# Azumaya loci and discriminant ideals of PI algebras

## TL;DR

This paper establishes a connection between discriminant ideals and Azumaya loci in prime PI algebras, providing a classification of Azumaya loci and linking discriminant zero sets to singular loci under certain conditions.

## Contribution

It proves the equivalence of discriminant ideals and their modified versions, characterizes the Azumaya locus via discriminant ideals, and applies these results to classify Azumaya loci and relate discriminant zero sets to singular loci.

## Key findings

- Discriminant ideal zero set equals modified discriminant ideal zero set.
- Zero set of discriminant ideal at PI-degree squared is the Azumaya locus complement.
- Discriminant zero set coincides with the singular locus of the center under certain conditions.

## Abstract

We prove that, under mild assumptions, for all positive integers $\ell$, the zero set of the discriminant ideal $D_{\ell}(R/Z(R); tr)$ of a prime polynomial identity (PI) algebra $R$ coincides with the zero set of the modified discriminant ideal $MD_{\ell}(R/Z(R); tr)$ of $R$. Furthermore, we prove that, when $\ell$ is the square of the PI-degree of $R$, this zero set is precisely the complement of the Azumaya locus of $R$. This description is used to classify the Azumaya loci of the mutiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of the top discriminant ideal of a prime PI algebra $R$ coincides with the singular locus of the center of $R$, provided that the discriminant ideal has height at least 2, $R$ has finite global dimension and $R$ is a Cohen-Macaulay module over its center.