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Proceedings of the American Mathematical Society

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Homological invariants of powers of an ideal


Author: Vijay Kodiyalam
Journal: Proc. Amer. Math. Soc. 118 (1993), 757-764
MSC: Primary 13H15; Secondary 13D40
DOI: https://doi.org/10.1090/S0002-9939-1993-1156471-5
MathSciNet review: 1156471
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Abstract: For any fixed nonnegative integer $ i$ and all sufficiently large $ n$, the following are shown to be polynomials in $ n$:

(1) The $ i$th Betti number, $ \beta _i^{{\mathcal{R}_0}}({\mathcal{M}_n})$, and the $ i$th Bass number, $ \mu _{{\mathcal{R}_0}}^i({\mathcal{M}_n})$, where $ \mathcal{M} = { \oplus _{n \geqslant 0}}{\mathcal{M}_n}$ is a finitely generated graded module over a Noetherian graded ring $ \mathcal{R} = {\mathcal{R}_0}[{\mathcal{R}_1}]$ with $ {\mathcal{R}_0}$ local.

(2) The lengths, $ {\lambda _R}(\operatorname{Tor} _i^R(M/{I^n}M,Q))$ and $ {\lambda _R}(\operatorname{Ext} _R^i(Q,M/{I^n}M))$, for an ideal $ I$ of a Noetherian ring $ R$ and finitely generated $ R$-modules $ M,\;Q$ with $ M{ \otimes _R}Q$ of finite length.

(3) The minimal number of generators, $ {\nu _R}(\operatorname{Tor} _i^R(M/{I^n}M,Q))$ and $ {\nu _R}(\operatorname{Ext} _R^i(Q,M/{I^n}M))$, where $ I$ is an ideal of a Noetherian local ring $ R$ and $ M,\;Q$ are finitely generated $ R$-modules.

It is also shown that the degrees of these polynomials are bounded by constants independent of $ i$.


References [Enhancements On Off] (What's this?)

  • [1] L. L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), 400-453. MR 485906 (81i:13009)
  • [2] M. Brodmann, The asymptotic nature of analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), 35-39. MR 530808 (81e:13003)
  • [3] J. Eagon and D. G. Northcott, Ideals defined by matrices and a complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188-204. MR 0142592 (26:161)
  • [4] T. H. Gulliksen, On the existence of minimal $ R$-algebra resolutions, Acta. Math. 120 (1968), 53-58. MR 0224607 (37:206)
  • [5] C. Huneke, On the symmetric and Rees algebra of an ideal generated by a $ d$-sequence, J. Algebra 62 (1980), 268-275. MR 563225 (81d:13016)
  • [6] G. Levin, Poincare series of modules over local rings, Proc. Amer. Math. Soc. 72 (1978), 6-10. MR 503520 (81b:13009)
  • [7] D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge Univ. Press, Cambridge and New York, 1968. MR 0231816 (38:144)
  • [8] -, Grade sensitivity and generic perfection, Proc. London Math. Soc. (3) 20 (1970), 597-618. MR 0272771 (42:7652)
  • [9] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 0059889 (15:596a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1156471-5
Article copyright: © Copyright 1993 American Mathematical Society

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