Wintersemester 2022

Übungen: M.Sc. Luca Terenzi

- The room is equipped with video and the lecture will be streamed via the BBB-room Steilkurs Schemes (It seems that the link only works once the room has actually been started.)

The exercise class will be held in English. Some information on the planned content can be found in the kommentiertes Vorlesungsverzeichnis kommentierten Vorlesungsverzeichnis.

Tuesday 08-10, SR 404, Ernst-Zermelo-Str. 1

Thursday 10-12; Friday 10-12

- 24.10.22: Let X=
**C**as topological space, O the sheaf of holomorphic functions, O^* the sheaf of nowhere vanishing holomorphic functions, exp:O\to O^* the exponential map. Show that exp is surjective as a morphism of sheaves, but not as a morphism of presheavesHartshorne II. 1.3, 1.8, 1.17, 1.19

- 8.11.22: Schemes and morphisms of schemes, first half of Hartshorne II.2
- Let $(X,\Oh_X)$ be a locally ringes space such that every point has a neighbourhood $U$ such that $(U,\Oh_X|_U)$ is isomorphic to $(U',\Ch)$ where $U\subset\R^n$ is open and $\Ch$ the sheaf of differentiable functions on $U$. Show that $X$ has a unique structure of a differentiable manifold such that $\Oh_X$ is the sheaf of differentiable functions on $U$.
- Work out $\Spec A$ and its topology in the following cases: $\Z$, $k$ field, $k[X]$ and $k[X,Y]$ for $k$ algebraically closed field, $\Z_p$ ($p$-adic numbers), $\Z/p^n$ for $p$ prime, $k[[X]]$ for $k$ a field.
- $\Spec(A\oplus B)=\Spec A\cup \Spec B$
- Hartshorne II.2.1, 2.3, 2.5, 2.7, 2.19

- 15.11.22: Varietäten, projektiver Raum, erste Eigenschaften
Hartshorne II 2.10, 2.11, 3.5, 3.6, 3.8

- 22.11.22:Immersions, T-valued points, fibre product Ha II.3
- Hartshorne II 3.9, 3.10, 3.13

- Let f:X\to Y be an $S$-morphism.
We define the graph G_f as

G_f(T)={ (x,f(x)) in X(T) x_{S(T)} Y(T) }

Show that the natural map G_f\to X is an isomorphism, in particular G_f is a scheme.

Let g:Y\to Z be another S-morphism. Show that

G_{fg}=G_f x_{Y}G_g

- Hartshorne II 3.9, 3.10, 3.13
- 29.11.22 Separated and proper, valuation rings Ha II.4
- Let A be a one-dimensional local noetherian integral domain with maximal ideal m and residue field k. The following are equivalent:
- A is a discrete valuation ring
- A is integrally closed
- m is a principal ideal
- dim_k(m/m^2)=1
- Every non-zero ideal is a power of m
- There is x in A such that every non-zero ideal is of the form (x^v) with v\geq 0.

- Ha II 4.1, 4.2, 4.3, 4.8

- Let A be a one-dimensional local noetherian integral domain with maximal ideal m and residue field k. The following are equivalent:
- 06.12.22 Valuative criteria Ha II.4
- Let k algebraically closed, C=V(XY)\subset\A^2_k, R=k[[t]]. Determine all morphisms of k-schemes \Spec k[[t]]\to C. What are the image of the special/generic point in each case?
- Ha II 4.4, 4.6, 3.17

- 13.12.22 Modulgarben Ha II.5

Ha II 5.1, 5.6, 5.7 Tipp: Nakyamas Lemma

Ha II 5.13, 5.16,5.18,7.5

- Ha II 8.3 (a)
- Compute the rank of \Omega_{E/k} for the projective curve V (y^2-x(x-1)(x-2)) in charakteristic different from 2.
- Ha Thm 8.13
- Read up on the proof of the assertions on commutative algebra!

Ha I 5.10, II 8.6, III 10.1 --

- Check that the morphism in H^1 does not depend on the choice of a refinement morphism
- Check the exactness of the beginning of the long exact cohomology sequence (H^0 and H^1) for a short exaxt sequence of sheaves
- Make Cartier-divisors and line bundles explicit on the spectrum of a Dedekind domain
- Ha II 6.10

Ha II 6.1, 6.6, using Examples 6.10.2, and in addition 6.5.2 and 6.11.3

- Studienleistung: Lösen von Übungaufgaben und Teilnahme an den Übungen und/oder Vorlesung (50 % Anwesenheit).
- Prüfungsleistung: mündliche Prüfung.

- R. Hartshorne. Algebraic Geometry. GTM52, Springer.