Student Algebra seminar

Speaker

Robert Garbary, Pure Math Department, University of Waterloo

Projective modules: abstract

For smooth projective varieties defined over \mathbb{C}, a famous result of Serre says that it doesn't really matter whether we work in the algebraic category or the complex analytic category: the two naturally coincide. Thus studying algebraic vector bundles over smooth projective varieties is equivalent to studying holomorphic vector bundles over projective complex manifolds, and so we may employ complex analysis to answer 'algebraic' questions.

Suppose we work with the non-projective variety X = \mathbb{C}^n, and ask to classify all the vector bundles on X. If we only care about topology, the answer is easy: we may homotope X to a point, and the only vector bundles over a point are trivial. In a complex analytic category, the analagous result is also true: use the \overbar{\partial}-Poincare lemma and the exponential sheaf sequence.

In an algebraic category, the analagous result is also true, but is much deeper. Its truth is equivalent to a modern problem. In his famous 1955 paper [FAC], Serre writes

... on ignore s'il existe des \mathbb{C}[x_1,...,x_n]-modules projectifs to type fini qui no soint pas libres

which reads 'it is unknown if every projective \mathbb{C}[x_1,...,x_n]-module is free'. This statement became known as (one of the many) Serre's conjecture. The conjecture was answered in the affirmative in 1976 by Suslin and Quillen.

This seminar will be about projective modules and local algebra, and should be of interest to people who like commutative algebra and/or algebraic geometry. We will discuss the basic properties of projectives, local rings, and localisations. The main goal is to describe the correspondence (here, A is a fixed base ring) between finitely generated projective A-modules and locally free \mathcal{O}_{Spec A}-modules of finite rank over the affine scheme Spec A. This will motivate the notion of a quasi-coherent sheaf of modules. Finally, we will describe a proof of the conjecture, probably modulo some heavy machinery.

In the first talk, I will give a rough outline of the plan for the term. I will then talk about the basics of projectives: several examples, non-examples, and about 31 equivalent definitions.