One quiver to rule them all

TL;DR

This paper introduces a quiver setting associated with a formally smooth algebra that captures local structure of its representation schemes, aiming to develop a non-commutative etale topology.

Contribution

It proposes a quiver setting that encodes local data of representation schemes and conjectures a non-commutative etale local isomorphism to path algebras.

Findings

Defines a quiver setting (Q,a) for formally smooth algebras

Shows the quiver setting determines local structure of representation schemes

Conjectures a non-commutative etale local equivalence to path algebras

Abstract

To a formally smooth algebra A we associate a quiver setting (Q,a) containing enough information to reconstruct all the local quiver settings determining the etale local structure of finite dimensional representation schemes of A, see math.AG/9904171. Conjecturally, in a (yet to be developed) non-commutative etale topology, A is locally isomorphic to an algebra B which in turn is Morita equivalent (via the dimension vector a) to the path algebra CQ of the quiver Q.