Homological invariants of powers of an ideal
HTML articles powered by AMS MathViewer
- by Vijay Kodiyalam
- Proc. Amer. Math. Soc. 118 (1993), 757-764
- DOI: https://doi.org/10.1090/S0002-9939-1993-1156471-5
- PDF | Request permission
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
- Luchezar L. Avramov, Small homomorphisms of local rings, J. Algebra 50 (1978), no. 2, 400–453. MR 485906, DOI 10.1016/0021-8693(78)90163-1
- M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 35–39. MR 530808, DOI 10.1017/S030500410000061X
- J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
- Tor Holtedahl Gulliksen, A proof of the existence of minimal $R$-algebra resolutions, Acta Math. 120 (1968), 53–58. MR 224607, DOI 10.1007/BF02394606
- Craig Huneke, On the symmetric and Rees algebra of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), no. 2, 268–275. MR 563225, DOI 10.1016/0021-8693(80)90179-9
- Gerson Levin, Poincaré series of modules over local rings, Proc. Amer. Math. Soc. 72 (1978), no. 1, 6–10. MR 503520, DOI 10.1090/S0002-9939-1978-0503520-2
- D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, London, 1968. MR 0231816
- D. G. Northcott, Grade sensitivity and generic perfection, Proc. London Math. Soc. (3) 20 (1970), 597–618. MR 272771, DOI 10.1112/plms/s3-20.4.597
- D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145–158. MR 59889, DOI 10.1017/s0305004100029194
Bibliographic Information
- © Copyright 1993 American Mathematical Society
- 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