Homological invariants of powers of an ideal

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 $i$.