Minimal γ-sheaves

Blickle, Manuel

doi:10.2140/ant.2008.2.347

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Minimal $\gamma$-sheaves

Manuel Blickle

Vol. 2 (2008), No. 3, 347–368

DOI: 10.2140/ant.2008.2.347

Abstract

In a seminal work Lyubeznik [1997] introduces a category $F$ -finite modules in order to show various finiteness results of local cohomology modules of a regular ring $R$ in positive characteristic. The key notion on which most of his arguments rely is that of a generator of an $F$ -finite module. This may be viewed as an $R$ finitely generated representative for the generally nonfinitely generated local cohomology modules. In this paper we show that there is a functorial way to choose such an $R$ -finitely generated representative, called the minimal root, thereby answering a question that was left open in Lyubeznik’s work. Indeed, we give an equivalence of categories between $F$ -finite modules and a category of certain $R$ -finitely generated modules with a certain Frobenius operation which we call minimal $γ$ -sheaves.

As immediate applications we obtain a globalization result for the parameter test module of tight closure theory and a new interpretation of the generalized test ideals of Hara and Takagi [2004] which allows us to easily recover the rationality and discreteness results for $F$ -thresholds of Blickle et al. [2008].

Keywords

positive characteristic, D-module, F-module, Frobenius operation

Mathematical Subject Classification 2000

Primary: 13A35

Milestones

Received: 10 December 2007

Revised: 13 February 2008

Accepted: 2 March 2008

Published: 1 May 2008

Authors

Manuel Blickle
Mathematik Essen Universität Duisburg-Essen 45117 Essen Germany
http://www.mabli.org

Vol. 2, No. 3, 2008