{Q}.

Proof.

P (A_{Q}, t)_{x,y} = ∑_{i} t^{i} dim Ext_{O^{𝔮}}^{i}(L_{y}^{𝔮}, L_{x}^{𝔮}) by definition
= ∑_{i} t^{i} dim Hom_{D}^{i}(L_{y}^{Q}, L_{x}^{Q}) by the preceding section
= ((P^{Q})^{T} P^{Q})_{x,y} by theorem 3.4.1.

3.7. Parabolic-singular duality. We want to establish the first isomorphism of theorem 1.1.3.

Theorem 3.7.1. Let λ, 𝔮 be as in the introduction and suppose S_{λ} = S_{𝔮}. Then there is an isomorphism of ℂ-algebras

End_{O_{λ}}(⊕P (x · λ)) ≅ Ext_{O^{𝔮}}^{•}(⊕L_{x}^{𝔮}, ⊕L_{x}^{𝔮})

such that the obvious idempotents 1_{x} on both sides correspond.

Proof. We will describe both sides in a combinatorial way to see that they are the same. In the case 𝔮 = 𝔟 this is done in [Soe90]. We will now reduce our more general situation to this case. Step 1: The dimensions of both sides coincide. Let (P (x · λ) : M(z · λ)) be the Verma flag multiplicities. They are related to Jordan-Hölder-multiplicities by the so-called BGG-reciprocity formula

(P (x · λ) : M(z · λ)) = [M(z · λ) : L(x · λ)].

Now we calculate

dim End_{O_{λ}}(⨁ P (x · λ)) = ∑_{x,y∈W^{Q}}[P (x · λ) : L(y · λ)]
= ∑_{x,y,z∈W^{Q}}(P (x · λ) : M(z · λ))[M(z · λ) : L(y · λ)]
= ∑_{x,y,z∈W^{Q}}[M(z · λ) : L(x · λ)][M(z · λ) : L(y · λ)]
= ∑_{x,y,z∈W^{Q}} P_{z,x}(1)P_{z,y}(1) by the KL-conjectures
= dim Ext_{O^{𝔮}}^{•}(⊕L_{x}^{𝔮}, ⊕L_{x}^{𝔮}) by proposition 3.6.1

Step 2: Combinatorial description of the left hand side.

Let S = S(𝔥) be the symmetric algebra over 𝔥, C = C(𝔥) = S/(S^{+})^{W}S the coinvariants for the action of the Weyl group and C_{λ} = C_{λ}(𝔥) = C^{W_{λ}} the W_{λ}-invariants in C. Let P_{λ} = P (w_{∘} · λ) ∈ O_{λ} be a projective cover of the simple Verma module. If θ^{out} : O_{λ} → O_{0} is the translation out of the walls, we have θ^{out}P_{λ} ≅ P_{0}. In [Soe90, Ber90] one may find the construction of isomorphisms C_{λ} = End_{O}P_{λ}. Under these isomorphisms the map End_{O}P_{λ} → End_{O}θ^{out}P_{λ} = End_{O}P_{0} corresponds to the inclusion C_{λ} ↪ C_{0} = C. The isomorphisms C_{λ} = End_{O}P_{λ} give rise to the functors 𝕍 = 𝕍_{λ} = Hom_{O}(P_{λ}, ) : O_{λ} → C_{λ}-Mof. In [Soe90] it is proved that 𝕍 induces isomorphisms Hom_{O}(P (x · λ), P (y · λ)) → Hom_{C_{λ}}(𝕍P (x · λ), 𝕍P (y · λ)) for all x, y ∈ W^{λ}. Thus End_{O_{λ}}(⊕P (x · λ)) ≅ End_{C_{λ}}(⊕𝕍P (x · λ)). Step 3: Combinatorial description of the right hand side. Analogously to C_{λ} = C_{λ}(𝔥) one may define C_{𝔮} = C_{𝔮}(𝔥^{∗}). Define a grading on C_{𝔮} by requiring that 𝔥^{∗} has degree 2. In [BGG73] one may find the construction of an isomorphism of graded ℂ-algebras C_{𝔮} ≅ H^{•}(G/Q), the cohomology ring of G/Q. Under this isomorphism the pull-back H^{•}(G/Q) → H^{•}(G/B) corresponds to the inclusion C_{𝔮} ↪ C_{𝔟} = C. In this way the hypercohomology gives rise to a functor ℍ^{•} : D(G/Q) → C_{𝔮}-mof. By proposition 3.4.2 hypercohomology induces injections Ext_{D}^{•}(L_{x}^{Q}, L_{y}^{Q}) → Hom_{C_{𝔮}}(ℍ^{•}L_{x}^{Q}, ℍ^{•}L_{y}^{Q}) of graded vector spaces over ℂ. Theorem 3.5.3 gives isomorphisms Ext_{O^{𝔮}}^{•}(L_{x}^{𝔮}, L_{y}^{𝔮}) ≅ Ext_{D}^{•}(L_{x}^{Q}, L_{y}^{Q}). Hence an injection Ext_{O^{𝔮}}^{•}(⊕L_{x}^{𝔮}, ⨁ L_{x}^{𝔮}) ↪ End_{C_{𝔮}}(⊕ℍ^{•}L_{x}^{Q}), which will soon turn out to be an isomorphism. Step 4: The isomorphism. Choose a W-equivariant isomorphism 𝔥 ≅ 𝔥^{∗}. Since S_{λ} = S_{𝔮} this gives an isomorphism C_{λ} ≅ C_{𝔮}. We just have to prove that there exist isomorphisms 𝕍P (x · λ) ≅ ℍ^{•}L_{x}^{Q} in C_{λ}-Mod ≅ C_{𝔮}-Mod. (Remark that by steps 1 and 2 this implies in particular that the injection at the end of the preceding step is indeed an isomorphism.) The existence of an isomorphism 𝕍P (x · 0) ≅ ℍ^{•}L_{x}^{B} is established in [Soe90]. The general case then follows directly from

Lemma 3.7.2. (i) For all x ∈ W^{λ} there exists an isomorphism in C_{λ}-Mod

res_{C_{λ}}^{C}𝕍_{0}P (x · 0) ≅ ⊕_{z∈W_{λ}}𝕍_{λ}P (x · λ).

(ii) For all x ∈ W^{𝔮} there exists an isomorphism in C_{𝔮}-mod

res_{C_{𝔮}}^{C}ℍ^{•}L_{x}^{B} ≅ ⊕_{z∈W_{𝔮}}(ℍ^{•}L_{x}^{Q})⟨2l(z) − d⟩.

Proof. (i) Let θ_{on} : O_{0} → O_{λ} be the translation onto the walls. It is known that θ_{on}θ^{out} is just a direct sum of |W_{λ}| copies of the identity functor on O_{λ}. Now for all x ∈ W^{λ} we have θ^{out}P (x · λ) ≅ P (x · 0), hence θ_{on}P (x · 0) ≅ ⨁_{z∈W_{λ}} P (x · λ). But, in addition, by the adjointness of the pair (θ^{out}, θ_{on})

𝕍_{λ}(θ_{on}M) = Hom_{O}(P_{λ}, θ_{on}M)
= Hom_{O}(P_{0}, M)
= res_{C_{λ}}^{C}(𝕍_{0}M)

for all M ∈ O_{0}. Hence res_{C_{λ}}^{C}𝕍_{0}P (x · 0) ≅ ⨁_{z∈W_{λ}} 𝕍_{λ}P (x · λ).

(ii) For all x ∈ W^{Q} we have

res_{C_{𝔮}}^{C}ℍ^{•}L_{x}^{B} = ℍ^{•}π_{∗}L_{x}^{B}
= ℍ^{•}π_{!}π^{!}L_{x}^{Q}[−d]
= ⊕_{z∈W_{Q}}ℍ^{•}(L_{x}^{Q}[2l(z) − d])
= ⊕_{z∈W_{Q}}(ℍ^{•}L_{x}^{Q})⟨2l(z) − d⟩

The theorem is proven.

Recall now the ring A^{Q} = Ext_{O}^{•}(⊕L(x · λ), ⊕L(x · λ)).

Corollary 3.7.3. We have a ring isomorphism A^{Q} ≅ E(A_{Q}) such that the obvious idempotents 1_{x} on both sides correspond.

Proof. As explained in section 1.3 the theorem gives an equivalence of categories

O_{λ} ≅ A_{Q}-Mof

identifying L(x · λ) ∈ O_{λ} with A_{Q}^{0}1_{x} ∈ A_{Q}-Mof. ¿From this we deduce an isomorphism of extension rings

Ext_{O_{λ}}^{•}(⊕L(x · λ), ⊕L(x · λ)) ≅ Ext_{A_{Q}}^{•}(A_{Q}^{0}, A_{Q}^{0}).

3.8. Canonicity of the grading on O_{λ}. A very annoying feature of our paper is that at lots of places we rather prove the existence of isomorphisms instead of really constructing them. This might be an indication that we not yet have the correct point of view. We want to explain in this section a part of our constructions that is canonical. Namely we construct (in the notations of the preceding section) a completely canonical equivalence of categories

Mof-(End_{D}^{•}(⊕L_{x}^{Q})) → O_{λ}.

This can be done as follows: First put P_{λ} = θ^{out}M(−ρ). This is a projective cover of the simple Verma module M(w_{∘} · λ) ∈ O_{λ} that doesn’t depend on any choices, and we have canonically End_{𝔤}P_{λ} = C_{λ}. Then we identify 𝔥 with 𝔥^{∗} via the Killing form and obtain an identification C_{λ} ≅ C_{𝔮}. Via this identification the intersection cohomology groups ℍ^{•}L_{x}^{Q} of the closures of B-orbits in G/Q can be considered as C_{λ}-modules and we may form the 𝔤-modules

P (x · λ) = Hom_{C_{λ}}(ℍ^{•}L_{x}^{Q}, P_{λ})

for each x ∈ W^{Q}. With the same arguments as in the proof of [Soe90, proposition 6], we see that these P (x · λ) are projective covers of the L(x · λ) in O_{λ}. (However we cannot canonically construct surjections P (x · λ) ↠ L(x · λ).) Let us put P = ⊕P (x·λ). This is a completely canonical projective generator of O_{λ}, and the obvious map

(End_{D}^{•}(⊕L_{x}^{Q})) → End_{𝔤}P

is an isomorphism. Now P has commuting structures as a left 𝔤-module and as a left End_{𝔤}P -module. For general reasons the functor

⊗_{(End_{𝔤}P)}P : Mof-(End_{𝔤}P) → O_{λ}

is an equivalence of categories, and via the above isomorphism this gives the soughtfor equivalence

Mof-(End_{D}^{•}(⊕L_{x}^{Q})) → O_{λ}.

3.9. The ring A_{B} is Koszul. The “correct” proof of this fact is geometric and will be given in section 4. Here we give two less conceptual but easier proofs via selfduality.

Theorem 3.9.1. A_{B} is Koszul.

First Proof. By the numerical Koszulity criterion we only need to check the matrix equation P (A_{B}, t)P (E(A_{B}), −t) = 1. Here P (A_{B}, t) = (P^{B})^{T} P^{B} by proposition 3.6.1. To compute the Hilbert polynomial of E(A_{B}) we use selfduality. Namely the corollary above gives us an isomorphism E(A_{B}) ≅ A^{B}, but since O^{𝔟} = O_{0} and L_{x}^{𝔟} = L(x^{−1}w_{∘} · 0) we have an identity A_{B} = A^{B} with 1_{x} corresponding to 1_{x^{−1}w_{∘}}. So up to a reindexation of our idempotents E(A_{B}) = A_{B}, and hence if we define the W × W-matrix P_{B} = P_{B}(t) by (P_{B})_{x,y} = P_{w_{∘}x^{−1},w_{∘}y^{−1}}^{B}, then P (E(A_{B}), t) = P_{B}^{T} P_{B}. But now the inversion formulas for KL-polynomials from [KL80a] yield P^{B}(t)P_{B}(−t)^{T} = 1. Hence

P (A_{B}, t)P (E(A_{B}), −t) = (P^{B}(t))^{T} P^{B}(t)P_{B}(−t)^{T} P_{B}(−t) = 1.

Second Proof. This is due to Roman Bezrukavnikov. One deduces the theorem from the following

Lemma 3.9.2. Suppose A is a positively graded ring over the semisimple ring A_{0} = k. Suppose furthermore A is an algebra over a field F and dim_{F} A_{i} < ∞ for i = 0, 1.

If there is an isomorphism of graded F -algebras A ≅ E(A), then A is Koszul.

Proof. Remark first that for any V ∈ k-Mod we have dim_{F} (V^{∗}) = dim_{F} V, for V^{∗} = Hom_{k}(V, k) as in section 2.7. Remark next that if M, N ∈ A-mod are graded A-modules and N_{i} = 0 for all but finitely many i, then Hom_{A}(M, N) = ∏_{ν} hom_{A}(M, N⟨ν⟩) and even Ext_{A}^{i}(M, N) = ∏_{ν} ext_{A}^{i}(M, N⟨ν⟩). Now consider I = A_{>0} and the short exact sequence I ↪ A ↠ k. ¿From this we get a commutative diagram

Ext_{A}^{1}(k, k) = (I/I^{2})^{∗}
↑  ↑
ext_{A}^{1}(k, k⟨1⟩) =  A_{1}^{∗}

where the vertical arrows are inclusions. Now our selfduality assumption tells us that both rows have the same dimension over F, hence Ext_{A}^{1}(k, k) = ext_{A}^{1}(k, k⟨1⟩) and (I/I^{2}) = A_{1}. But from (I/I^{2}) = A_{1} follows that A is generated by A_{1} over k, hence using selfduality again E(A) is generated by Ext_{A}^{1}(k, k) = ext_{A}^{1}(k, k⟨1⟩) over k^{opp}. This in turn implies Ext_{A}^{i}(k, k) = ext_{A}^{i}(k, k⟨i⟩) for all i, hence A is Koszul by proposition 2.1.3.

Now remark as before that there is an isomorphism of graded ℂ-algebras A_{B} ≅ E(A_{B}). The theorem follows immediately from the lemma.

Remark . We can now run the first proof backwards and deduce the inversion formulas for Kazhdan-Lusztig polynomials from the selfduality of A_{B}.

3.10. The rings A_{Q} and A^{Q} are Koszul. Here our argument avoiding mixed geometry is really screwed-up. By a “Koszul grading” on a ring we mean a grading such that the corresponding graded ring is Koszul. Consider first A^{𝔮} = End_{O}(⊕P_{x}^{𝔮}).

Proposition 3.10.1. The ring A^{𝔮} can be given a Koszul grading.

Proof. Consider the equivalence of categories O^{𝔟} ≅ Mof-A^{B}. Since

O^{𝔮} = {M ∈ O^{𝔟} | [M : L_{x}^{𝔟}] ≠ 0 ⇒ x ∈ W^{Q}}

this induces an equivalence of categories O^{𝔮} ≅ Mof-(A^{B}/I_{Q}), where I_{Q} ⊂ A^{B} is the two-sided ideal generated by all 1_{x} with x ∈ W \ W^{Q}. Since both A^{B}/I_{Q} and A^{𝔮} are basic rings with the same category of right modules O^{𝔮}, we find that they are isomorphic.

To prove the proposition we will show that the graded ring A^{B}/I_{Q} is Koszul. So we need to show that extensions between pure objects of mof-(A^{B}/I_{Q}) vanish unless our objects have the correct weight. But since we know already that A^{B} is Koszul it will be sufficient to show that the restriction functor Mof-(A^{B}/I_{Q}) → Mof-(A^{B}) induces injections on extensions. In other words, we have to show that the inclusion O^{𝔮} → O^{𝔟} induces injections on extensions. But this is just the “faithful”-part of theorem 3.5.3.

Theorem 3.10.2. 1. The ring A_{Q} is Koszul.

2. The ring A^{Q} is Koszul.

3. We have A^{𝔮} ≅ A^{Q}.

Proof. (1) Under the equivalence of categories O^{𝔮} ≅ Mof-A^{𝔮} the objects ⊕L_{x}^{𝔮} and A_{0}^{𝔮} correspond, both being just the direct sum of “all” simples (i.e. taking one from each isomorphism class). Hence we get a ring isomorphism

Ext_{O^{𝔮}}^{•}(⊕L_{x}^{𝔮}, ⊕L_{x}^{𝔮}) ≅ Ext_{−A^{𝔮}}^{•}(A_{0}^{𝔮}, A_{0}^{𝔮})

which can be read as A_{Q} ≅ E((A^{𝔮})^{opp}). Hence by proposition 2.2.1 and theorem 2.10.2 the ring A_{Q} is Koszul.

(2) We know already A^{Q} ≅ E(A_{Q}) from corollary 3.7.3. Hence A^{Q} is Koszul by theorem 2.10.2.

(3) The isomorphism A_{Q} ≅ E((A^{𝔮})^{opp}) leads to E(A_{Q}) ≅ (A^{𝔮})^{opp} using theorem 2.10.2. But from corollary 3.7.3 we deduce E(A_{Q}) ≅ A^{Q}, and we know A^{Q} ≅ (A^{Q})^{opp}.

3.11. Graded representation categories and Koszul duality. Given λ, 𝔮 as in the introduction with S_{λ} = S_{𝔮} we constructed finite dimensional graded ℂ-algebras A_{Q}, A^{Q} which are Koszul rings with degree zero part the semisimple ring k(Q) = ⊕_{x∈W^{Q}}ℂ1_{x}. We furthermore defined isomorphisms (A_{Q})^{opp} = A_{Q}, (A^{Q})^{opp} = A^{Q} over k(Q) and in section 1.3 we deduced from theorem 1.1.3 (whose proof is already complete):

1. There exists an equivalence of ℂ-categories A_{Q}-Mof ≅ O_{λ} identifying A_{Q}^{0}1_{x} with L(x · λ).

2. There exists an equivalence of ℂ-categories A^{Q}-Mof ≅ O^{𝔮} identifying A_{0}^{Q}1_{x} with L_{x}^{𝔮}.

3. There exists an isomorphism A_{Q}^{!} ≅ A^{Q} of graded ℂ-algebras over k(Q). Let us put O_{Q} = A_{Q}-mof. The forgetting of grading v together with a choice of an equivalence as in (1) gives us an exact functor v : O_{Q} → O_{λ}. We have v(M⟨i⟩) = vM for every M ∈ O_{Q}, and for any M, N ∈ O_{Q} the obvious maps

⊕Ext_{O_{Q}}^{n}(M⟨i⟩, N) → Ext_{O_{λ}}^{n}(vM, vN)

are isomorphisms. For x ∈ W^{Q} let us define the objects L_{Q}^{x} = A_{Q}^{0}1_{x} and P_{Q}^{x} = A_{Q}1_{x} in O_{Q}. The L_{Q}^{x} are simple, and every simple object of O_{Q} is isomorphic to exactly one of the L_{Q}^{x}⟨i⟩. The P_{Q}^{x} are projective covers of the L_{Q}^{x}. Obviously P_{Q}^{x}, L_{Q}^{x} ∈ O_{Q} are lifts of P (x · λ), L(x · λ) ∈ O_{λ}, i.e. vP_{Q}^{x} ≅ P (x · λ), vL_{Q}^{x} ≅ L(x · λ). We also want to lift the Verma modules to O_{Q}. To this end remark that M(x · λ) is the projective cover of L(x · λ) in the truncated category

(O_{λ})_{≮x} = {M ∈ O_{λ} | y < x ⇒ [M : L(y · λ)] = 0}.

This means that under our equivalence O_{λ} ≅ A_{Q}-Mof the Verma M(x · λ) has to correspond to (A_{Q}/I_{<x})1_{x}, where I_{<x} ⊂ A_{Q} is the two-sided ideal generated by all 1_{y} with y < x in the Bruhat order. So let us just define M_{Q}^{x} = (A_{Q}/I_{<x})1_{x} ∈ O_{Q}. We also define a duality d : O_{Q} → O_{Q} on O_{Q} by the formula dM = Hom_{ℂ}(M, ℂ), using the isomorphism A_{Q} = A_{Q}^{opp}. Then d(M⟨i⟩) = (dM)⟨−i⟩ and dL_{Q}^{x} ≅ L_{Q}^{x}.

Analogously we define the category O^{Q} = A^{Q}-mof, the “forgetting of the grading” v : O^{Q} → O^{𝔮} and the lifts P_{x}^{Q}, M_{x}^{Q}, L_{x}^{Q} ∈ O^{Q} of P_{x}^{𝔮}, M_{x}^{𝔮}, L_{x}^{𝔮} ∈ O^{𝔮}. Remark that all of these are to a certain extent unique: Up to isomorphism there is a unique basic algebra A^{𝔮} describing the category O^{𝔮}. There are several possible Koszul gradings on A^{𝔮}, but by corollary 2.5.2 all of these give rise to a graded ring isomorphic to A^{Q}. And since by lemma 2.5.3 the lifts of indecomposable objects are unique up to shift of grading and isomorphism, L_{x}^{Q} can be characterized (up to isomorphism) as the unique lift of L_{x}^{𝔮} that is pure of weight zero, and then P_{x}^{Q}, M_{x}^{Q} are the unique lifts of P_{x}^{𝔮}, M_{x}^{𝔮} surjecting onto L_{x}^{Q}. Similar remarks apply to O_{Q}. With all these notations, we can state

Theorem 3.11.1. There exists a contravariant equivalence of triangulated categories over ℂ

Kd : D^{b}(O_{Q}) ⟶ D^{b}(O^{Q})

such that P_{Q}^{x}, M_{Q}^{x}, L_{Q}^{x} ∈ O_{Q} get mapped to L_{x}^{Q}, M_{x}^{Q}, P_{x}^{Q} ∈ O^{Q}, for all x ∈ W^{Q} and Kd(M[n]) = (Kd(M))[−n], Kd(M⟨n⟩) = (Kd(M))[n]⟨n⟩ for all M.

Proof. We may define Kd as the composition

D^{b}(O_{Q}) ⟶^{d} D^{b}(O_{Q}) = D^{b}(A_{Q}-mof) ⟶^{K} D^{b}(A_{Q}^{!}-mof) ≅ D^{b}(A^{Q}-mof) = D^{b}(O^{Q}).

Then the only assertion which doesn’t follow directly from theorem 2.12.6 and the definitions is the formula KdM_{Q}^{x} ≅ M_{x}^{Q}. But this is readily deduced from the following characterizations of M_{x}^{Q} and M_{Q}^{x}.

Lemma 3.11.2. M_{x}^{Q} is—up to isomorphism—the unique object M ∈ D^{b}(O^{Q}) such that

1. Hom(P_{y}^{Q}⟨i⟩, M[n]) ≠ 0 ⇒ y ≤ x.

2. Hom(P_{x}^{Q}⟨i⟩, M[n]) = ℂ if i = n = 0, zero else.

3. Hom(M⟨i⟩, L_{y}^{Q}[n]) ≠ 0 ⇒ y ≥ x.

4. Hom(M⟨i⟩, L_{x}^{Q}[n]) = ℂ if i = n = 0, zero else.

Lemma 3.11.3. M_{Q}^{x} is—up to isomorphism—the unique object M ∈ D^{b}(O_{Q}) such that

1. Hom(P_{Q}^{y}⟨i⟩, M[n]) ≠ 0 ⇒ y ≥ x.

2. Hom(P_{Q}^{x}⟨i⟩, M[n]) = ℂ if i = n = 0, zero else.

3. Hom(M⟨i⟩, L_{Q}^{y}[n]) ≠ 0 ⇒ y ≤ x.

4. Hom(M⟨i⟩, L_{Q}^{x}[n]) = ℂ if i = n = 0, zero else.

Proof. We will only prove the first lemma. Certainly M = M_{x}^{Q} satisfies the stated properties. On the other hand suppose M ∈ D^{b}(O^{Q}) also satisfies (1) through (4). We deduce further properties of M.

5. By (1) all cohomology groups H^{ν}(M) have only composition factors L_{y}^{Q}(i) with y ≤ x. By (2) only H^{0}(M) has a composition factor of the form L_{x}^{Q}(i).

6. ¿From (5) and (3) we deduce H^{ν}(M) = 0 for ν > 0. By (4) then H^{0}(M) has the unique simple quotient L_{x}^{Q}.

7. By (5) and (6) there is a surjection M_{x}^{Q} ↠ H^{0}(M). By (5) and (6) again this can be lifted to a nonzero map M_{x}^{Q} → M.

Now complete this map to a distinguished triangle M_{x}^{Q} → M → E →^{[1]} . We have to show E = 0. Let H^{ν}(E) be the highest nonvanishing cohomology group. Then there is a nonzero map E → L_{y}^{Q}⟨i⟩[ν] for suitable y, i. But y < x by (5) and construction, and y ≥ x by (3). This contradiction shows E = 0.

For any M ∈ O let M = rad^{0}M ⊃ rad^{1}M ⊃ … be the radical filtration and put \overline{ra}d^{i} M = rad^{i}M/rad^{i+1}M. Parts (i) and (ii) of the following theorem are to be seen as definition of the n_{Q}^{x,y}(i), n_{x,y}^{Q}(i).

Theorem 3.11.4. (i)
dim Ext_{O}^{i}(M(x · λ), L(y · λ)) = [\overline{ra}d^{i} M_{x}^{𝔮} : L_{y}^{𝔮}]
∥(1)  ∥(3)
dim Ext_{O_{Q}}^{i}(M_{Q}^{x}, L_{Q}^{y}⟨i⟩)  =^{(2)} [M_{x}^{Q} : L_{y}^{Q}⟨i⟩]
∥  ∥(4)
n_{Q}^{x,y}(i)  = (P_{y}^{Q} : M_{x}^{Q}⟨i⟩)

(ii)

dim Ext_{O^{𝔮}}^{i}(M_{x}^{𝔮}, L_{y}^{𝔮}) = [\overline{ra}d^{i} M(x · λ) : L(y · λ)]
∥(5)  ∥(7)
dim Ext_{O^{Q}}^{i}(M_{x}^{Q}, L_{y}^{Q}⟨i⟩) =^{(6)}  [M_{Q}^{x} : L_{Q}^{y}⟨i⟩]
∥  ∥(8)
n_{x,y}^{Q}(i)  =  (P_{Q}^{y} : M_{Q}^{x}⟨i⟩).

(iii) If we collect the n’s into W^{Q} ×W^{Q}-matrices P^{Q}(t), P_{Q}(t) with entries in ℤ[t] given by (P^{Q}(t))_{x,y} = ∑ n_{x,y}^{Q}(i)t^{i}, (P_{Q}(t))_{x,y} = ∑ n_{Q}^{x,y}(i)t^{i} then the matrix equation

(P_{Q}(t))^{T} P^{Q}(−t) = 1

holds (where the upper index T transposes the matrix). (iv) In terms of the Kazhdan-Lusztig polynomials P_{x,y} defined in [KL80a] we have

\begin{aligned}(P^{Q}(t))_{x,y} & = & P_{x,y}(t^{−2})t^{l(y)−l(x)} \\ (P_{Q}(t))_{x,y} & = & ∑_{z∈W_{Q}}(−1)^{l(z)}P_{xzw_{∘},yw_{∘}}(t^{−2})t^{l(x)−l(y)} \end{aligned}

for all x, y ∈ W^{Q}. Our polynomials are of degree l(y) − l(x) and l(x) − l(y) respectively with leading coefficient one.

Remark . The equalities (1) (resp. (5)) mean that extensions of Vermas by simples are “pure”, i.e. Ext_{O_{Q}}^{i}(M_{Q}^{x}, L_{Q}^{y}⟨n⟩) = 0 unless i = n (resp. …). This means that our graded Vermas are “Koszul modules” in the terminology of section 2.14.

Proof. (i) We start with (3). We deduce from proposition 2.4.1 that [\overline{ra}d^{i} M_{x}^{𝔮} : L_{y}^{𝔮}] = [M_{x}^{Q} : L_{y}^{Q}⟨i⟩]. Next we prove (1) and (2) in one argument. Namely

Ext_{O_{Q}}^{i}(M_{Q}^{x}, L_{Q}^{y}⟨n⟩) = Hom_{D^{b}(O_{Q})}(M_{Q}^{x}, L_{Q}^{y}⟨n⟩[i])
= Hom_{D^{b}(O^{Q})}(P_{y}^{Q}, M_{x}^{Q}[i − n]⟨−n⟩).

But clearly this vanishes if i ≠ n and in case i = n it has dimension [M_{x}^{Q} : L_{y}^{Q}⟨i⟩].

To make clear the structure of the following arguments let us develop some generalities. For any abelian category A let [A] be its Grothendieck group. Any A ∈ A gives [A] ∈ [A]. We often just write A instead of [A].

Now consider the Grothendieck group [O^{Q}] of O^{Q}. We let ℤ[t, t^{−1}] act on it via t[M] = [M⟨1⟩]. Then it becomes a free ℤ[t, t^{−1}]-module of rank |W^{Q}| with three bases [P_{x}^{Q}], [M_{x}^{Q}] and [L_{x}^{Q}]. Furthermore [O^{Q}] comes with a symmetric ℤ[t, t^{−1}]- bilinear form given by

⟨M, N⟩ = ∑_{i,ν}(−1)^{i} dim Ext_{O^{Q}}^{i}(M, dN⟨ν⟩)t^{ν}.

With respect to this form the [P_{x}^{Q}] and the [L_{x}^{Q}] are dual bases, whereas the [M_{x}^{Q}] form an orthonormal basis. It is then clear that

[M] = ∑_{z}⟨P_{z}^{Q}, M⟩[L_{z}^{Q}] = ∑_{z}⟨M, M_{z}^{Q}⟩[M_{z}^{Q}]

for any M ∈ O^{Q}. In other words, ⟨P_{z}^{Q}, M⟩ = ∑_{i}[M : L_{z}^{Q}⟨i⟩]t^{i} and ⟨M, M_{z}^{Q}⟩ = ∑_{i}(M : M_{z}^{Q}⟨i⟩)t^{i}, where the “virtual Verma flag multiplicities” (M : M_{z}^{Q}⟨i⟩) are just defined by [M] = ∑_{z,i}(M : M_{z}^{Q}⟨i⟩)[M_{z}^{Q}⟨i⟩] and coincide with the actual Verma flag multiplicities in case M has a Verma flag. Now (4) follows from

∑_{i}[M_{x}^{Q} : L_{y}^{Q}⟨i⟩]t^{i} = ⟨P_{y}^{Q}, M_{x}^{Q}⟩ = ∑_{i}(P_{y}^{Q} : M_{x}^{Q}⟨i⟩)t^{i}

and we established (i).

(ii) is proved the same way.

(iii) Remark that (P_{Q}(t))_{x,y} = ⟨P_{y}^{Q}, M_{x}^{Q}⟩ and (P^{Q}(−t))_{x,y} = ⟨M_{x}^{Q}, L_{y}^{Q}⟩. This reduces our statement to linear algebra.

(iv) This follows from the literature [Soe89a, Irv90, Irv88, CC87, Bar83, BB86, ES87]. The first equality is also a direct consequence of the results exposed in section 3.5. Namely we can compute

dim Ext_{O^{𝔮}}^{i}(M_{x}^{𝔮}, L_{y}^{𝔮}) = dim Hom_{D}^{i}(M_{x}^{Q}, L_{y}^{Q}) by section 3.5
= dim Hom_{D}^{i}(j_{x!}C_{x}, IC_{y}) in other notations
= dim Hom_{D}^{i}(C_{x}, j_{x}^{!}IC_{y}) by adjointness

and this certainly is a suitable coefficient of IC(G/Q, t)_{x,y} = (P^{Q}(t))_{x,y}.

We also want to show that our graded versions of projectives admit graded Verma flags. In the case of O_{Q} this follows directly from the following proposition, whose proof is stolen from [Jan87, II, 4.16]:

Proposition 3.11.5. Let M ∈ O_{Q} be such that Ext_{O_{Q}}^{1}(M, dM_{Q}^{x}⟨j⟩) = 0 for all x, j. Then M has a filtration with subquotients M_{Q}^{y}⟨j⟩.

Proof. Without restriction we may suppose M ≠ 0. Choose y ∈ W maximal such that Hom(M, L_{Q}^{y}⟨i⟩) ≠ 0 for some i. It follows that Hom(M, L_{Q}^{z}⟨j⟩) = 0 for all z > y, all j. ¿From the assumptions we know in particular that Ext^{1}(M, dM_{Q}^{z}⟨j⟩) = 0 for all z > y, all j. These two statements together give Ext^{1}(M, L_{Q}^{z}⟨j⟩) = 0 for all z > y, all j, and hence Ext^{1}(M, radM_{Q}^{y}⟨i⟩) = 0. Thus any nonzero morphism M → L_{Q}^{y}⟨i⟩ lifts to a surjection M → M_{Q}^{y}⟨i⟩. But the kernel of this surjection satisfies again the conditions of the proposition and we are done by induction.

The case of O^{Q} is similar. As another direct consequence of our results we obtain a new proof of the following result from [Irv88, Bar83, BB86].

Proposition 3.11.6. Verma modules are rigid, i.e. for any μ ∈ 𝔥^{∗} the socle and radical filtrations on M(μ) coincide.

Proof. We know that M(μ) has simple socle and simple top. By some equivalence of categories [Soe90] we may suppose μ is integral, say μ = x·λ. Write M(x·λ) = vM_{Q}^{x}. By proposition 2.4.1 both the socle and radical filtration on M(x · λ) correspond to the grading filtration on M_{Q}^{x}.

Remark . The same argument proves that projectives in O with a simple socle are rigid. This applies in particular to the projective covers of simple Verma modules. In fact, the argument proves that any object in O^{𝔮} or O_{λ} that has simple socle and top and admits a graded version is rigid.

Continuing in the spirit of the above results one might for two parabolics P ⊂ Q define graded versions of translation functors θ_{!}, θ_{∗} : O_{P} → O_{Q} and θ^{!}, θ^{∗} : O_{Q} → O_{P} . Their derived functors should correspond under Koszul duality to the graded derived versions of the embedding O^{𝔮} ↪ O^{𝔭} and its adjoints, the “parabolic truncation functors”. However we didn’t work this out.

4. Koszul rings arising from ‘mixed’ geometry.

One knows that various categories of representations may be interpreted geometrically as certain categories of perverse sheaves. For example the category O^{𝔮} (for regular integral weight) is equivalent to the category P_{B}(G/Q) of perverse sheaves on G/Q which are smooth along the Schubert stratification (see 3.5). In this section we show that the graded representation category O^{Q} may be identified with an appropriate full subcategory of the corresponding category P_{B}(G/Q)_{mixed} of mixed perverse sheaves. As a bonus, we also get a natural explanation of the Koszulity of O^{Q}.

The categories of (mixed) sheaves do not arise naturally as the categories of modules over some algebra. In the preliminary sections we develop some category language to overcome this nuisance.

4.1. Mixed categories and graded algebras. (see [BMS87, BGS88]). For simplicity we fix a ground field F and assume that all categories are F -categories and all functors are F -linear. By an “artinian category” we mean an abelian category with objects of finite length. For an artinian category A we denote by IrrA the set of isomorphism classes of irreducible objects of A.

Definition 4.1.1. A mixed category is an artinian category M equipped with a map w_{M} = w : IrrM → ℤ (called weight) such that for any two irreducible objects M, N one has Ext^{1}(M, N) = 0 if w(M) ≤ w(N).

Let (M, w) be a mixed category. An object M of M is called pure of weight i if all its irreducible components have same weight i. Any such object is semisimple.

Denote by M_{i} the full subcategory of pure objects of weight i. Clearly, we have M^{ss} = ⊕M_{i} is the full subcategory formed by all semisimple objects of M.

Lemma 4.1.2. Any object L ∈ M has a unique finite increasing filtration W_{•} = W_{•}L such that gr_{i}^{W} L = W_{i}L/W_{i−1}L is a pure object of weight i for all i ∈ ℤ. Any morphism in M is strictly compatible with W_{•}.

Proof. Left to the reader.

The filtration W_{•} is called weight filtration. Note that any full subcategory M′ ⊂ M closed under direct sums and subquotients is itself a mixed category (with respect to w_{M′} = w_{M′}|_{M}). Given any set of objects S in M we denote by M_{S} the smallest among such subcategories that contain S.

Definition 4.1.3. A functor π : M^{(1)} → M^{(2)} between mixed categories is called pure if π is exact and sends M_{i}^{(1)} to M_{i}^{(2)}, for each i ∈ ℤ.

For example the above embedding M_{S} ↪ M is pure, and any fully faithful pure embedding comes this way. The mixed categories form a 2-category with 1-morphisms being pure functors.

Examples . 1. The category H_{F} of polarizable F -Hodge structures is a mixed category (we assume F ⊂ ℂ and take the usual weight w). More generally, the category H(X) of mixed Hodge modules on an algebraic variety X over ℂ is a mixed category (see [Sai90]).

2. The category M(X) of mixed ℚ_{l}-sheaves on an algebraic variety X over a finite field is not mixed. However its full subcategory that consists of those perverse sheaves F that gr_{i}^{W} F is semisimple for all i ∈ ℤ is mixed. Unfortunately, one still does not know if this mixed category is stable with respect to various standard functors (e.g. if the action of the Frobenius on H^{•}(X, ℚ_{l}) is semisimple, the so-called “standard conjectures”).

Remark . The above examples explain the origin of the name mixed categories. The, yet conjectural, categories of mixed motivic F -sheaves on an algebraic variety should also be a mixed category if charF = 0. Beware that if charF ≠ 0 then these categories are not mixed categories in the sense of 4.1.1.

Definition 4.1.4. For an integer d a degree d Tate twist on a mixed category M is an automorphism ⟨d⟩ of M, written M ↦ M⟨d⟩, with the property w(M⟨d⟩) = w(M) + d for all M ∈ IrrM.

Then for any M one has W_{i}(M⟨d⟩) = (W_{i−d}M)⟨d⟩. For any n ∈ ℤ we denote by ⟨nd⟩ the n − th power of ⟨d⟩, so that ⟨−d⟩ is the inverse to ⟨d⟩ and ⟨nd⟩ ∘ ⟨md⟩ = ⟨(m + n)d⟩. If n ∈ ℤ then ⟨nd⟩ is a degree nd Tate twist. If M^{1}, M^{2} are mixed categories equipped with Tate twists ⟨d⟩^{i} of the same degree d, then for any pure functor ϕ : M^{1} → M^{2} the functor ⟨d⟩^{2} ∘ ϕ ∘ ⟨−d⟩^{1} is also pure. We’ll say that ϕ is compatible with Tate twists if we are given an isomorphism ϵ : ϕ ≅ ⟨d⟩^{2} ∘ ϕ ∘ ⟨−d⟩^{1}. Thus the mixed categories equipped with degree d Tate twists form 2-category with 1-morphisms being pairs (ϕ, ϵ) as above. The mixed categories occurring in the above examples are equipped with a standard Tate twist of degree −2. We write F⟨−2n⟩ = F(n).

Sometimes it is convenient to represent ⟨d⟩ as d-th power of a degree 1 Tate twist ⟨1⟩. One may do this in a universal manner as follows. Put M^{1/d} := ⊕_{0≤a≤d−1}M^{(a)} (the sum of d copies of M). Denote by ⟨1⟩ the automorphism of M^{1/d} that shifts

M^{(a)} ⊂ M^{1/d} to M^{(a+1)} for a < d − 1, and coincides with M^{(d−1)} = M →^{⟨d⟩}

M = M^{(0)} on M^{(d−1)}. Let us identify M with M^{(0)} ⊂ M^{1/d}. Then M^{1/d} is a mixed category with respect to the weight w defined as w(M⟨i⟩) = w(M) + i for M ∈ IrrM ⊂ IrrM^{1/d}, M ⊂ M^{1/d} is a pure fully faithful embedding, and ⟨d · 1⟩ coincides with ⟨d⟩ on M.

The most trivial example of a mixed category with degree 1 Tate twist is the category vect_{F} of graded finite dimensional vector spaces over a field F. Although this is not necessary for us from a strictly logical point of view, let us show in the remainder of this subsection that any mixed F -category with Tate twist and finite dimensional Hom’s is equivalent to the category of graded vector spaces equipped with certain extra symmetries.

Definition 4.1.5. A mixed algebra A is a projective limit of a family of graded finite dimensional algebras A_{α}^{•} such that A_{α}^{i} = 0 for i < 0 and that the A_{α}^{0} are semisimple F -algebras.

In other words, a mixed algebra A = ∏ A^{i} is a ring object in the abelian tensor category \underleftarrow{lim}(vect_{F}) of graded profinite dimensional vector spaces such that A^{i} = 0 for i < 0 and A^{0} is a projective limit of semisimple algebras. For a mixed algebra A we denote by A-mof the category of F -finite dimensional continuous A-modules, i.e. for A = \underleftarrow{lim}(A_{α}) one has A-mof = ⋃_{α} A_{α}-mof. If M ∈ A-mof is an irreducible module, then A^{>0}M = 0, i.e. M is actually an irreducible A^{0} = A/A^{>0}-module. It is supported in a single degree i, i.e. M^{j} = 0 for j ≠ i; set w(M) := −i. It is clear that (A-mof, w) is a mixed category. The shift of degrees functor M ↦ M⟨1⟩, (M⟨1⟩)^{i} = M^{i−1}, is a degree 1 Tate twist on A-mof. Note that the category \underrightarrow{lim}(A-mof)_{0} (which is the category of all continuous non-graded A^{0}-modules) has a distinguished object T defined by property Hom(T, M) = M for any M ∈ (A-mof)_{0}; clearly T contains any irreducible object with finite non-zero multiplicity.

Proposition 4.1.6. The correspondence A ↦ (A-mof, w, ⟨1⟩, T) is a 1-1 correspondence between mixed algebras and collections (M, w, ⟨1⟩, T) where (M, w) is a mixed category, ⟨1⟩ is a degree 1 Tate twist on M, and T is an object of \underrightarrow{lim}M_{0} such that T contains any irreducible object of M_{0} with finite non-zero multiplicity.

Proof. Let us construct the inverse correspondence. Take (M, w, ⟨1⟩, T) as above. To define the mixed algebra A consider the functor ϕ : M → vect_{F} , ϕ(P)^{i} := Hom(T, gr_{0}^{w}(P ⟨−i⟩)). This is a pure faithful functor compatible with Tate twist, i.e. one has ϕ(M⟨a⟩)^{i} = ϕ(M)^{i−a}. Consider the components ϕ^{i} : M → vect_{F}. The above compatibility shows that the space of morphisms of functors Hom(ϕ^{a}, ϕ^{a+i}) is independent of a ∈ ℤ; denote it by A^{i}. Note that A^{0} = EndT and A^{i} = 0 for i < 0. The composition of morphisms defines a product A^{i} × A^{j} → A^{i+j}, therefore A = ∏ A^{i} is a ring. It acts canonically on the graded vector space ϕ(P) for any P ∈ M. Clearly A coincides with projective limit of its images A_{P} in Endϕ(P). Therefore A is a mixed algebra, and our ϕ may be considered as a pure functor M → A-mof compatible with Tate twists. It is easy to check that this is an equivalence of categories.

Exercises . 1. Describe the pure functors between the mixed categories in terms of the corresponding mixed algebras.

2. Spell the analog of the above proposition for arbitrary mixed categories (without Tate twists).

4.2. Projective objects in artinian categories. Below we’ll use several times the following lemma of folklore. Its proof is left to the reader.

Let C be any artinian category. Denote by {L} a collection of representatives of isomorphism classes of irreducible objects in C, and put F_{L} = EndL. Let P ∈ C be an object equipped with a filtration P = P^{0}⊃P^{1}⊃ … ⊃ P^{N} = 0 for N ≥ 0, with semisimple successive quotients.

Lemma 4.2.1. The following properties are equivalent:

1. P is a projective object and the filtration is the radical filtration.

2. For any i ≥ 1 and any semisimple object M ∈ C the map

Hom(P^{i}/P^{i+1}, M) → Ext^{1}(P/P^{i}, M)

coming from the short exact sequence P^{i}/P^{i+1} ↪ P/P^{i+1} ↠ P/P^{i} is bijective.

3. For any i ≥ 1 and L ∈ IrrC the group E^{i}(L) = Ext^{1}(P/P^{i}, L) is a finitely generated F_{L}-module, E^{i}(L) = 0 for all but finitely many L, and there exists an isomorphism

P^{i}/P^{i+1} → ⨁_{L∈IrrC} E^{i}(L)^{∗} ⊗_{F_{L}} L

where E^{i}(L)^{∗} := Hom_{F_{L}}(E^{i}(L), F_{L}), such that under the chain of isomorphisms

Ext^{1}(P/P^{i}, P^{i}/P^{i+1}) → Ext^{1}(P/P^{i}, ⨁_{L} E^{i}(L)^{∗} ⊗_{F_{L}} L)
= ⨁_{L} E^{i}(L)^{∗} ⊗_{F_{L}} E^{i}(L)
= ⨁_{L} End_{F_{L}}(E^{i}(L))

the class of the extension P^{i}/P^{i+1} ↪ P/P^{i+1} ↠ P/P^{i} goes to ∑_{L} id_{E^{i}(L)}.

4.3. Gradings on artinian categories. Let C be an artinian category (hence an F -category by our conventions) with finite dimensional Hom’s, and let M be a mixed category with a degree d Tate twist. Let v : M → C be an exact faithful functor that sends semisimple objects of M to semisimple objects of C, and let ε : v(M) ↦ v(M⟨d⟩), M ∈ M, be a natural isomorphism. We’ll call such a pair (v, ε) a degrading functor.

For M, N ∈ M consider the map v_{M,N}^{i} : ⊕_{n}Ext_{M}^{i}(M, N⟨nd⟩) → Ext_{C}^{i}(vM, vN) defined by v and ε. We’ll be interested in the condition

(∗)_{M,N}^{i}: The map v_{M,N}^{i} is bijective.

Definition 4.3.1. 1. A grading on an artinian category C is a pair (M, (v, ε)), where M is a mixed category with a Tate twist and (v, ε) : M → C is a degrading functor, such that (a) any irreducible object in C is isomorphic to v(M) for some M ∈ M, and (b) the condition (∗)_{M,N}^{i} holds for all i and all M, N ∈ M.

2. An equivalence between two gradings (M, (v, ε)), and (M′, (v′, ε′)) on C is a pure functor ϕ : M → M′ compatible with Tate twists together with an isomorphism v′ϕ = v compatible with ε, ε′.

Remarks . 1. As follows from our definitions any equivalence ϕ as above is actually an equivalence between the categories M and M′.

2. If (∗)_{M,N}^{i} holds for all irreducible M, N and all i ∈ ℤ, then (∗)_{M,N}^{i} is true for all M, N, i.

3. Any degrading functor (v, ε) : M → C extends in a unique manner to a degrading functor (v^{1/d}, ε^{1/d}) : M^{1/d} → C, and (v, ε) is a grading on C if and only if such is (v^{1/d}, ε^{1/d}). Therefore we may always pass to the situation with d = 1.

Examples . 1. Let A be a mixed F -algebra, M = A-mof, C = A-Mof the category of all finite dimensional continuous non-graded A-modules. Then the forgetting of grading functor v : M → C (with the obvious ε) is a grading on C. Actually any grading arises in this way.

2. The functor v : H_{F} → Vect_{F} that assigns to a Hodge structure its underlying vector space is a degrading functor. To define ε we must fix a generator in v(F (1)). The restriction of v to the subcategory of those Hodge structures that are isomorphic to direct sums of Tate ones is a grading on Vect_{F}.

3. Consider the categories of mixed perverse sheaves in either Hodge or ℚ_{l}- version. The forgetting of mixed structure functor that sends a mixed sheaf to the usual underlying geometric one is a degrading functor.

Lemma 4.3.2. 1. Let (v, ε) : M → C be a grading on C. Then

(a) v sends irreducible objects in M to irreducible objects in C and the corresponding map IrrM/⟨ℤd⟩ → IrrC is bijective.

(b) The same holds when we replace the word “irreducible” by “indecompos-

able projective”.

2. Assume that an artinian category C has enough projective objects and for any L ∈ IrrC one has EndL = F. Then any degrading functor (v, ε) : M → C that satisfies the properties (a), (b) above is a grading on C.

Proof. (1) For (a) use (∗)_{M,N}^{0} for irreducible M and N. For (b) note that an object P in M is projective if and only if v(P) is projective in C (use (∗)_{P,N}^{1} with P our projective, N any irreducible, and (a)). An argument with (∗)_{P,P}^{0} shows that such P is indecomposable if and only if such is v(P). It remains to show that any indecomposable projective object P_{C} in C can be lifted to M. To do this consider the radical filtration P_{C} = P_{C}^{0} ⊃ P_{C}^{1} ⊃ …, and lift P_{C}/P_{C}^{i} step by step using 4.2.1 (3) and the above statement (a).

(2) We need to check (∗)_{M,N}^{i}. It obviously holds when M is projective. For arbitrary M use a projective resolution.

From now on let us assume that C admits a projective generator P, so in particular IrrC is a finite set. Put A = (EndP)^{opp}. This is a finite dimensional F -algebra, and we have a canonical equivalence of categories C → A-Mod, M ↦ Hom(P, M). Let θ be a positive grading A = ⊕A_{θ}^{i} on A such that A_{θ}^{0} is semisimple (i.e. A_{θ}^{>0} = radA) or, equivalently, that the graded ring A_{θ} is a mixed algebra. Set C̃_{θ} = A_{θ}-mof. This is a mixed category with a degree 1 Tate twist; the forgetting of the grading functor C̃_{θ} = A_{θ}-mof → A-Mof = C is a grading on C.

Lemma 4.3.3. Any grading on C with a degree 1 Tate twist is equivalent to some C̃_{θ}. The gradings C̃_{θ}, C̃_{θ′} are equivalent if and only if θ′ = a^{−1}θa for some a ∈ A^{×}.

Proof. Let M → C be a grading with Tate twist of degree 1. To identify M with some C̃_{θ} let us choose a lifting (P̃_{M}, α) of P (where P̃_{M} is an object of M and α : P → v(P̃_{M}) is an isomorphism), such that the maximal semisimple quotient T of P̃_{M} is pure of weight 0. Such a lifting exists and is uniquely defined up to a change α ↦ αa, a ∈ A^{×} = (AutP)^{opp}.

The ring Ã = (Endv(P̃_{M}))^{opp} is a mixed algebra with respect to its natural grading Ã^{i} = Hom(P̃_{M}, P̃_{M}⟨−i⟩). Note that Ã coincides with the mixed algebra attached to (M, T) by 4.1.6. The isomorphism α identifies A with Ã, hence defines the grading θ = θ(P̃_{M}, α) on A. It is clear from proposition 4.1.6 that the grading M on C is equivalent to C̃_{θ}. This is the first statement of the lemma. The second one follows since θ(P̃_{M}, αa) = a^{−1}θ(P̃_{M}, α)a for any a ∈ A^{×}.

Remark . In the situation the lemma it may happen that for a couple of gradings θ, θ′ there exists an automorphism ϕ of A such that ϕ(θ) = θ′. Such a ϕ defines an obvious pure equivalence C̃_{θ} ≅ C̃_{θ′} which is compatible with projections to C (i.e. ϕ is an equivalence of gradings) iff ϕ is an inner automorphism. Note that such ϕ always exists if we know that our graded algebras A_{θ}, A_{θ′} are generated by their components of degree 1 and 0: Then they are isomorphic to grA = ⊕(radA)^{i}/(radA)^{i+1}.

4.4. Mixed category O: ℚ_{l}-version. Let X be a variety over a finite field 𝔽_{q} equipped with a stratification by smooth affine strata {X_{α}}. Set n_{α} = dim X_{α} and denote by i_{α} : X_{α} ↪ X the embeddings. Let L_{α} = i_{α!∗}ℚ_{l,X_{α}}[n_{α}] be the intersection cohomology sheaf of X_{α} and M_{α} = i_{α!}ℚ_{l,X_{α}}[n_{α}] be the corresponding “standard” perverse sheaf. Note that L_{α} is a mixed sheaf in a natural way ( actually, a pure one of weight n_{α}). If we regard L_{α} as a mixed sheaf, we denote it by L̃_{α}.

Assume each stratum X_{α} is isomorphic to an affine space 𝔸_{𝔽_{q}}^{n_{α}}. We consider the following condition:

(∗): For any α, β and j ∈ ℤ the sheaf H^{j}i_{β}^{∗}L̃_{α} on X_{β} vanishes if n_{α} + j is odd and is isomorphic to a direct sum of copies of ℚ_{l,X_{β}}((−n_{α} − j)/2) if n_{α} + j is even.

To check this condition one may use:

Lemma 4.4.1. Condition (∗) is satisfied provided the following two properties hold.

1. For each β, H^{j}i_{β}^{∗}L̃_{α} is a (geometrically) constant sheaf on X_{β}; this sheaf vanishes whenever n_{α} + j is odd.

2. The geometric Frobenius acts on H^{j}(X, L̃_{α}) via multiplication by q^{(j+n_{α})/2}.

Proof. There is a standard spectral sequence with E_{2}^{p,q} = ⨁_{n_{β}=−p} H_{c}^{p+q}(X_{β}, i_{β}^{∗}L̃_{α}) that converges to H_{c}^{p+q}(X, L̃_{α}) (H_{c}^{•} stands here for the cohomology with compact support). Property (1) yields the parity condition: E_{2}^{p,q} = 0 whenever n_{α} + p + q is odd. Hence all the differentials d_{r}, r ≥ 2 vanish, and the spectral sequence degenerates at the E_{2}-term. Thus, the group H_{c}^{j}(X_{β}, i_{β}^{∗}L̃_{α}) gets identified in a canonical way with a subquotient of

H_{c}^{j}(X, L̃_{α}) ≃ H^{j}(X, L̃_{α})^{∗}(−n_{α})

The property (2) ensures now that Frobenius acts via multiplication by q^{(j+n_{α})/2} on the groups involved. Note finally, that the first property yields an isomorphism: H_{c}^{j}(X_{β}, i_{β}^{∗}L̃_{α}) = H^{0}(X_{β}, H^{j−2n_{β}}i_{β}^{∗}L̃_{α})(−n_{β}), and condition (∗) follows.

To apply the lemma in the situation we need to prove the following result.

Lemma 4.4.2. Assume that for any α there exists a proper surjective morphism π_{α} : Y_{α} → \overline{X_{α}} such that

1. Y_{α} is a smooth scheme.

2. For any X_{β} ⊂ \overline{X_{α}} the projection π_{α}^{−1}(X_{β}) → X_{β} is smooth.

3. The ℚ_{l}-cohomology groups of Y_{α} are generated by classes of algebraic cycles

defined over 𝔽_{q}. Then the property (2) of lemma 4.4.1 holds.

Proof. Form the direct image complex F = π_{α ∗}ℚ_{l,Y_{α}}[n_{α}] ∈ D_{mixed}^{b}(X, ℚ_{l}). Clearly, L̃_{α} is a subquotient of the mixed perverse sheaf ^{p}H^{0}F. On the other hand, the complex F is a semisimple perverse sheaf (with Frobenius action disregarded), due to the decomposition theorem. Hence the vector space H^{j}(X, L̃_{α}) with the Frobenius action on it is isomorphic to a subquotient of H^{j}(X, F) = H^{j+n_{α}}(Y_{α}, ℚ_{l}). The claim follows.

Corollary 4.4.3. Condition (∗) holds for the stratification of the flag variety G/B by Schubert cells.

Proof. Let U be the maximal unipotent subgroup of G so that U-orbits are the Schubert cells. Then, clearly, L̃_{α} is an U-equivariant perverse sheaf. Hence, for all β and j ∈ ℤ, the sheaf H^{j}i_{β}^{∗}L̃_{α} is U-equivariant, hence, a geometrically constant sheaf. Now, the first property of lemma 4.4.1 is proved by Kazhdan-Lusztig [KL80b]. The second property follows from lemma 4.4.2 applied to the Demazure-Hansen desingularization Y_{α} → \overline{X_{α}} of Schubert varieties.

From now on assume that (X, {X_{α}}) satisfies condition (∗) above. Let P be the Serre subcategory of the category of ℚ_{l}-perverse sheaves on X ⊗ \overline{𝔽_{q}} generated by the L_{α}, and denote by P̃ the full subcategory of the category of mixed perverse sheaves on X that consists of those F̃ for which any gr_{j}^{W} F̃ is isomorphic to a direct sum of copies of L̃_{α}((n_{α}−j)/2) with n_{α}−j even. Then P is an artinian ℚ_{l}-category and P̃ is a mixed ℚ_{l}-category with a degree −2 Tate twist. One has IrrP = {L_{α}}, IrrP̃ = {L̃_{α}(n)}. We have the forgetting of mixed structure functor v : P̃ → P. Choose a generator ε of ℚ_{l}(1). Then (v, ε) is a degrading functor. The following theorem is the main result of this paragraph. Its proof will occupy the rest of it.

Theorem 4.4.4. 1. (v, ε) : P̃ → P is grading on P.

2. P̃ is a Koszul category.

Let us start the proof with

Lemma 4.4.5. For a constructible complex G ∈ D(X ⊗ \overline{𝔽_{q}}, ℚ_{l}) the following conditions are equivalent:

1. ^{p}H^{j}(G) ∈ P for all j ∈ ℤ.

2. The H^{j}i_{β}^{∗}(G) are constant sheaves on X_{β} for all β and j ∈ ℤ.

Proof. The sheaves that satisfy (1) or (2) form the full triangulated subcategories D_{1}, D_{2} of D(X ⊗ \overline{𝔽_{q}}, ℚ_{l}). The category D_{1} is generated by the \{L_{α}\}, the category D_{2} by the {M_{α}}. By (∗) we have L_{α} ∈ D_{2} and therefore D_{1} ⊂ D_{2}. To prove the converse inclusion it suffices to show that all M_{α} lie in P. We’ll do it using induction on the number of strata. Let X_{ω} be an open stratum. By induction we know that any perverse sheaf G supported on X−X_{ω} that satisfies (2) automatically lies in P, in particular this holds for M_{α} whenever α ≠ ω. Consider the canonical surjection M_{ω} → L_{ω}. Let K_{ω} be its kernel. Then K_{ω} is supported on X − X_{ω} and H^{j}i_{β}^{∗}K_{ω} = H^{j−1}i_{β}^{∗}L_{ω} for any β ≠ ω. From (∗) we deduce K_{ω} ∈ P, hence M_{ω} ∈ P.

Remark . Actually the above arguments prove the following general statement. Let X be any variety stratified by smooth strata X_{α}, put n_{α} = dim X_{α}, and assume that for each X_{α} we picked a Serre subcategory C_{α} of the category of lisse sheaves on X_{α}. Assume that for any α, β and F ∈ IrrC_{α} the sheaves H^{j}i_{β}^{∗}i_{α!∗}F, j ∈ ℤ, lie in C_{β}. Then the full triangulated subcategory of D(X, ℚ_{l}) generated by the perverse sheaves i_{α!∗}(F), F ∈ IrrC_{α}, consists precisely of those complexes G which satisfy the condition H^{j}(i_{β}^{∗}G) ∈ C_{β} for all β and j ∈ ℤ.

Lemma 4.4.6. 1. The category P has enough projectives.

2. The obvious functor D^{b}(P) → D(X ⊗ \overline{𝔽_{q}}, ℚ_{l}) is a fully faithful embedding.

Proof. The parallel statement in the situation when X is a complex variety (with condition (∗) dropped) was proven in 3.3. The reader may easily check that the same proof works in our present situation, provided we know that the perverse sheaves M_{α} lies in P. And this was the preceding lemma.

If F, G are perverse sheaves on X ⊗ \overline{𝔽_{q}} we set

Ext^{i}(F, G) := Hom_{D(X⊗\overline{𝔽_{q}},ℚ_{l})}(F, G[i]).

This group also coincides with the Yoneda Ext in the category of all perverse sheaves on X. The above lemma claims, in particular, that for any F, G ∈ P the obvious map Ext_{P}^{i}(F, G) → Ext^{i}(F, G) is an isomorphism. For a pair of irreducibles L_{α}, L_{β} consider the ℚ_{l}-vector space Ext^{1}(L_{α}, L_{β}). The mixed structure L̃_{α}, L̃_{β} defines a Frobenius action on it.

Lemma 4.4.7. If n_{α} − n_{β} is even then Ext^{1}(L_{α}, L_{β}) = 0. If n_{α} − n_{β} is odd, then the Frobenius acts on Ext^{1}(L_{α}, L_{β}) as multiplication by q^{(n_{β}−n_{α}+1)/2}.

Proof. We have a canonical isomorphism

Ext^{1}(L_{α}, L_{β}) = Ext^{1}(DL_{β}, DL_{α}) = Ext^{1}(L_{β}, L_{α})(n_{α} − n_{β}),

the latter since DL_{α} = L(n_{α}). Therefore we may assume that \overline{X_{α}} − X_{α} ⊅ X_{β} (otherwise interchange α and β by the above formula). The short exact sequence K_{α} ↪ M_{α} ↠ L_{α} provides the exact sequence Hom(K_{α}, L_{β}) → Ext^{1}(L_{α}, L_{β}) → Ext^{1}(M_{α}, L_{β}). Since our condition implies that Hom(K_{α}, L_{β}) = 0, we see that Ext^{1}(L_{α}, L_{β}) ↪ Ext^{1}(M_{α}, L_{β}) = H^{1}(X_{α} ⊗\overline{𝔽_{q}}, i_{α}^{∗}(L_{β})[−n_{α}]), so it suffices to check the statement of our lemma for this larger group. Since the X_{α} are supposed to be just affine linear spaces this coincides with H^{0}(X_{α} ⊗ \overline{𝔽_{q}}, H^{1−n_{α}}i_{α}^{∗}L_{β}), and by (∗) we are done.

Lemma 4.4.8. Let P be an indecomposable projective in P. Then

1. P can be lifted to P̃, i.e. there exists P̃ ∈ P̃ together with an isomorphism v(P̃) → P.

2. Any such P̃ is a projective object of P̃.

Proof. We’ll use the following general fact. Let F̃_{1}, F̃_{2} be mixed perverse sheaves. Then one has a canonical short exact sequence

Hom(F_{1}, F_{2})_{F r} ↪ Ext^{1}(F̃_{1}, F̃_{2}) ↠ Ext^{1}(F_{1}, F_{2})^{F r}

where F_{i} = v(F̃_{i}) are the underlying usual perverse sheaves, and the middle Ext^{1} is the group of extensions in the category of all mixed sheaves (not nexessary gr_{•}^{W} - semisimple ones).

We now prove (1). Let P = P^{0} ⊃ P^{1} ⊃ … be the radical filtration on P. We’ll find a lifting P̃ such that P^{i} = v(W_{w−i}(P̃)) for some w ∈ ℤ. Namely, we’ll define inductively liftings { \atop P/P^{i}} of P/P^{i} such that { \atop P/P^{i+1}} is an extension of { \atop P/P^{i}} by a pure sheaf { \atop P^{i}/P^{i+1}} of weight w − i. To start the induction remark that P/P^{1} is irreducible, hence can be lifted to some { \atop P/P^{1}}. Set w = w({ \atop P/P^{1}}). Now assume we already have { \atop P/P^{i}}. Consider the isomorphism

P^{i}/P^{i+1} → ⨁_{α}Ext^{1}(P/P^{i}, L_{α})^{∗}⊗_{ℚ_{l}}L_{α}

from 4.2.1 (3). The mixed structure { \atop P/P^{i}} and the standard mixed structure L̃_{α} on L_{α} define a Frobenius action on Ext^{1}(P/P^{i}, L_{α}). Since our filtration is the radical filtration, the restriction map Ext^{1}(P/P^{i}, L_{α}) → Ext^{1}(P^{i−1}/P^{i}, L_{α}) is injective. By our induction hypothesis { \atop P^{i−1}/P^{i}} = W_{w−i+1}({ \atop P/P^{i}}) is a pure (hence semisimple) object of P̃ of weight w − i + 1. By 4.4.7 the Frobenius acts on Ext^{1}(P^{i−1}/P^{i}, L_{α}) as multiplication by q^{(n_{α}−w+i)/2} (and it vanishes if n_{α} − w + i is an odd integer). Therefore the same is true for Ext^{1}(P/P^{i}, L_{α}) and we may consider ⨁_{α}Ext^{1}(P/P^{i}, L_{α})^{∗} ⊗ L_{α} as a pure object of P̃ of weight w − i.

This defines the pure structure { \atop P^{i}/P^{i+1}} on P^{i}/P^{i+1}. The class of the extension P^{i}/P^{i+1} ↪ P/P^{i+1} ↠ P/P^{i} is invariant with respect to the Frobenius-action on Ext^{1}(P/P^{i}, P^{i}/P^{i+1}) defined by the mixed structures { \atop P/P^{i}} and { \atop P^{i}/P^{i+1}} (see 4.2.1 (3)), hence it comes from an extension { \atop P^{i}/P^{i+1}} ↪ { \atop P/P^{i+1}} ↠ { \atop P/P^{i}}. Clearly { \atop P/P^{i+1}} ∈ P̃ and we are done.

Next we prove (2). We need to show that Ext1 ; P̃(P̃, L̃) = 0 for any L̃ ∈ IrrP̃. Note that Ext1 ; P̃(P̃, L̃) ⊂ Ext^{1}(P̃, L̃). Since Ext^{1}(P, L) = 0 (by 4.4.6 (2)) we see that Ext1 ; P̃(P̃, L̃) ⊂ Hom(P, L)_{F r} = Hom(P/P^{1}, L)_{F r}. This group is non-zero if

and only if { \atop P/P^{1}} ≅ L̃. In this case Ext1 ; P̃(P̃, L̃) = 0 (the corresponding extensions are not gr^{W} -semisimple), and we are done.

Proof of 4.4.4. (1) We may use the criterium 4.3.2 (2) by 4.4.6 (1). The condition 4.3.2 (a) is obvious, while 4.3.2 (b) follows from 4.4.8.

(2) We have to show that for any L̃_{1}, L̃_{2} ∈ IrrP̃ of weights w_{1}, w_{2} the extension Exti ; P̃(L̃_{1}, L̃_{2}) vanishes unless i = w_{1} − w_{2}. For this it is sufficient, by 3.4.2, to establish an injection Exti ; P̃(L̃_{1}, L̃_{2}) ↪ Ext^{i}(L_{1}, L_{2})^{F r}. This we deduce from the following commutative diagram:

Exti ; P̃(L̃_{1}, L̃_{2})  ↪  Ext_{P}^{i}(L_{1}, L_{2})
↓  ∥
Ext^{i}(L̃_{1}, L̃_{2}) ↠ Ext^{i}(L_{1}, L_{2})^{F r} ↪ Ext^{i}(L_{1}, L_{2}).

Here the upper horizontal is an injection by 4.4.4 (1), the right vertical is an isomorphism by 4.4.6(2), the left vertical comes from the fact that the upper left corner is a universal ∂-functor, and left map in the lower horizontal is the end of the short exact sequence from the proof of lemma 4.4.8.

4.5. Mixed category O : Hodge version. Let X be an algebraic variety over ℂ. Fix a subfield F ⊂ ℝ. Denote by P the category of perverse F -sheaves on X and by P_{H} the category of Hodge F -modules (i. e. mixed Hodge sheaves) on X. We have the forgetting of Hodge structure functor v : P_{H} → P. If we choose a generator ε of the 1-dimensional vector space F (1) = F · 2π\sqrt{−1}, then (v, ε) is a degrading functor (see 4.3).

Let C ⊂ P be a Serre subcategory. We consider the following problem: Find a mixed subcategory C̃ ⊂ P_{H} closed with respect to Tate twist such that C̃ → C is a grading on C. This problem (for given C) seems to be quite difficult. For example, we have no idea if C exists in the generality of 4.4 or 3.4 (i.e. in case when C consists of sheaves smooth along a stratification by affine planes). However in the very particular case when X = G/B and C coincides with category O of perverse sheaves lisse along the Schubert stratification {X_{α}} the category C̃ does exist. We’ll construct it below.

Let X = G/B, {X_{α}}, O be as above and let O_{H} := v^{−1}O be the category of Hodge modules lisse along {X_{α}}. Note that each L_{α} carries a canonical Hodge structure (of weight n_{α}). Denote this irreducible object of O_{H} by L̃_{α}.

Lemma 4.5.1. Any Hodge structure on L_{α} is isomorphic to the twist L̃_{α}(j) for some j ∈ ℤ.

Proof. Let L̃_{α}^{1} be a Hodge structure on L_{α}. Let us consider the canonical isomorphism Hom(L_{α}, L_{α}) ⊗ L_{α} ≅ L_{α} (note that Hom(L_{α}, L_{α}) = F). If we put on F = Hom(L_{α}, L_{α}) the Hodge structure F̃ defined by L̃_{α} and L̃_{α}^{1}, then this will give the isomorphism of Hodge sheaves F̃ ⊗ L̃_{α} ≅ L̃_{α}^{1}. Since dim F = 1 the only possible F̃ is F (j).

Recall some general facts about Hodge sheaves (see [Sai90]). If F̃ is a Hodge sheaf on X and F is the underlying perverse sheaf, then the groups H^{j}(X, F) carry a natural Hodge structure. We denote it by H^{j}(X, F̃) ∈ H_{F}. More generally, for F̃_{1}, F̃_{2} the groups Ext^{j}(F_{1}, F_{2}) carry a natural Hodge structure. We denote it by Ext^{j}(F̃_{1}, F̃_{2}) ∈ H_{F}. We have a canonical spectral sequence converging to Ext_{H}^{•}(F̃_{1}, F̃_{2})(:= Ext groups in the category of Hodge sheaves) with first term E_{1}^{p,q} = H_{H}^{2p+q}(Ext^{−p}(F̃_{1}, F̃_{2})). Here H_{H}^{j} is the absolute Hodge cohomology functor: For a Hodge structure M ∈ H_{F} one has H_{H}^{j}(M) := Ext_{H_{F}}^{j} (F (0), M). Since H_{H}^{j} = 0 for j ≠ 0, 1 this spectral sequence degenerates at E_{2}. In particular we have a short exact sequence

H_{H}^{1}Hom(F̃_{1}, F̃_{2}) ↪ Ext_{H}^{1}(F̃_{1}, F̃_{2}) ↠ H_{H}^{0}Ext^{1}(F̃_{1}, F̃_{2}).

Lemma 4.5.2. 1. For any α, β and j ∈ ℤ the sheaf H^{j}i_{β}^{∗}L̃_{α} on X_{β} vanishes if j+n_{α} is odd. It is isomorphic to a direct sum of some copies of F ((−j−n_{α})/2) if j + n_{α} is even.

2. The cohomology group H^{j}(X, L̃_{α}) vanishes if j + n_{α} is odd. It is isomorphic to a direct sum of some copies of F ((−j − n_{α})/2) if j + n_{α} is even.

3. The group Ext^{j}(L̃_{α}, L̃_{β}) vanishes if j − n_{α} + n_{β} is odd. It is isomorphic to a direct sum of some copies of F ((−j − n_{α} + n_{β})/2) if j − n_{α} + n_{β} is even.

Proof. (1) (cf 4.4.2.) Use the Demazure-Hansen desingularization Y_{α} of \overline{X_{α}} and the decomposition theorem.

(2) follows from (1), since the terms of the spectral sequence for computing H^{j}(X, L̃_{α}) via the stratification {X_{α}} have the properties listed in (2) and any extension between Tate (or any Hodge) structures of same weight is trivial. Alternatively use directly the demcomposition theorem to identify H^{•}(X, L_{α}) with pieces of the cohomology of Y_{α}.

(3) To compute Ext^{•}(L̃_{α}, L̃_{β}) = H^{•}(X, D(L̃_{α}⊗DL̃_{β})) = H^{•}(X, L̃_{α}⊗L̃_{β})^{∗}(−n_{β}) one may use, as in (2), the spectral sequence for the stratification {X_{α}}. Again, by (1), its terms satisfy the properties listed in (3), hence this is true for the Ext^{•}-groups themselves. Alternative: By 3.4.2 the canonical map Ext^{j}(L_{α}, L_{β}) → Hom(H^{•}(X, L_{α}), H^{•+j}(X, L_{β})) is injective. Since it is obviously compatible with Hodge structures, we are done by (2).

Lemma 4.5.3. Let P ↠ L_{α} be a projective cover in O and P = P^{0} ⊃ P^{1} ⊃ … the radical filtration of P.

1. There exists a Hodge structure P̃ on P such that P̃ ↠ L̃_{α} is a morphism of Hodge sheaves.

2. For any such P̃ one has P^{i} = v(W_{n_{α}−i}P̃), and { \atop P^{i}/P^{i+1}} := gr_{n_{α}−i}^{W}(P̃) is a direct sum of some objects of the form L̃_{β}(j) (of weight n_{α} − i).

Proof. Let us start with (2). By induction on i we may assume that P^{j} is a Hodge subsheaf of P̃ and P^{j−1}/P^{j} is a direct sum of Tate twists of some L̃_{β} of weight n_{α} − j + 1 for any j ≤ i. We need to show that P^{i+1} is a Hodge subsheaf and P^{i}/P^{i+1} is a sum of some objects of the form L̃_{β}(j) of weight n_{α} −i. First consider the canonical morphism γ : P̃ → ⊕_{β}Hom(P̃^{i}, L̃_{β})^{∗} ⊗ L̃_{β} coming from the obvious maps Hom(P̃^{i}, L̃_{β}) ⊗ P̃^{i} → L̃_{β}. Its kernel is P^{i+1}, hence this is a Hodge subsheaf of P̃^{i}. The canonical isomorhism 4.2.1(3)

P̃^{i}/P̃^{i+1} → ⊕_{β}Ext^{1}(P̃/P̃^{i}, L̃_{β})^{∗} ⊗_{F} L̃_{β}

is obviously compatible with Hodge structures. Since our filtration is the radical one, the restriction map Ext^{1}(P/P^{i}, L_{β}) → Ext^{1}(P^{i−1}/P^{i}, L_{β}) is injective. Since, by induction, P̃^{i−1}/P̃^{i} is a direct sum of Tate twists of L̃_{γ}’s of weight n_{α} − i + 1, we are done by part (3) of the preceding lemma.

To prove existence of P̃ we use again induction over i. Namely we build successively the { \atop P/P_{i}}. The procedure is identical with the one used in the proof of 4.4.8 (1). We should only replace the words “Frobenius action” by “Hodge structure” and use the short exact sequence preceding lemma 4.5.2 instead of the one from the beginning of the proof of 4.4.8.

Now consider the skyscraper sheaf L_{e} supported at the point orbit. Let P̃_{e} → L̃_{e} be a morphism of Hodge sheaves such that P_{e} → L_{e} is a projective covering of L_{e}. Define ÕP̃_{e} to be the smallest full abelian subcategory of O_{H} containing all P̃_{e}(j). It consists of all subquotients of direct sums of objects of the form P̃_{e}(j).

Theorem 4.5.4. 1. ÕP̃_{e} → O is a grading on O.

2. ÕP̃_{e} is a Koszul category.

Proof. First note that (2) follows from (1) . Namely by (1) and 4.5.1 we know that IrrÕP̃_{e} = {L̃_{α}(n)} where w(L̃_{α}(n)) = n_{α} − 2n, therefore Koszulity means that Exti ; ÕP̃e (L̃_{α}(a), L̃_{β}(b)) vanishes if i ≠ n_{α} − n_{β} − 2a + 2b. By (1) the map-

ping Exti ; ÕP̃e ((L̃_{α}(a), L̃_{β}(b)) → Ext^{i}(L_{α}, L_{β}) is injective. Since it factors through

Ext_{H}^{i}(L̃_{α}(a), L̃_{β}(b)) its image lies in H_{H}^{0}(Ext^{i}(L̃_{α}(a), L̃_{β}(b)), and we are done by 4.5.2 (3).

To prove (1) we will use the following lemma. Let C̃^{•} be the maximal W - invariant quotient algebra of SymH_{1}(H, F). Here H is a Cartan subgroup of G and we consider H_{1}(H, F) as a Hodge structure. Clearly H_{1}(H, F) = Γ_{H} ⊗ F (1), where Γ_{H} is the lattice of 1-parameter subgroups in H. Therefore C̃^{•} is a commutative algebra in H_{F} graded in an obvious manner. Consider the corresponding graded F -algebra C^{•} = v(C̃^{•}). According to [Soe90] we have a canonical isomorphism C^{•} ≅ EndP_{e}.

Lemma 4.5.5. This isomorphism is compatible with Hodge structures, i.e. C̃ ≅ EndP̃_{e}.

Proof. Proof postponed to the next subsection.

We’ll say that F̃ ∈ ÕP̃_{e} is a special object if it is isomorphic to a quotient of a direct sum of copies of P̃_{e}(n)’s.

Step 1. Let F̃ and G̃ be objects of ÕP̃_{e} and suppose F̃ is special. Then Hom(F̃, G̃) is a direct sum of Tate structures F (i). This follows immediately from projectivity of P_{e} and the preceding lemma, since our Hom(F̃, G̃) is a subquotient of a sum of Hodge structures isomorphic to (EndP̃_{e})(i).

Step 2. For any α there exists a special object Ĩ_{α} such that I_{α} := v(Ĩ_{α}) is an injective envelope (:= an indecomposable injective object) of L_{α}. Indeed, according to [Soe90] there exists a graded C^{•}-module V_{α}^{•} such that I_{α} := V_{α} ⨂_{C} P_{e} is an injective envelope of L_{α} (here V_{α} is V_{α}^{•} with its grading forgotten). Now consider the Hodge structure Ṽ_{α} := ⊕V_{α}^{i}(i), and set Ĩ_{α} := Ṽ_{α} ⊗C̃ P̃_{e}. Note that dim Hom(L_{α}, I_{α}) = 1, therefore we may assume, as in the proof of 4.5.1, after possible twisting of I_{α}, that L̃_{α} ⊂ Ĩ_{α}.

Step 3. The category ÕP̃_{e} consists precisely of those sheaves F̃ ∈ O_{H} that Hom(F̃, Ĩ_{α}) is a direct sum of Tate structures for any α. Indeed, if F̃ is a special object of ÕP̃_{e}, then we may embed it into a special one and use injectivity of I_{α} to get our claim. Conversely, assume that F̃ has the property that all Hom(F̃, Ĩ_{α}) are direct sums of Tate structures. The canonical embedding F̃ ↪ ⊕_{α}Hom(F̃, Ĩ_{α})^{∗}⊗Ĩ_{α} is compatible with Hodge structures, and the right hand side obviously lies in ÕP̃_{e}, so we are done.

Step 4. As follows from Step 3, the Ĩ_{α}(n) are injective objects of ÕP̃_{e}. Since IrrÕP̃_{e} = {L̃_{α}(n)} by 4.5.3 (2), our theorem follows from 4.3.2 (2). (Strictly speaking, we need to apply it to C = O^{opp} since in 4.3.2 (2) we spoke about projective objects, and here we deal with injective ones.)

4.6. Monodromy actions. We still have to prove lemma 4.5.5. We need some general facts about monodromic sheaves. Let H be our torus, π : X → Y an H-principal bundle, F ∈ P (X) a monodromic mixed Hodge module.

Proposition 4.6.1. The logarithm of the unipotent part of monodromy 𝔥(1) → EndF is compatible with Hodge structures.

Proof. Recall that a perverse sheaf F ∈ P(X) is called monodromic if h^{∗}F is isomorphic to F for all h ∈ H. Consider the projection and the multiplication p, m : H × X → X and the other pojection q : H × X → H. Consider

A = A_{F} = H^{−n}q_{∗}RHom(p^{∗}F, m^{!}F)[−n].

If F is monodromic, this is a local system on H. The fibre A_{1} of this local system at one is canonically A_{1} = EndF. Put Γ = X_{∗}(H) ⊂ 𝔥. We regard this as an integral Hodge structure and consider also its Tate twist Γ(1) = 2πiΓ ⊂ 𝔥(1). Then Γ(1) = π_{1}(H, 1) acts on A_{1} = EndF by monodromy, and the action of γ ∈ Γ(1) is in fact left multiplication with some M(γ) ∈ AutF. Now let N(γ) ∈ EndF be the logarithm of the unipotent part of M(γ). Then N : Γ(1) → EndF extends to a homomorphism of Lie algebras N_{F} : 𝔥(1) → EndF and we claim that in case F has a mixed structure this is compatible with the mixed structures.

But it is known that if A is a smooth mixed Hodge sheaf on a torus, then the logarithm of monodromy N_{A} : 𝔥(1) → EndA_{1} is compatible with Hodge structures. And in case A = A_{F} (so that A_{1} = EndF) this factorizes as 𝔥(1) → EndF → End(EndF) where the first map is N_{F} and the second left multiplication.

Proof of Lemma 4.5.5. Let U ⊂ B be the unipotent radical. Then we may identify

U-equivant perverse sheaves on G/B ∥ U × B-equivariant perverse sheaves on G ∥ B-equivariant perverse sheaves on U/G.

Certainly the latter are H-monodromic. Thus monodromy gives a homomorphism S(𝔥(1)) → EndF for all F ∈ O, and for F ∈ Õ this morphism is even compatible with Hodge structures. Now using [BB86] one may see that under localization O^{𝔟} ≅ P_{B}(G/B) the monodromy action of S(𝔥(1)) on the right corresponds to the action of the center of U on the left, via the Harish-Chandra homomorphism. In particular for P̃_{e} ∈ Õ a lift of the antidominant projective by translating [Soe90, Bea83] into geometry we get a surjection S(𝔥(1)) ↠ EndP̃_{e}. Thus indeed EndP̃_{e} is a split Tate structure.

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Massachusetts Institute of Technology, Department of Mathematics, Cambridge MA 02139, USA

E-mail address: sasha@math.mit.edu

The University of Chicago, Department of Mathematics, Chicago IL 60637, USA

E-mail address: ginzburg@zaphod.uchicago.edu

Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, D-53 Bonn 3, Germany

Current address: Universität Freiburg, Mathematisches Institut, Albertstraße 23b, D-79104 Freiburg, Germany

E-mail address: soergel@sun1.mathematik.uni-freiburg.de