Algebra & Number Theory

Minimal $\gamma$-sheaves

Manuel Blickle

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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

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]

positive characteristic D-module F-module Frobenius operation


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

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  • J. Alvarez-Montaner, M. Blickle, and G. Lyubeznik, “Generators of $D$-modules in positive characteristic”, Math. Res. Lett. 12:4 (2005), 459–473.
  • M. Blickle, “The intersection homology $D$-module in finite characteristic”, Math. Ann. 328:3 (2004), 425–450.
  • M. Blickle and G. B öckle, “Cartier crystals”. In preparation.
  • M. Blickle, M. Mustaţǎ, and K. E. Smith, “F-thresholds of hypersurfaces”, Trans AMS (2008). To appear.
  • M. Emerton and M. Kisin, “The Riemann–Hilbert correspondence for unit $F$-crystals”, Astérisque 293 (2004), vi+257.
  • N. Hara and S. Takagi, “On a generalization of test ideals”, Nagoya Math. J. 175 (2004), 59–74.
  • R. Hartshorne and R. Speiser, “Local cohomological dimension in characteristic $p$”, Ann. of Math. $(2)$ 105:1 (1977), 45–79.
  • N. M. Katz, Rigid local systems, Annals of Mathematics Studies 139, Princeton University Press, Princeton, NJ, 1996.
  • M. Katzman, G. Lyubeznik, and W. Zhang, “On the discreteness and rationality of jumping coefficients”, preprint, 2007.
  • E. Kunz, “Characterizations of regular local rings for characteristic $p$”, Amer. J. Math. 91 (1969), 772–784.
  • G. Lyubeznik, “$F$-modules: applications to local cohomology and $D$-modules in characteristic $p>0$”, J. Reine Angew. Math. 491 (1997), 65–130.
  • C. Miller, Cohomology of p-torsion sheaves on characteristic-p curves, PhD Thesis, UC Berkeley, 2007.
  • S. Takagi and R. Takahashi, “$D$–modules over rings of $F$–finite representation type”, preprint, 2007.