BIREP — Representations of finite dimensional algebras at Bielefeld

# Categories of Sheaves (eKVV: 241212)

Lecture in the winter term 2014/2015 (2 SWS)
• Lecturer: Dr. Julia Sauter
• Tuesday 10-12 Uhr, in V4-116 starting at the 14th of October and ending at the 3rd of February.

# Content

Please have a look at the announcement (pdf).
We start with a quick background course for the necessary notions from algebraic geometry. This is the material that I want to cover but changes are possible.
Part I: A quick introduction into algebraic geometry
• Chapter 1: Schemes and Varieties, outline (pdf)
The spectrum of a commutative ring, Zariski topology. Naive varieties. Sheaves. Regular maps. The structure sheaf. Schemes and varieties, also the functorial point of view. Naive Varieties as schemes. Dimension.
• Chapter 2: Categories of sheaves (pdf)
The category of modules over the structure sheaf. Coherent sheaves. The equivalence of coherent sheaves over an affine scheme to the finitely generated modules over the coordinate ring. The equivalence of coherent sheaves on a projective schemes to the quotient given by finitely generated graded modules over the coordinate ring modulo the finite dimensional graded modules. Vector bundles. The equivalence of vector bundles to locally free sheaves.
• Chapter 3: Cohomology and Comparism
5 functor calculus. sheaf cohomology. Analytification. Theorem of Grothendieck (implying finite dimensionality of cohomology groups for projective varieties). The cotangent sheaf, smoothness. GAGA-comparism - change to analytic topology. Local systems and connections on vector bundles. Monodromy. Constructible sheaves.
• Chapter 4: Derived categories of sheaves
Derived categories and the 6 functors. Spectral sequences. t-structures. Recollements. Semi-orthogonal decompositions. Perverse sheaves. The decomposition theorem. Shortly mention: The Riemann-Hilbert correspondence
Part II: Representation theoretic and triangulated category methods
• Chapter 5: Tilting in Db(Coh(X))
• Chapter 6: Quiver descriptions of perverse sheaves
• Chapter 7: Balmers support theory
• Chapter 8: dg-methods

# Lectures

This is my (tentative) planning of the lectures but I am open for suggestions from your side.
• Oct 14: Chapter 1 - Schemes and Varieties (part 1)
• Oct 21: Chapter 1 - Schemes and Varieties (part 2)
• Oct 28: Chapter 2 - Categories of Sheaves (part 1)
• Nov 4: Chapter 2 - Categories of Sheaves (part 2)
• Nov 11: Chapter 3 - Comparism and Cohomology (part 1)
• Nov 18: Chapter 3 - Comparism and Cohomology (part 2)
• Nov 25: Chapter 4 - Derived categories of Sheaves (part 1)
• Dec 2: Chapter 4 - Derived categories of Sheaves (part 2)
• Dec 9: Chapter 5 - Tilting in Db(Coh(X)) (part 1)
• Dec 16: Chapter 5 - Tilting in Db(Coh(X)) (part 2)
• Jan 6: no lecture
• Jan 13: Chapter 6 - Quiver descriptions of perverse sheaves (part 1)
• Jan 20: Chapter 6 - Quiver descriptions of perverse sheaves (part 2)
• Jan 27: Chapter 7 - Balmer's support theory
• Feb 3: Chapter 8 - Relationships between coherent and constructible sheaves

# Literature

For chapter 1: Schemes and Varieties
• Robin Hartshorne: Algebraic Geometry, chapter 1 and 2
• Qing Liu: Algebraic Geometry and Arithmetic Curves, chapter 2 and 3
• Lecture notes of Oliver Roendigs (in German): Vorlesung ueber Schemata , WS 2004/2005.
• For the definition of schemes as functors with extra properties, see for example
Demazure: Lectures on p-divisible groups, chapter I
For chapter 2: Categories of Sheaves

# Exams

If you want or need to be examined in this course, please contact Julia Sauter for it directly.