## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, May 10, 2012 — 1:00 PM EDT

Robert Garbary, Pure Math Department, University of Waterloo

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.

Location

MC - Mathematics & Computer Building

5136B

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

200 University Avenue West

Waterloo, ON N2L 3G1

Canada

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.