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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
DOI: https://doi.org/10.1090/S0002-9947-97-01704-2
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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  • [BE] D. Buchsbaum and D. Eisenbud, What makes a complex exact? J. of Algebra 25 (1973), 259-268. MR 47:3369
  • [Cc] A. Conca, Hilbert-Kunz function of monomial ideals and binomial hypersurfaces, preprint.
  • [Ct] M. Contessa, On the Hilbert-Kunz function and Koszul homology, J. of Algebra 175 (1995), 757-766; 180 (1966), 321-322. MR 96e:13018; CMP 1996:8
  • [D1] S. P. Dutta, Frobenius and multiplicities, J. of Algebra 85 (1983), 424-448. MR 85f:13022
  • [D2] -, Exact and Frobenius, J. of Algebra 127 (1989), 163-177. MR 91g:13018
  • [F] W. Fulton, Intersection Theory, Springer-Verlag, 1984. MR 85k:14004
  • [GH] A. Grothendieck (notes by R. Hartshorne), Local Cohomology, Lecture Notes in Math. No. 41, Springer-Verlag, 1967. MR 37:219
  • [He] J. Herzog, Ringe der Charakteristik p und Frobeniusfunktoren, Math. Z. 140 (1974), 67-78. MR 50:4569
  • [HH1] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. A.M.S. 3 (1990), 31-116. MR 91g:13010
  • [HH2] -, Tight closure and strong F-regularity, Mém. Soc. Math. France 38 (concré au colloque en l'honneur de P. Samuel) (1989), 119-133. MR 91i:13025
  • [HH3] -, Phantom homology, Mem. Amer. Math. Soc. 103 (1993), no. 490. MR 93j:13020
  • [HH4] -, F-regularity, test elements and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1-62. MR 95d:13007
  • [HM] C. Han and P. Monsky, Some surprising Hilbert-Kunz functions, Math. Z. 214 (1993), 119-135. MR 94f:13008
  • [K1] E. Kunz, Characterizations of regular local rings for characteristic p, Amer. J. Math. 91 (1969), 772-784. MR 40:5609
  • [K2] -, On Noetherian rings of characteristic p, Amer. J. Math. 98 (1976), 999-1013. MR 55:5612
  • [Mac] S. MacLane, Homology, Classics in Mathematics, Springer-Verlag, 1995. MR 96d:18001
  • [Mat1] H. Matsumura, Commutative Algebra, Benjamin/Cummings, Reading, Ma., second edition, 1980. MR 82i:13003
  • [Mat2] -, Commutative Ring Theory, Cambridge Univ. Press, 1986. MR 88h:13001
  • [Mo1] P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), 43-49. MR 84k:13012
  • [Mo2] -, Fine estimates for the growth of $e_{n}$ in $\mathbb {Z} ^{d}_{p}$-extensions, Algebraic Number Theory (I. Iwasawa 70th Birthday Vol.), Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, and Kinokuniya, Tokyo, 1989, pp. 309-330. MR 90e:11120
  • [N] D.G. Northcott, Finite Free Resolutions, Cambridge Univ. Press, 1976. MR 57:377
  • [PS] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47-119. MR 51:10330
  • [R1] P. Roberts, Le théorème d'intersection, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 177-180. MR 89b:14008
  • [R2] -, Intersection theorems, Commutative Algebra, (M. Hochster et al., eds.), Math. Sci. Res. Inst. Publ., vol. 15, Springer-Verlag, 1989, pp. 417-436. MR 90j:13024
  • [Sei] G. Seibert, Complexes with homology of finite length and Frobenius functors, J. of Algebra 125 (1989), 278-287. MR 90j:13012
  • [Ser] J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Math., no. 11, Springer-Verlag, 1965. MR 34:1352
  • [St] J. Strooker, Homological Questions in Local Algebra, London Math. Soc. Lecture Note Ser., No. 145, Cambridge Univ. Press, 1990. MR 91m:13013
  • [Sz] L. Szpiro, Sur la théorie des complexes parfaits, Commutative Algebra (Durham, 1981), London Math. Soc. Lecture Note Ser., No. 72, Cambridge Univ. Press, 1982, pp. 83-90. MR 84m:13015

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Additional Information

Shou-Te Chang
Affiliation: Department of Mathematics, National Chung Cheng University, Minghsiung, Chiayi 621, Taiwan, R.O.C.
Email: stchang@math.ccu.edu.tw

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