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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Homological invariants of powers of an ideal


Author: Vijay Kodiyalam
Journal: Proc. Amer. Math. Soc. 118 (1993), 757-764
MSC: Primary 13H15; Secondary 13D40
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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1156471-5
PII: S 0002-9939(1993)1156471-5
Article copyright: © Copyright 1993 American Mathematical Society