On Hilbert-Kunz functions of some hypersurfaces. (English) Zbl 0898.13011
Let \((A,\mathfrak m)\) denote a complete local \(\mathbb Z/(p)\)-algebra, where \(p\) denotes a prime number. Let \(e_n(A)=\text{length}(A/\mathfrak m_n),\) where \(\mathfrak m_n\) is the ideal of \(A\) generated by all \(a^q\), \(q=p^n\), with \(a\in\mathfrak m\), denote the (local) Hilbert-Kunz function. It was shown by P. Monsky in Math. Ann. 263, 43-49 (1983; Zbl 0509.13023) that \(e_n(A)=cp^{dn}+\Delta_n\), where \(d=\dim A\) and \(\Delta_n=O(p^{n(d-1)})\). Put \(B=\mathbb Z/(p)[[X_1, \ldots, X_s]]\) and \(A=B/fB\). In the case of \(f=X_1^{d_1}-\prod_{i=2}^s X_i^{d_i}\) (resp. \(f = \sum_{i=1}^s X_i^{d_i}\)) it was shown by E. Kunz in Am. J. Math. 98, 999-1013 (1976; Zbl 0341.13009), resp. by C. Han and P. Monsky in Math. Z. 214, 119-135 (1993; Zbl 0788.13008) that \(c\) is rational and \(\Delta_n\) is eventually periodic. In the main result of the paper under review this is generalized to the case of \(f\) a certain sum of powers of squarefree monomials. As a main technical tool the authors use the representation ring developed by C. Han and P. Monsky (loc. cit.). Moreover this generalization covers also the a result of A. Conca shown in Manuscr. Math. 90, No. 3, 287-300 (1996; Zbl 0882.13019).
Reviewer: P.Schenzel (Halle)
MSC:
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
| 14J70 | Hypersurfaces and algebraic geometry |
References:
| [1] | S. T. Chang, The Asymptotic Behaviour of Hilbert-Kunz Functions and Their Generalizations, University of Michigan, 1992; S. T. Chang, The Asymptotic Behaviour of Hilbert-Kunz Functions and Their Generalizations, University of Michigan, 1992 |
| [2] | L. Chiang, Hilbert-Kunz Functions, National Taiwan Normal University, 1996; L. Chiang, Hilbert-Kunz Functions, National Taiwan Normal University, 1996 |
| [3] | Chiang, L.; Hung, Y. C., On Hilbert-Kunz function and representation ring, Bull. Inst. Math. Acad. Sinica (1998) · Zbl 0904.13007 |
| [4] | Conca, A., Hilbert-Kunz function of monomial ideals and binomial hypersurfaces, Manuscripta Math., 90, 287-300 (1996) · Zbl 0882.13019 |
| [5] | Contessa, M., On the Hilbert-Kunz function and Koszul homology, J. Algebra, 175, 757-766 (1995) · Zbl 0829.13004 |
| [6] | Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms (1992), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelburg · Zbl 0756.13017 |
| [7] | C. Han, The Hilbert-Kunz Function of a Diagonal Hypersurface, Brandeis University, 1992; C. Han, The Hilbert-Kunz Function of a Diagonal Hypersurface, Brandeis University, 1992 |
| [8] | Han, C.; Monsky, P., Some surprising Hilbert-Kunz functions, Math. Z., 214, 119-135 (1993) · Zbl 0788.13008 |
| [9] | Kunz, E., Characterizations of regular local rings of characteristic \(p\), Amer. J. Math., 41, 772-784 (1969) · Zbl 0188.33702 |
| [10] | Kunz, E., On Noetherian rings of characteristic \(p\), Amer. J. Math., 98, 999-1013 (1976) · Zbl 0341.13009 |
| [11] | Monsky, P., The Hilbert-Kunz function, Math. Ann., 263, 43-49 (1993) · Zbl 0509.13023 |
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