Test elements in equal characteristic semianalytic algebras

TL;DR

This paper proves that the Jacobian ideal is contained within the test ideal in various algebraic settings, advancing understanding of singularities in equal characteristic semianalytic algebras.

Contribution

It establishes new results linking Jacobian ideals and test ideals in equal characteristic settings, including characteristic zero and positive characteristic.

Findings

Jacobian ideal is contained in the test ideal in characteristic p for complete rings.

Jacobian ideals are contained in test ideals in characteristic zero for various algebraic structures.

New techniques for analyzing singularities in semianalytic algebras.

Abstract

We establish a series of results showing that the Jacobian ideal is contained in the test ideal. We first prove a new result in characteristic $p$ for complete rings over a field $K$. Then we prove some results showing that Jacobian ideals are contained in test ideals in equal characteristic 0 for affine, analytic, affine-analytic and semianalytic $K$-algebras.