Principal gradient schemes have regular reduced closed subschemes

TL;DR

This paper proves that principal gradient schemes possess regular reduced subschemes and establishes a regularity criterion for reduced quotient rings, advancing understanding of their structural properties.

Contribution

It introduces a new regularity criterion for reduced quotient rings and proves that principal gradient schemes have regular reduced subschemes.

Findings

Principal gradient schemes have regular reduced subschemes.

A new regularity criterion for reduced quotient rings is established.

The results deepen the understanding of the structure of gradient schemes.

Abstract

We prove that principal gradient schemes have regular reduced subschemes. We also obtain a regularity criterion for reduced quotient rings.