Lecturer: Damiano Testa
Term(s): Term 2
Commitment: 30 lectures
Assessment: Oral exam
Good familiarity with the topics covered in: MA3G6 Commutative Algebra
Recommended, not compulsory: MA4A5 Algebraic Geometry
Recommended: MA3D5 Galois Theory
What are schemes and what are they good for? More importantly, why should I bother?
Schemes are geometric objects patched together from (commutative, unital) rings in very much the same way that differentiable manifolds are patched from open subsets of . The local models are called affine schemes, and are really the same thing as commutative rings, but from a geometric vantage point.
Whereas general schemes can be rather unwieldy and pathological, there are several important subclasses that are better behaved and amenable to deeper study (this is analogous to restricting from general topological spaces to CW complexes in topology). In particular, the algebraic varieties over algebraically closed ground fields which make up the basic objects of the introductory algebraic geometry module MA4A5 can be naturally viewed as schemes. However, it is common to study algebraic varieties not just in isolation, but in families, in order to distinguish between special properties of a single variety from general properties shared by all varieties that "nearby" the starting one. A general setup consists of a family of algebraic varieties depending algebraically on a parameter , where we want to relate properties of a general to the properties of the "limit variety" . It turns out that endowing the limit with a scheme structure can ensure reasonable continuity properties in the family. In good situations, this allows us to either deduce properties about a general member of the family from properties of , or, vice versa, to deduce properties of from properties of the members of the family . This is a first indication of the usefulness of schemes.
From this point of view, allowing nilpotent elements in the structure sheaf is the main advantage of schemes over varieties. Nilpotents usually play a more or less explicit role in every statement that is true "with multiplicities".
Thus, even within classical algebraic geometry over the complex numbers, schemes provide an advantageous theoretical set-up: the theory of deformations (and degenerations) of algebraic varieties. Studying properties of varieties via degeneration goes way back to at least the classical Italian school of algebraic geometry, but has recently reached new levels of sophistication and seen a boost of activity.
Another main area where schemes are useful is number theory: classical algebraic varieties are essentially solution sets, in some fixed algebraically closed ground field, of systems of polynomial equations (in many variables). But polynomial equations make sense over any ring R, and we may be interested in solutions in R, or any overring S of R, or, more generally, any R-algebra. For example, R may be the integers. We can reduce the defining polynomial equations modulo various primes, and try to relate the properties of the solutions of the reductions to that of the initial equations (essentially, the idea of degeneration again). Schemes provide a way to talk about these processes in a geometric way. In particular, they allow to establish and pursue useful analogies, for example between the rings of integers in an algebraic number field and algebraic curves. It can be argued that schemes do not quite capture everything number theorists care for (they tend to miss "what is going on at Archimedean primes", André Weil’s criticism of scheme theory), but schemes go a long way towards a unified perspective on number theory and geometry ("Kronecker's Jugendtraum").
This Module is not a first introduction to algebraic geometry. For this, the Module MA4A5 is more suitable. Indeed, it would be useful, but not indispensable, if you had knowledge at roughly the level of MA4A5 before taking this Module. However, some solid background in Commutative Algebra, as in MA3G6, is indispensable. Some prior acquaintance with basic algebraic number theory can be helpful to appreciate parts of the course more fully, but is absolutely not necessary. Of course, there is nothing that prevents you from taking this as a first module in algebraic geometry, but it is likely to corrupt your mathematical character in that case, in the sense that you may in future be more attracted to vacuous generalities and fancy language than actual mathematical substance (although I will try hard to minimize that risk). So it is at your own risk if this is a first course for you.
Possible topics include, but need not be limited to:
- Schemes and sheaves: definitions and first examples
- Basic attributes: reduced, irreducible, finite type; separated schemes, proper morphisms
- Fibre products; uses of generic points
- Proj and blow-ups; some invariants of projective schemes: Hilbert function and Hilbert polynomial
- Flat limits and basics about deformations and degenerations
- Schemes in arithmetic: ground fields, base rings; connections to Galois theory; the Frobenius morphism
- Group schemes; the functor of points
- Kaehler differentials and smooth morphisms
The “applied” in the title of the Module refers to the fact that we will be interested throughout in applications of schemes to problems in other areas that can be formulated without using scheme-theoretic language, but for which scheme theory supplies valuable tools. This is similar to the meaning of the qualifier in “Calculus for the practical person”! We are not interested in learning tons of definitions in the arid generality of general scheme theory. The goal is to understand how things work and only know the properties the properties that help us solve problems!
By the end of the module students should be able to:
- thoroughly command the language of basic scheme theory and have a good idea for the “look and feel”, i.e. for the geometric peculiarities of some classes of schemes; for example, the student should have some intuition for the significance of non-reducedness and should be able to draw pictures of some “arithmetic surfaces”;
- know about some important applications of scheme theory, for example, in deformation theory and number theory;
- apply the theory of schemes to solve unseen problems in easy new contexts.
D. Eisenbud, J. Harris: The Geometry of Schemes, Graduate Texts in Math. 197, Springer (2000).
D. Mumford: Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, (1966).
Y. Manin: Introduction to the Theory of Schemes, Moscow Lectures 1, Springer (2018).
D. Mumford, T. Oda: Algebraic Geometry II, Texts and Readings in Mathematics 73, Hindustan Book Agency (2015).