Separably closed fields with Hasse derivations

Martin Ziegler

September 11, 2002

Abstract

In [6] Messmer and Wood proved quantifier elimination for separably closed fields of finite Ershov invariant e equipped with a (certain) Hasse derivation. We propose a variant of their theory, using a sequence of e commuting Hasse derivations. In contrast to [6] our Hasse derivations are iterative.

1 Introduction

Definition. Let R be a commutative ring. A Hasse derivation is a family D = (D_{0}, D_{1}, …) of additive maps D_{n} : R → R such that^{1}

D_{0}(x) = x  (1.1)
D_{n}(xy) = ∑_{a+b=n} D_{a}(x)D_{b}(y)  (1.2)

D_{a}D_{b} = \binom{a + b}{a}D_{a+b} .  (1.3)

Two Hasse derivations D and E commute if D_{m}E_{n} = E_{n}D_{m} for all m, n.

We fix for the rest of the paper a natural number e and a prime p.

The following notion was introduced by Okugawa in [7]: A D–field is a pair (K, D), where K is a field of characteristic p and D = (D_{1}, …, D_{e}) is a sequence of commuting Hasse derivations on K. The field of constants^{2} C consists of those elements of K on which all derivations D_{i,1} (i = 1, …, e) vanish. Clearly C contains K^{p}. (K, D) is a strict D–field if C = K^{p}.

Definition. Let L_{e} be the natural language for D–fields, which contains symbols {0, 1, +, −, ·} for the field operations and unary function symbols D_{i,n} (i ∈ {1, …, e}, n ∈ ℕ). We denote by SCH_{p,e} the L_{e}–theory of all separably closed, strict D–fields which have degree of imperfection e.^{3}

1Equation (1.3) means that we consider only iterative Hasse derivations.

2The definition used here differs from the definition given in [7], where the constants are killed by all D_{i,j} (j > 0)

The aim of this article is to prove the following theorem:

Theorem 1.1.

1. SCH_{p,e} is complete and has quantifier elimination.

2. Every D–field can be extended to a model of SCH_{p,e}.

3. Every separably closed field of degree of imperfection e can be expanded to a model of SCH_{p,e}.

Our theory is a variant of the theory given by M. Messmer and C. Wood in [6], where a single, non–iterative Hasse derivation was used. For e = 1 our two approaches coincide and Theorem 1.1 was proved (slightly differently) in [6].^{4}

We will prove the theorem in Section 3. The main algebraic ingredient is the amalgamation property of the class of D–fields, which we prove in Section 2, Proposition 2.6. In Section 4 we give an alternative proof for quantifier elimination.

I thank Anand Pillay for various helpful discussions.

2 Amalgamation

We will prove in this section that the class of D–fields has the amalgamation property: Any two extensions of a D–field K can be jointly embedded in a third extension of K.

Lemma 2.1. For any D–field K the index of its field C of constants is bounded by p^{e}. Let K′ be an extension of K with constant field C′. Then C′ and K are linearly disjoint over C.

Proof. We write d_{i} for the derivation D_{i,1} and C_{i} for its field of constants in K. By reordering D we may assume that C is the irredundant intersection of the first f of the C_{i}. So the B_{i} = C_{1} ∩ ⋯ ∩ C_{i} form a properly descending sequence

K = B_{0} ⊃ B_{1} ⊃ ⋯ ⊃ B_{f} = C.

The formula (1.3) implies that d_{i}^{p} = 0 for all i. Since the d_{i} commute, each d_{i} maps B_{i−1} into itself. By Theorem 27.3 of [3] we find elements x_{i} ∈ B_{i−1} with d_{i}(x_{i}) = 1 and for any such choice 1, x_{i}, …, x_{i}^{p−1} is a basis of B_{i−1} over B_{i}.

3I.e. [K : K^{p}] = p^{e}

4M. Messmer and C. Wood have asked me to point out, that, for e > 1, there is a gap in the proof of the main theorem [6], as well as a false claim about the product rule in the non-iterative case.

Whence the x = x_{1}^{e_{1}} ⋯ x_{f}^{e_{f}} , (e_{i} < p) form a basis of K over B_{f} = C. For the same reason these elements form a basis of K′ over B′_{f} ⊃ C′. Thus the x are independent over C′.

An alternative proof uses the Wronskian matrix: Let θ_{1}, …, θ_{p^{e}} be an enumeration of all operators of the form D_{1,n_{1}}D_{2,n_{2}} ⋯ D_{e,n_{e}}, (n_{i} < p) (or, equivalently, d_{1}^{n_{1}}d_{2}^{n_{2}} ⋯ d_{e}^{n_{e}}). It can easily be proved by a standard argument that a sequence x_{1}, …, x_{N} is linearly independent over C iff the matrix (θ_{α}(x_{β})) has rank N (see [7, Proposition 5.1]). The Lemma follows immediately from this.

Corollary 2.2. Let K be a strict D–field and F a D–field which extends K. Then F is a separable extension of K. If [K : K^{p}] = p^{e} , F is strict iff K and F have a common p–basis.

(See [1] for the definition of p–basis and its basic properties.)

Proof. By the Lemma K and the field C constants of F are linearly disjoint over K^{p}. This implies that K and F^{p} are linearly disjoint over K^{p}. Thus F is separable over K. We have [K : K^{p}] ≤ [F : C] ≤ p^{e}. So, if [K : K^{p}] = p^{e}, we have [F : C] = p^{e}. Therefore C = F^{p} iff [F : F^{p}] = p^{e}, which proves the second part.

Lemma 2.3. Let (K, D) be a D–field and F a field extension of K. Assume that K and F have a common p–basis. Then D extends uniquely to a sequence E of commuting Hasse derivations on F. Furthermore, if (F′, E′) is an extension of (K, D) which contains F, the functions in E′ map F into itself, so that E = E′ ↾ F.

In the special case that F is a separably algebraic extension of K the Lemma is due to F.K. Schmidt ([2]) for e = 1 and to Okugawa ([7, Proposition 2.8]) for arbitrary e. We will deduce the general case from a theorem of Matsumara.

Proof. For a single Hasse derivation the lemma follows from the fact that field extensions with a common p–basis are 0–étale , see [3, 26.7 and 27.2]. So, if E is a sequence of Hasse derivations of F which extends D, it remains to show that the E_{i} commute. Let us prove that E_{1} and E_{2} commute, i.e. that E_{1,i} and E_{2,j} commute for all i, j, by induction on i + j. Fix m and n and assume that E_{1,i} and E_{2,j} commute for all i + j < m + n. It is easy to check (use (1.2)) that then E_{1,m}E_{2,n} −E_{2,n}E_{1,m} is a derivation. Since D_{1} and D_{2} commute, this derivation vanishes on K and therefore also on F.

The uniqueness stated in the Lemma follows also from the following recursive formula, which shows that D can be computed from its values on a basis^{5} of F/F^{p}: Let D be any Hasse derivation. Then (2.2) below implies for r < p that

5Even the values on a p–basis would suffice.

D_{pn+r}(x^{p}b) = ∑_{m≤n} D_{m}(x)^{p}D_{p(n−m)+r}(b).  (2.1)

Lemma 2.4. Any D–field K has a smallest strict extension K^{strict}, which is a purely inseparable extension of K.

Proof. Consider an arbitrary Hasse derivation D. We note first that (1.2) implies

D_{n}(x^{p}) = \begin{cases}D_{\frac{n}{p}} (x)^{p} & if p|n \\ 0 & otherwise \end{cases}  (2.2)

Also, by (1.3), if D_{1}(x) = 0, we have D_{m}(x) = 0 for all m which are not divisible by p. It follows that

D′ = (D_{0}, D_{p}, D_{2p}, …)

is a Hasse derivation on the constant field of D_{1}.^{6}

Let C be the constant field of (K, D). Since the D_{i} commute, all D_{i,n} map C into itself. By the last remark D′ = (D′_{1}, …, D′_{e}) is a sequence of commuting

Hasse derivations on C. We transport D′ from C to K^{∗} = C^{\frac{1}{p}} via the Frobenius map:

D_{i,n}^{∗}(x) = D_{pn,i}(x^{p})^{\frac{1}{p}} .

D^{∗} extends D by (2.2). We repeat this process and get an infinite sequence of purely inseparable extensions. The union of this sequence is K^{strict}.

Note that K^{strict} is separably closed if K is separably closed.

Lemma 2.5. Let F and L be D–fields which both extend the D–field K. Assume that, in a common field extension, F and L are linearly disjoint over K. Then F L has a unique D–structure which extends the D–structures of F and L.

Proof. A D–module over K is a K–vector space V with a family D_{i,n} (i ∈ {1, …, e}, n ∈ ℕ) of commuting additive maps V → V such that for all D = D_{i}, x ∈ K and v ∈ V.

D_{0}(v) = v  (2.3)
D_{n}(xv) = ∑_{a+b=n} D_{a}(x)D_{b}(v)  (2.4)

D_{a}D_{b}(v) = \binom{a + b}{a}D_{a+b}(v) .  (2.5)

A commutative K–algebra R is a D–algebra if it is a D–module and the D_{i} are Hasse derivations on R.

The following statements are easy to check (cf. [4]):

6Note that \binom{n}{i} ≡ \binom{pn}{pi} mod p.

• If V and W are D–modules over K, the tensor product V ⊗_{K} W becomes a D–module by the definition

D_{i,n}(v ⊗ w) = ∑_{a+b=n} D_{i,a}(v) ⊗ D_{i,b}(w).

• If R and S are D–algebras over K, their tensor product is also a D–algebra.

If R and S have unit–elements, R and S are subrings of R ⊗_{K} S. It is clear that the D–structure of R ⊗_{K} S is the only common extension of the D–structures of R and S^{7}.

If F and L are linearly disjoint over D, F L is the quotient field of F ⊗_{K} L. By [2] and [7, Proposition 2.3] a sequence of commuting Hasse derivations on a domain extends uniquely to the quotient field. This proves the Lemma.

Proposition 2.6. The class of D–fields has the amalgamation property.

Proof. Let F and L be D–fields which both extend the D–field K. If we apply Lemma 2.3 and Lemma 2.4 to the separable algebraic closures of F and L, we see that we may assume that F and L are separably closed and strict. Then (K^{sep})^{strict} is a D–subfield of F and L (again by Lemmas 2.3 and 2.4), so we may assume that K is separably closed and strict. We may also assume that F and L are situated in a common extension field and are algebraically independent over K. By Corollary 2.2 F is a separable extension of K and therefore a regular extension, since K is separably closed. This implies that F and L are linearly independent over K and that we can extend the D–structure of F and L to F L.

3 Proof of the Theorem

1. Quantifier elimination and completeness

To prove that SCH_{p,e} has quantifier elimination, we have to show that the following is true: If F and L are models of SCH_{p,e} with a common substructure R, we can embed F over R in an elementary extension of L. Let K be the quotient field of R in F and K′ the copy of K in L.

For all Hasse derivations D we have the recursion formula

D_{n}(\frac{r}{s}) = \frac{D_{n}(r) − ∑_{a<n} D_{a}(\frac{r}{s})D_{n−a}(s)}{s}

This shows that K and K′ are D–subfields of F and L and are, over R, isomorphic as D–fields. So we can assume that R = K.

7Note that D_{n}(1) = 0 for any Hasse derivation D and n > 0.

By amalgamation we find a D–field F′ which extends F and L. We may assume that F′ is strict. By Corollary 2.2 F′ is a separable extension of L. Since L is separably closed, we can embed F′ over L in an elementary extension L′ of L (see [1, Claim 2.2]). Let F′′ be the copy of F′ in L′.

[figure: L′; F′; F′′; F; L; K; R]

It remains to show, that F′′ is a D–subfield of L′ which, over L, is isomorphic to F′. But this follows immediately from Lemma 2.3, since F′ and L have a common p–basis by Corollary 2.2. Note that the assumption that F is a model of SCH_{p,e} was not used.

SCH_{p,e} is complete since it is consistent by part 3 below and since all models contain the trivial D–field 𝔽_{p}.

2. Every D–field is contained in a model

Let F be a D–field and L be any model of SCH_{p,e}. (We will see below that SCH_{p,e} is consistent.). By the proof of quantifier elimination we can embed F (over 𝔽_{p}) in an elementary extension of L.

3. Every separably closed field with degree of imperfection e can be expanded to model

Let F be a separably closed field of imperfection degree e. Let b_{1}, …, b_{e} be a p–basis of F. Then the b_{i} are algebraically independent over 𝔽_{p} and form a p– basis of K = 𝔽_{p}(b_{1}, …, b_{e}). Define a sequence of commuting Hasse derivations on K by

f(b_{1}, …, b_{i} + t, …, b_{e}) = ∑_{n=0}^{∞} D_{i,n}(f(b_{1}, …, b_{e})) t^{n}.  (3.1)

or, equivalently, by

D_{i,n}(b_{1}^{k_{1}} ⋯ b_{e}^{k_{e}}) = \binom{k_{i}}{n}b_{1}^{k_{1}} ⋯ b_{i}^{k_{i}−n} ⋯ b_{e}^{k_{e}}  (3.2)

It is easy to check, and well–known, that this definition turns K into a strict D–field (see [7, Section I.1]). By Lemma 2.3 we can extend D to F. F is strict since [F : F^{p}] = p^{e} (Corollary 2.2). So (F, D) is a model of SCH_{p,e}.

4 Remarks

Stability and elimination of imaginaries

Using the methods of [1] and [6] it is easy to prove that SCH_{p,e} is stable and has elimination of imaginaries. The stability of SCH_{p,e} can also be derived directly from the stability of separably closed fields ([8]) as follows: Let F be a separably closed field with p–basis b_{1}, …, b_{e} and D a Hasse derivation of F. The formula (2.1) shows that all D_{n} are definable in the field F using the parameters D_{n}(b_{1}^{k_{1}} ⋯ b_{e}^{k_{e}}).^{8} This implies that F together with any sequence of Hasse derivations is stable.

Let me also indicate why SCH_{p,e} has elimination of imaginaries, following [5]. One notes first, that, working in fields, it suffices to show that SCH_{p,e} has weak elimination of imaginaries (see [5, Fact 5.5]). By a theorem of Evans, Pillay and Poizat (see [5, Proposition 5.8]) it is enough to show that every type q(x_{1}, …, x_{m}) over a model (F, D) has a canonical base. Let θ_{1}, θ_{2}, … be an enumeration of all operators of the form D_{1,n_{1}}D_{2,n_{2}} ⋯ D_{e,n_{e}}, (n_{i} = 0, 1, …) and let I_{q} be the ideal of all polynomials f ∈ F [X_{α,j}]_{α=1,2,...; j=1,...,m} such that the formula f(θ_{α}(x_{j})) =^{.} 0 belongs to q. By quantifier elimination q is determined by I_{q}. Thus the field of definition of I_{q} serves as a canonical base of q.

Canonical p-bases

Let (F, D) be a D–field with degree of imperfection e. A p–basis b_{1}, …, b_{e} is canonical if for all n > 0

D_{i,n}(b_{j}) = \begin{cases}1 & if n = 1 and i = j \\ 0 & otherwise \end{cases} .  (4.1)

Lemma 4.1. Let F be a field with with degree of imperfection e. Every p–basis of F is a canonical p–basis of a uniquely determined sequence of commuting Hasse derivations.

Proof. A canonical p–basis b_{1}, …, b_{e} determines D uniquely: D_{i,n}(b_{1}^{k_{1}} ⋯ b_{e}^{k_{e}}) is given by (3.2). To compute D_{i,n}(x) for arbitrary x, write

x = ∑_{0≤k_{1},...,k_{e}<p^{m}}x_{k_{1}...k_{e}}^{p^{m}}b_{1}^{k_{1}} ⋯ b_{e}^{k_{e}}

for some m with n < p^{m}. Then

D_{i,n}(x) = ∑_{0≤k_{1},...,k_{e}<p^{m}}x_{k_{1}...k_{e}}^{p^{m}}D_{i,n}(b_{1}^{k_{1}} ⋯ b_{e}^{k_{e}}).  (4.2)

Now let b_{1}, …, b_{e} be any p–basis. The construction at the end of the last section shows that (3.2) and (4.2) define a sequence D of commuting Hasse derivations with canonical p–basis b_{1}, …, b_{e}.

8Actually the parameters b_{i} and D_{p^{m}}(b_{i}) suffice.

The D constructed in in the last part of the proof is strict. So we conclude, that only a strict sequence D can have a canonical p–basis. The converse is true if (F, D) is ω-saturated:

Remark. Every ω–saturated strict D–field has a canonical p–basis.

I will give the proof only in the following special case, which will be used later.

Corollary 4.2. Every ω–saturated model of SCH_{p,e} has a canonical p–basis.

Proof. To have a canonical p–basis means that a certain countable set Σ(x_{1}, …, x_{e}) of formulas is realized. Since SCH_{p,e} is complete, it is enough to show that some model of SCH_{p,e} has a canonical p–basis. For this take a separably closed field F of imperfection degree e and fix a p–basis b̄. Let D be the unique sequence which has b̄ as a canonical p–basis. (F, D) is a model of SCH_{p,e}.

Lemma 4.1 and the last remark allow us to determine all strict sequences of commuting Hasse derivations of an ω–saturated field F. We note first that, if b_{1}, …, b_{e} is a canonical p–basis for D, then b′_{1}, …, b′_{e} is a canonical p–basis for D iff the differences b_{i} − b′_{i} belong to

F^{p^{∞}} = ⋂_{k=1}^{∞} F^{p^{k}} = {a ∈ F | D_{i,n}(x) = 0 (i = 1, …, e; n = 1, 2, …)}.

This gives

Remark. Let F be an ω–saturated field with degree of imperfection e. There is a natural 1–1–correspondence between the set of all strict sequences of commuting Hasse derivations and the set of all p–bases modulo F^{p^{∞}}.

Lambda functions

Let b_{1}, …, b_{e} be a p–basis of F. The functions λ_{k_{1}...k_{e}}^{m} are defined by

x = ∑_{0≤k_{1},...,k_{e}<p^{m}}λ_{k_{1}...k_{e}}^{m}(x)^{p^{m}} b_{1}^{k_{1}} ⋯ b_{e}^{k_{e}}

Fix a natural number m. For a multi-index κ = (k_{1}, …, k_{e}) ∈ {0, …, p^{m} − 1}^{e} and a sequence D of Hasse derivations let us use the notations

b^{κ} = b_{1}^{k_{1}} ⋯ b_{e}^{k_{e}}  and  D_{κ} = D_{1,k_{1}}D_{2,k_{2}} ⋯ D_{e,k_{e}}.

If we apply D_{κ} to the equation

x = ∑_{μ} λ_{μ}^{m}(x)^{p^{m}}b^{μ},

we obtain

D_{κ}(x) = ∑_{μ} λ_{μ}^{m}(x)^{p^{m}}D_{κ}(b^{μ}).

If D is strict, the Wronski matrix (D_{κ}(b^{μ})) is always regular. If b_{1}, …, b_{e} is canonical for D, its entries are, up to factors from 𝔽_{p}, monomials in the b_{i}. It is also easy to see that the determinant is 1. This yields

Lemma 4.3. Let (F, D) be a D–field with canonical p–basis b_{1}, …, b_{e}. Then the functions (λ_{μ}^{m}(x))^{p^{m}} are polynomials in b_{1}, …, b_{e} and the D_{κ}(x).

Quantifier elimination

Let T_{p,e} denote the theory of separably closed fields F of characteristic p with a named p–basis b_{1}, …, b_{e}. It is shown in [1] that T_{p,e} is complete and has quantifier elimination if one adds function symbols for the λ_{μ}^{m} to the language.^{9}

This fact can be used to give an alternative proof for the quantifier elimination of SCH_{p,e}: Let ϕ(x̄) be an L_{e}–formula and (F, D) a saturated model of SCH_{p,e}. By Corollary 4.2 we can find a canonical p–basis b_{1}, …, b_{e}. Since we can define the D_{i,n} in (F, b_{1}, …, b_{e}), ϕ(x̄) is equivalent to a Boolean combination of polynomial equations between b_{1}, …, b_{e} and terms of the form λ_{μ}^{m}(x_{i}), for sufficiently large m. By taking p^{m}-th powers we can replace the λ_{μ}^{m}(x_{i}) by λ_{μ}^{m}(x_{i})^{p^{m}}. By the last lemma we obtain an equivalent Boolean combination of equations of the form

∑_{κ} q_{κ}(x̄) b^{κ} =^{.} 0  (4.3)

where the q_{κ}(x̄) are terms in the D_{κ}(x_{i}). The equivalence holds for any choice of the canonical p–basis b_{1}, …, b_{e}. Since F^{p^{∞}} is infinite, we can find the b_{1}, …, b_{e} algebraically independent over any given tuple x̄. This shows that we can replace each equation (4.3) by ⋀_{κ} q_{κ}(x̄) =^{.} 0. We observe finally that the resulting quantifier free L_{e}–formula does not depend on the choice of F.

References

[1] Françoise Delon. Separably closed fields. In Bouscaren, editor, Model theory and algebraic geometry: An Introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture, volume 1696 of Lecture Notes in Mathematics, pages 143–176. Springer, Berlin, 1998.

[2] Helmut Hasse and F.K. Schmidt. Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper mit einer Unbestimmten. J. Reine Angew. Math., 177:215–237, 1937.

9Only the λ_{μ}^{1} are needed for quantifier elimination. As Françoise Delon has explained to me, also the constants for the p–basis can be disposed of.

[3] Hideyuki Matsumara. Commutative ring theory. Cambridge University Press, 1986.

[4] B. Heinrich Matzat. Differential galois theory in positive characteristic. Lecture Notes, October 2001.

[5] Margit Messmer. Some model theory of separably closed fields. In D. Marker, M. Messmer, and A. Pillay, editors, Model Theory of Fields, volume 5 of Lectures Notes in Logic, pages 135–152. Springer, Berlin, 1996.

[6] Margit Messmer and Carol Wood. Separably closed fields with higher derivation I. The Journal of Symbolic Logic, 60(3):898–910, September 1995.

[7] K. Okugawa. Basic properties of differential fields of arbitrary characteristic and the Picard–Vessiot theory. J. Math. Kyoto Univ., 2(3):294–322, 1963.

[8] Carol Wood. Notes on the stability of separably closed fields. J. Symbolic Logic, 44(3):412–416, September 1979.

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ziegler@uni-freiburg.de