96734 - Scheme Theory

Learning outcomes

At the end of this course, the student knows the basic notions of scheme theory. These can be applied in their research field in algebra and geometry.

Course contents

If there are people who are interested in this class and do not speak Italian, I am happy to deliver this class in English.

The theory of schemes, developed by Alexander Grothendieck in the 1960s, is the modern and rigorous language used to study, do, and write algebraic geometry. It unifies classical algebraic geometry (i.e. the study of the zero loci of polynomial equations with coefficients in an algebraically closed field) and algebraic number theory (i.e. the study of the behaviour of prime ideals under finite field extensions of the field of rational numbers), allowing for geometric intuition to be brought into commutative algebra.

The following analogy holds: differential calculus (i.e. the study of differentiable functions between open subsets of R^n, as done in Calculus II) is to differential geometry (i.e. the study of differentiable manifolds), as commutative algebra (i.e. the study of commutative rings) is to the theory of schemes. Indeed, in a very rough sense, one can say that a scheme is a "geometric object" that locally behaves like a ring.

Topics covered in the course include: sheaves, schemes, global and local properties of schemes, and coherent sheaves.

For prerequisites (i.e. the preliminary knowledge required at the beginning of the course), please refer to the Italian version of this web page.

Readings/Bibliography

Testi consigliati:

Hartshorne, Algebraic geometry, GTM 52, Springer

Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics

 

Altri testi per la consultazione:

Mumford, The Red Book of Varieties and Schemes, Springer

Eisenbud & Harris, The geometry of schemes, GTM 197, Springer

Görtz & Wedhorn, Algebraic geometry I: Schemes, Springer

Görtz & Wedhorn, Algebraic geometry II: Cohomology of schemes, Springer

Vakil, The rising sea, Foundations of algebraic geometry, Princeton University Press or https://math.stanford.edu/~vakil/216blog/

Teaching methods

Blackboard lectures

Assessment methods

Homework + Oral exam

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students ) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Office hours

See the website of Andrea Petracci