Vol. 2, No. 3, 2008

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Minimal $\gamma$-sheaves

Manuel Blickle

Vol. 2 (2008), No. 3, 347–368

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].

positive characteristic, D-module, F-module, Frobenius operation
Mathematical Subject Classification 2000
Primary: 13A35
Received: 10 December 2007
Revised: 13 February 2008
Accepted: 2 March 2008
Published: 1 May 2008
Manuel Blickle
Mathematik Essen
Universität Duisburg-Essen
45117 Essen