# Informationen zum Steilkurs Schemata/Fast track course schemes Sommersemester 2017

Dozentin: Prof. Dr. Annette Huber-Klawitter
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.

## Dates

### Lecture

Monday 14-16, SR 125, Eckerstr. 1

### Excercise and discussion classes

Thursday 10-12, SR403, Friday 12-14

## Material covered and excercises

• 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 presheaves

Hartshorne 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
• 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:
1. A is a discrete valuation ring
2. A is integralla closed
3. m is a principal ideal
4. dim_k(m/m^2)=1
5. Every non-zero ideal is a power of m
6. 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
• 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

• 3.7.17 Differential forms Ha II 8, Matsumura: Commutative algebra

• 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!
• 10.7.17 Regularity and smoothness Ha I 5, II 8 and III 10

Ha I 5.10, II 8.6, III 10.1

• 17.7.17 Line bundles, Chech-cohomology and Cartier-divisors Ha II 6, III 4

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

### Studien/Prüfungsleistung

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

## Literature

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

## Mailingliste

We are about to start a mailing list for the participants of the class. Please sign up my sending an email to hartshorne-on@math.uni-freiburg.de. Mails to hartshorne@math.uni-freiburg.de will reach everyone on the list.