Test ideals via algebras of $p^{-e}$-linear maps

TL;DR

This paper introduces a new perspective on test ideals using $p^{-e}$-linear maps, simplifying their study and extending their definition to non-reduced rings, with elementary proofs of key properties in affine $k$-algebras.

Contribution

It provides a generalized, simplified framework for understanding test ideals via $p^{-e}$-linear maps, including in non-reduced settings, and offers elementary proofs of their properties.

Findings

Test ideals can be characterized as minimal $F$-pure modules in a new algebraic framework.

The approach extends the definition of test ideals to non-reduced rings.

Elementary proofs of discreteness of jumping numbers in affine $k$-algebras are obtained.

Abstract

Continuing ideas of a recent preprint of Schwede arXiv:0906.4313 we study test ideals by viewing them as minimal objects in a certain class of $F$-pure modules over algebras of p^{-e}-linear operators. This shift in the viewpoint leads to a simplified and generalized treatment, also allowing us to define test ideals in non-reduced settings. In combining this with an observation of Anderson on the contracting property of p^{-e}-linear operators we obtain an elementary approach to test ideals in the case of affine k-algebras, where k is an F-finite field. It also yields a short and completely elementary proof of the discreteness of their jumping numbers extending most cases where the discreteness of jumping numbers was shown in arXiv:0906.4679.