Introductory description

This is a PhD module on cohomology of coherent sheaves following Serre's FAC.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Sheaves provide a language for making global constructions from local
data. It includes cases when the local construction is trivial, but
passing between local pieces involves transition function to take
account of coordinate changes. The module is a first introduction to the
notion of coherent sheaves in algebraic geometry and their cohomology.
The content aims to cover the results of the famous paper [Jean-Pierre
Serre, Faisceaux algébriques cohérents. Ann. of Math. (2) 61 (1955),
197-278].

The first sections define sheaves, including the structure sheaf of
regular functions on an algebraic variety. The first basic result is the
equivalence of categories between coherent sheaves on an affine
algebraic variety and finite modules over its coordinate ring. This
elementary result means that affine varieties have zero higher
dimensional coherent cohomology, so that affine varieties play the role
of contractible topological spaces.

Coherent cohomology groups can be thought of a finite dimensional vector
spaces given by polynomials of given degree. Working with them includes
practically all the calculations in elementary geometry, such as the
linear systems of plane curves vanishing or satisfying tangency
conditions at marked points. Serre's theorems provide indispensable
foundations for work with algebraic varieties, including the theory of
algebraic curves and the classification results of algebraic surfaces
and threefolds.

Learning outcomes

By the end of the module, students should be able to:

• Demonstrate understanding of the basic concepts, theorems and calculations related to rings of functions on algebraic varieties and modules over them, including sheaves of functions or vector fields on curves and surfaces. == Demonstrate understanding of the basic definitions, calculations and results concerning modules over the coordinate ring of an affine variety and the associated coherent sheaves of modules. Examples related to the ideal sheaves defining varieties and the restriction of functions from an ambient space to a subvariety. == Acquire practical understanding of the usefullness of Cech cohomology related to transition functions used to pass between local coordinate pieces in the study of bundles on manifolds == Demonstrate knowledge and understanding of the statement of Serre's theorems on coherent cohomology and an understanding of their applications.

Indicative reading list

Reading lists can be found in Talis

Subject specific skills

Understand the use of sheaves in different areas of geometry, both
in theory and practical use. This includes translating between
high-flown notions in sheaf theory and category theory and their
simple and more advanced applications to geometry such as the
intersection of curves in the plane or the construction of varieties
with given singularities.

Transferable skills

• sourcing research material
• prioritising and summarising relevant information
• absorbing and organizing information
• presentation skills (both oral and written)