# Projective modules and A1-homotopy theory

May 5 to May 9, 2014

at the

American Institute of Mathematics, Palo Alto, California

organized by

Aravind Asok, Jean Fasel, and Satyagopal Mandal

This workshop, sponsored by AIM and the NSF, will be devoted to studying recent interactions between the classical theory of projective modules and $A^1$-homotopy theory.

Recent work of F. Morel provides an analog of Steenrod's classification of topological vector bundles in terms of homotopy classes of maps to a Grassmannian. More precisely, he identifies the set of isomorphism classes of projective modules of fixed rank over a smooth affine algebra $R$ with the maps in the $A^1$-homotopy category between the associated affine scheme Spec $R$ and an algebro-geometric Grassmannian. This result provides a dictionary by which to translate topological results about vector bundles to results about projective modules. The goal of this workshop is to introduce practitioners of the theory of projective modules to modern ideas about $A^1$-homotopy theory and vice versa.

During the workshop, we will focus on the following questions/problems:

• If $P$ is a projective module of rank below the dimension, beyond the primary Euler class" obstruction, what can we say about secondary (and higher) obstructions to splitting off a free rank 1 summand? When are such higher obstructions sufficient to splitting?
• Can $A^1$-homotopy theory shed light on classical problems regarding complete intersections, e.g., the question of whether a curve in affine space is a complete intersection, or Murthy's problem: given an ideal $I$ in a polynomial ring, is it the case that the number of generators of $I/I^2$ is the same as the number of generators of $I$?

• How does one reconcile the explicit" and functorial approaches to Euler classes (via Euler class groups in the first case, and Chow-Witt groups in the second), especially in higher codimension.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.