Test module filtrations for unit $F$-modules

TL;DR

This paper extends the concept of test module filtration to unit F-modules, compares it with Stadnik's V-filtration, and explores their behavior under various morphisms, providing new insights into their properties.

Contribution

It introduces a generalized test module filtration for unit F-modules and compares it with Stadnik's V-filtration, highlighting cases of coincidence and divergence.

Findings

Test module filtrations coincide with Stadnik's V-filtration in certain cases.

Filtrations do not always coincide, showing differences in general.

Test modules are preserved under smooth morphisms but not necessarily under finite flat, tamely ramified morphisms.

Abstract

We extend the notion of test module filtration introduced by Blickle for Cartier modules. We then show that this naturally defines a filtration on unit $F$-modules and prove that this filtration coincides with the notion of $V$-filtration introduced by Stadnik in the cases where he proved existence of his filtration. We also show that these filtrations do not coincide in general. Moreover, we show that for a smooth morphism $f: X \to Y$ test modules are preserved under $f^!$. We also give examples to show that this is not the case if $f$ is finite flat and tamely ramified along a smooth divisor.