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

Blickle, Manuel

doi:10.1090/S1056-3911-2012-00576-1

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

Author: Manuel Blickle
Journal: J. Algebraic Geom. 22 (2013), 49-83
DOI: https://doi.org/10.1090/S1056-3911-2012-00576-1
Published electronically: March 6, 2012
MathSciNet review: 2993047
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Abstract | References | Additional Information

Abstract:

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 $F$-pure modules, over algebras of $p^{-e}$-linear operators. We develop the basics of a theory of $F$-pure modules and show an important structural result, namely that $F$-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 $p^{-e}$-linear operators yields an elementary approach to test ideals in the case of affine $k$-algebras, where $k$ is an $F$-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 $\mathbb {Q}$-Gorenstein case.

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Additional Information

Manuel Blickle
Affiliation: Johannes Gutenberg Universität Mainz, FB08 Institut für Mathematik, 55099 Mainz, Germany
Email: blicklem@uni-mainz.de

Received by editor(s): March 29, 2010
Received by editor(s) in revised form: August 25, 2010, and September 20, 2010
Published electronically: March 6, 2012
Additional Notes: The research for this paper was conducted while I was supported by a Heisenberg Fellowship of the DFG and by the SFB/TRR45.