Slicing criterion for ind-smooth ring maps

TL;DR

This paper introduces a method to test ind-smoothness of flat ring maps using constructible stratifications, extending the concept to animated rings and providing a practical criterion for non-Noetherian cases.

Contribution

It generalizes ind-smoothness and ind-lci to ind-$d$-smoothness, enabling testing on constructible stratifications for a broader class of rings.

Findings

Ind-smoothness can be tested on constructible stratifications.

Generalization of ind-smoothness to ind-$d$-smoothness.

Applicable to non-Noetherian rings.

Abstract

We show that ind-smoothness of flat ring maps can be tested on constructible stratifications, even for maps of non-Noetherian rings. We prove this by generalizing ind-smoothness and ind-lci to a sequence of conditions on animated ring maps called ind-$d$-smoothness, and showing all at once that they can be tested on constructible stratifications.