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MA948 - Applied Scheme Theory

Lecturer: Christian Boehning

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Prerequisites: Familiarity with topics covered in:

MA4A5 (recommended, not compulsory), MA3G6, MA3D5 (recommended)

Content:

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 R^n. 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, and this is a first indication that schemes are useful, when one considers families of algebraic varieties, and in particular limits of varieties {X_t} depending algebraically on a parameter t, it turns out that the limit X_0 must usually be viewed as a scheme to ensure reasonable continuity properties in the family (meaning, X_0 should under favourable conditions inherit properties from X_t). Viewing X_0 as a scheme amounts to imagining it being endowed with some “infinitesimal fluff” that remembers the limiting procedure in a certain sense. This can be made precise using nilpotent elements in the structure sheaves. Thus this is a first area (entirely within classical algebraic geometry, over the complex numbers, say) where 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 a bunch of polynomial equations (in many variables). But polynomial equations make sense over any ring R, and then you may be interested in solutions in R, or any overring S of R. For example, R may be the integers. You can then also 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”, Andr\’{e} 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 some 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.

We aim to cover, in particular, the following topics:

-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 recurse to scheme 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 and only know the names of things, we want to understand how things work and 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

References:

D. Mumford, T. Oda: Algebraic Geometry II, Texts and Readings in Mathematics 73, Hindustan Book Agency (2015)

Y. Manin: Introduction to the Theory of Schemes, Moscow Lectures 1, Springer (2018)

D. Eisenbud, J. Harris: The Geometry of Schemes, Graduate Texts in Math. 197, Springer (2000)