Sommersemester 2017

Übungen: Thomas Spittler

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

Monday 14-16, SR 125, Eckerstr. 1

Thursday 10-12, SR403, Friday 12-14

- 24.04.17: Sheaves and Presheaves, Hartshorne II.1
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

- 1.5.17: Holiday! This will give you a week to work on the basics of category theory. Here is an excercise sheet to help you. (In German, sorry) hier
- 8.05.17: 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

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

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

- Let f:X\to Y be an $S$-morphis.
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
- 29.05.17 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 integralla 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:
- 12.6.17 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

- 16.6.17 Modulgarben Ha II.5
Ha II 5.1, 5.6, 5.7 Tipp: Nakyamas Lemma

- 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 morphis in H^1 does not depend on the choice of a refinement morphism
- Check the exact 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

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

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