The gamma-construction and permanence properties of the (relative) $F$-rational signature

TL;DR

This paper investigates the behavior of the relative F-rational signature under gamma-construction, extending known results from the F-finite case and exploring properties of the gamma-construction itself.

Contribution

It demonstrates the compatibility of the relative F-rational signature with gamma-construction and extends results from the F-finite case to a broader setting.

Findings

F-rational signature is compatible with gamma-construction

Extended results from F-finite to more general cases

Explored properties of gamma-construction that may be of independent interest

Abstract

We study some permanence properties of the relative $F$-rational signature defined and studied by Smirnov--Tucker. We show that this invariant is compatible with the gamma-construction, and then derive other main results from the $F$-finite case established by Smirnov--Tucker. We also obtain limited results about the $F$-rational signature defined and studied by Hochster--Yao. We explore some features of the gamma-construction along the way, which may be of independent interest.