$p^{-1}$-linear maps in algebra and geometry

TL;DR

This paper surveys the properties of $p^{-e}$-linear endomorphisms in algebraic geometry, highlighting their significance in commutative algebra, local cohomology, test ideals, and geometric applications like vanishing theorems.

Contribution

It provides a comprehensive overview of $p^{-e}$-linear maps, emphasizing their roles in both algebraic and geometric contexts, and discusses recent developments and applications.

Findings

$p^{-e}$-linear maps are crucial in understanding test ideals and Frobenius actions.

These maps have significant implications for vanishing theorems and lifting sections.

The survey connects algebraic properties with geometric applications in positive characteristic.

Abstract

In this article we survey the basic properties of $p^{-e}$-linear endomorphisms of coherent $\O_X$-modules, i.e. of $\O_X$-linear maps $F_* \sF \to \sG$ where $\sF,\sG$ are $\O_X$-modules and $F$ is the Frobenius of a variety of finite type over a perfect field of characteristic $p > 0$. We emphasize their relevance to commutative algebra, local cohomology and the theory of test ideals on the one hand, and global geometric applications to vanishing theorems and lifting of sections on the other.