OXFORD C2.6 INTRODUCTION TO SCHEMES 2020-2022
Prof. Alexander F. Ritter, Associate Professor, University of Oxford.
LECTURE NOTES AND EXERCISES
♦ Lecture notes, 1 page per side (version 56, Sep 2021)
♦ Lecture notes, 2 pages per side (version 56, Sep 2021)
(You may find the last chapter from my algebraic geometry notes a useful prelude to this course: notes here, or two-pages-per-side here)
♦ Exercise Sheets:
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.
Cech cohomology. Vanishing of higher cohomology groups of quasi-coherent sheaves on affine schemes. Independence of Cech cohomology on the choice of open cover. Line bundles, examples on projective n-space. Cartier divisors. Weil divisors. Computations of Cech cohomology on projective space.
Sheaf cohomology. Acyclic resolutions. Comparison of sheaf cohomology and Cech cohomology.
Brief discussion of (quasi-)coherent sheaves on projective n-space, graded modules, and Proj of a graded ring.
Prof. Alexander F. Ritter. Contact
Associate Professor in Geometry, Mathematical Institute, Oxford.
The Roger Penrose Fellow and Tutor, Wadham College, Oxford.