SF2704 Theory of Schemes

Welcome to the course SF2704 Theory of Schemes!

This course will give an introduction to schemes. When Grothendieck introduced the modern foundations of Algebraic Geometry in the 1950s, he replaced the earlier theory of varieties with schemes. The classical theory of varieties is concerned with zero-sets of polynomials over algebraically closed fields. Scheme theory gives a more general and flexible setting which allows for nilpotents and can also be used to study arithmetic questions (over rings of integers and non-algebraically closed fields).

Consider for example the equation $x^2+ay^2=b$ in the affine plane where a and b are parameters. Fix an algebraically closed field k (not of char 2). When $a,b\in k$ are non-zero, the equation then defines a smooth curve in the affine plane — an ordinary variety. When b is zero, the curve becomes singular but is still an ordinary variety (the intersection of two lines). When both a and b are zero, we obtain the equation $x^2=0$. The corresponding variety, i.e., the vanishing locus, is just the y-axis and does not remember the square. The corresponding scheme, however, will be different and have a non-zero “function” $x$ whose square is zero. To accomplish this, schemes are modelled as a topological space together with a sheaf of functions.

With schemes, we can also consider the equation above with solutions in say integers instead of a field. This leads to diophantine equations and arithmetic geometry.

The second half of the course will consider quasi-coherent sheaves on schemes and their cohomology. This yields a powerful tool to understand geometric properties by homological computations, and leads to vast generalizations of classical results such as the Riemann–Roch theorem for curves.

Prerequisites

Suitable prerequisites for the course is basic knowledge of commutative algebra (rings, modules, localization, tensor products, noetherian rings, etc.) and affine varieties (Nullstellensatz etc) such as in the course MM7042 Commutative Algebra and Algebraic Geometry.

Course content

• sheaves, locally ringed spaces and schemes
• affine and projective schemes and examples
• coherent sheaves and their cohomology

Contact information

In this course, you will meet:

Lecturer: David Rydh, KTH <dary@kth.se>
Lecturer: Georg Oberdieck, KTH <georgo@kth.se>
Tutor: Michele Pernice, KTH <mpernice@kth.se>

The first lecture is on Jan 17, 2023, 10:15–12:00 in Q26.

Course literature

The main reference for the course is the book:

• Introduction to schemes by G. Ellingsrud and J.C. Ottem (Jan 21 2023). It is freely available here Links to an external site.. (Unfortunately a previous version was used earlier but this is now a link that will be stable during the course.)

Other good references are:

• Algebraic Geometry by R. Hartshorne, Springer-Verlag, 1977 (available on SpringerLink Links to an external site.).
• Algebraic geometry I. Schemes with examples and exercises by U. Görtz and T. Wedhorn, 1st edition 2010 and 2nd edition 2020 (available on SpringerLink: first edition Links to an external site. and second edition Links to an external site.; for errata see the book's webpage Links to an external site.).

Preliminary lecture plan

1. [Ch 1.1, 3.1] Introduction: affine varieties, schemes. Presheaves and sheaves. [notes lecture 1 Download notes lecture 1]
2. [Ch 3.2, 12.0–12.2] Stalks, saturation, sheafification, kernels, images, quotients, cokernels, exact sequences. [notes lecture 2 Download notes lecture 2]
3. [Ch 2, 4.1] Prime spectrum of a ring and structure sheaf of an affine scheme.
4. [Ch 3.3–3.4, 4.2–4.6] Push-forward of sheaves. Locally ringed spaces. Affine schemes. General schemes.
5. [Ch 5] Gluing sheaves and schemes.
6. [Ch 6] Examples of schemes.
7. [Ch 7-8] Noetherian schemes and dimension of schemes. Fiber products.
8. [Ch 8-9] Examples, base change, fibers. Separated and proper schemes.
9. [Ch 10] Projective schemes. Examples.
10. [Ch 12.3, 13] Sheaves of modules, push-forward, pull-back, quasi-coherent sheaves, coherent sheaves.
11. [Ch 14] Categorical properties, closed immersions, induced reduced scheme structure
12. [Ch 16] Projective modules, locally free sheaves, Picard group, zero locus of section, global generation.
13. [Ch 15] Graded-Tilde functor, twisting sheaf O(1), category of quasi-coherent sheaves on Proj(A), closed subschemes, Segre embedding, hypersurface/complete intersection
14. [Ch 17] Čech cohomology
15. [Ch 18] Cohomology of affine schemes. Cohomology of O(n).
16. [Ch 18] Cohomology of coherent sheaves on projective spaces, Euler characteristic, Bezout's theorem.
17. [Ch 22, 24.1] Differentials, smoothness, statement of Serre duality.

Examination

Homework and oral/written exam. More details to be given later.