On the Azumaya locus of almost commutative algebras

TL;DR

This paper establishes conditions under which the Azumaya and smooth loci of the center of an algebra coincide in positive characteristic, especially for symplectic reflection algebras, linking geometric properties to algebraic structure.

Contribution

It proves a general criterion connecting the Azumaya and smooth loci of algebra centers with the symplectic structure of associated graded spectra, especially for symplectic reflection algebras.

Findings

Azumaya and smooth loci coincide under certain conditions

Symplectic reflection algebras are smooth and Azumaya loci coincide

Spectrum of associated graded algebra has a large symplectic leaf

Abstract

We prove a general statement which implies the coincidence of the Azumaya and smooth loci of the center of an algebra in positive characteristic, provided that the spectrum of its associated graded algebra has a large symplectic leaf. In particular, we show that for a symplectic reflection algebra smooth and the Azumaya loci coincide.