6533b834fe1ef96bd129d55e

RESEARCH PRODUCT

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

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description

Building on previous work of Schwede, Böckle, and the author, we study test ideals by viewing them as minimal objects in a certain class of modules, called F F -pure modules, over algebras of p − e p^{-e} -linear operators. We develop the basics of a theory of F F -pure modules and show an important structural result, namely that F F -pure modules have finite length. This result is then linked to the existence of test ideals and leads to a simplified and generalized treatment, also allowing us to define test ideals in non-reduced settings. Combining our approach with an observation of Anderson on the contracting property of p − e p^{-e} -linear operators yields an elementary approach to test ideals in the case of affine k k -algebras, where k k is an F F -finite field. As a byproduct, one obtains a short and completely elementary proof of the discreteness of the jumping numbers of test ideals in a generality that extends most cases known so far; in particular, one obtains results beyond the Q \mathbb {Q} -Gorenstein case.