# C2.6 Introduction to Schemes - Material for the year 2019-2020

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 (see the beginning of the lecture notes for precise references). C3.4 Algebraic Geometry is recommended but not technically necessary.

C3.1 Algebraic Topology contains many homological techniques also used in this course.

16 hours.

### Assessment type:

- Written Examination

Scheme theory is the foundation of modern algebraic geometry. 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, with an emphasis on cohomological techniques.

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 coherent sheaves and their cohomology groups.

Sheaves and cohomology of sheaves.

Affine schemes: points, topology, structure sheaf. Schemes: definition, subschemes, morphisms, glueing. Relative schemes: fibred products, Cohomological characterisation of affine schemes.

Projective schemes, morphisms to projective space. Ample line bundles. Cohomological characterisation of ampleness.

Flat morphisms, semicontinuity, Hilbert polynomials. Cohomological characterisation of flatness.

Constructibility and irreducibility. Images of constructible sets.

Separatedness, properness and valuative criteria. Hilbert and Quot schemes.

- Robin Hartshorne,
*Algebraic Geometry*. - Ravi Vakil,
*Foundations of Algebraic Geometry*, online notes on the website of Stanford University (open access).

- David Mumford,
*The Red Book of Varieties and Schemes*. - David Eisenbud and Joe Harris,
*The Geometry of Schemes*.