## Algebra & Number Theory

### Minimal $\gamma$-sheaves

Manuel Blickle

#### 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].

#### Article information

Source
Algebra Number Theory, Volume 2, Number 3 (2008), 347-368.

Dates
Revised: 13 February 2008
Accepted: 2 March 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797254

Digital Object Identifier
doi:10.2140/ant.2008.2.347

Mathematical Reviews number (MathSciNet)
MR2407119

Zentralblatt MATH identifier
1183.13005

#### Citation

Blickle, Manuel. Minimal $\gamma$-sheaves. Algebra Number Theory 2 (2008), no. 3, 347--368. doi:10.2140/ant.2008.2.347. https://projecteuclid.org/euclid.ant/1513797254

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