C2.6 Introduction to Schemes - Material for the year 2020-2021

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2020-2021
Lecturer(s): 
Prof. Alexander Ritter
General Prerequisites: 

B2.2 Commutative Algebra is essential. C2.2 Homological Algebra is highly recommended and C2.7 Category Theory is recommended but the necessary material from both courses can be learnt during the course. C3.4 Algebraic Geometry is strongly recommended but not technically necessary. C3.1 Algebraic Topology contains many homological techniques also used in this course.

Course Term: 
Hilary
Course Lecture Information: 

16 hours.

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course Overview: 

Scheme theory is the foundation of modern algebraic geometry, whose origins date back to the work from the 1950s and 1960s by Jean-Pierre Serre and Alexander Grothendieck. It unifies algebraic geometry with algebraic number theory. This unification has led to proofs of important conjectures in number theory such as the Weil conjecture by Deligne and the Mordell conjecture by Faltings.

This course will cover the basics of the theory of schemes.

Learning Outcomes: 

Students will have developed a thorough understanding of the basic concepts and methods of scheme theory. They will be able to work with affine and projective schemes, as well as with (quasi-)coherent sheaves and their cohomology groups.

Course Synopsis: 

The Spec of a ring, Zariski topology, comparison with classical algebraic geometry.

Pre-sheaves and stalks, sheaves, sheafification. The abelian category of sheaves of abelian groups on a topological space. Direct and inverse images of sheaves. Sheaves defined on a topological basis.

Ringed spaces and morphisms of ringed spaces. Affine schemes, construction of the structure sheaf, the equivalence of categories defined by Spec.

Schemes, closed subschemes. Global sections. The functor of points.

Properties of schemes: (locally) Noetherian, reduced, irreducible, and integral schemes. Properties of morphisms of schemes: finite type, open/closed immersions, flatness. Simple examples of flat families of schemes arising from deformations.

Gluing sheaves. Gluing schemes. Affine and projective n-space viewed as schemes.

Products, coproducts and fiber products in category theory. Existence of products of schemes. Fibers and pre-images of morphisms of schemes. Base change.

Further properties of morphisms of schemes: separated, universally closed, and proper morphisms. Projective n-space and projective morphisms. Abstract varieties. Complete varieties. Scheme structure on a closed subset of a scheme.

Sheaves of modules. Vector bundles and coherent sheaves. The abelian category of sheaves of modules over a scheme. Pull-backs.

Quasi-coherent sheaves. Gluing sheaves of modules. Classification of (quasi-)coherent sheaves on Spec of a ring.

Čech cohomology. Vanishing of higher cohomology groups of quasi-coherent sheaves on affine schemes. Independence of Čech cohomology on the choice of open cover. Line bundles, examples on projective n-space.

Sheaf cohomology. Acyclic resolutions. Comparison of sheaf cohomology and Čech cohomology.

Brief discussion of (quasi-)coherent sheaves on projective n-space, graded modules, and Proj of a graded ring.

Reading List: 
  1. Robin Hartshorne, Algebraic Geometry.
  2. Ravi Vakil, Foundations of Algebraic Geometry, online notes on the website of Stanford University (open access)
Further Reading: 
  1. David Mumford, The Red Book of Varieties and Schemes.
  2. David Eisenbud and Joe Harris, The Geometry of Schemes.
  3. George R. Kempf, Algebraic Varieties.