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Hilbert-Kunz functions and Frobenius functors

Author: Shou-Te Chang
Journal: Trans. Amer. Math. Soc. 349 (1997), 1091-1119
MSC (1991): Primary 13A35; Secondary 13D03, 13D05, 13D25, 13D45, 18G15, 18G40
MathSciNet review: 1370637
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Abstract: We study the asymptotic behavior as a function of $e$ of the lengths of the cohomology of certain complexes. These complexes are obtained by applying the $e$-th iterated Frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module. The rings involved here all have positive prime characteristic. For the zeroth homology, these functions also contain the class of Hilbert-Kunz functions that a number of other authors have studied. This asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper: these are defined in terms of the Krull dimensions of the Matlis duals of the local cohomology of the module. There is a more detailed study of this behavior when the given complex is a Koszul complex.

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Additional Information

Shou-Te Chang
Affiliation: Department of Mathematics, National Chung Cheng University, Minghsiung, Chiayi 621, Taiwan, R.O.C.

Received by editor(s): August 20, 1995
Additional Notes: Part of this work was done at the University of Michigan. The author would like to thank Professor Melvin Hochster for his many useful suggestions. The author is also partially supported by a grant from the National Science Council of R. O. C
Article copyright: © Copyright 1997 American Mathematical Society

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