Algebra & Number Theory
- Algebra Number Theory
- Volume 2, Number 3 (2008), 347-368.
In a seminal work Lyubeznik  introduces a category -finite modules in order to show various finiteness results of local cohomology modules of a regular ring in positive characteristic. The key notion on which most of his arguments rely is that of a generator of an -finite module. This may be viewed as an 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 -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 -finite modules and a category of certain -finitely generated modules with a certain Frobenius operation which we call minimal -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  which allows us to easily recover the rationality and discreteness results for -thresholds of Blickle et al. .
Algebra Number Theory, Volume 2, Number 3 (2008), 347-368.
Received: 10 December 2007
Revised: 13 February 2008
Accepted: 2 March 2008
First available in Project Euclid: 20 December 2017
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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