Explicitly Extending Frobenius Splittings over Finite Maps

TL;DR

This paper provides an explicit criterion and proof for extending Frobenius splittings over finite maps of normal varieties in characteristic p, focusing on tamely ramified cases and exploring additional examples.

Contribution

It offers a highly explicit, term-by-term proof of Frobenius splitting extension criteria for tamely ramified finite maps, enhancing previous theoretical results.

Findings

Explicit extension criterion for Frobenius splittings in tamely ramified cases

Term-by-term proof method for Frobenius splitting extension

Additional examples illustrating the criterion

Abstract

Suppose that $\pi \: Y \to X$ is a finite map of normal varieties over a perfect field of characteristic $p > 0$. Previous work of the authors gave a criterion for when Frobenius splittings on $X$ (or more generally any $p^{-e}$-linear map) extend to $Y$. In this paper we give an alternate and highly explicit proof of this criterion (checking term by term) when $\pi$ is tamely ramified in codimension 1. Some additional examples are also explored.