Hilbert-Kunz functions and Frobenius functors

Chang, Shou-Te

doi:10.1090/S0002-9947-97-01704-2

Hilbert-Kunz functions and Frobenius functors
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by Shou-Te Chang

Trans. Amer. Math. Soc. 349 (1997), 1091-1119

DOI: https://doi.org/10.1090/S0002-9947-97-01704-2

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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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