Coherent functors, powers of ideals, and asymptotic stability

Abstract

Let $R$ be a Noetherian ring, $I_1,\dots ,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S={⨁}_{\underline{n}\in {\mathbb{N}}^r}{S}_{\underline{n}}$ be a Noetherian standard ${\mathbb{N}}^r$-graded ring with $S_{\underline{0}}=R$, and $\mathcal{M}$ be a finitely generated ${\mathbb{Z}}^r$-graded $S$-module. For $\underline{n}=(n_1,\dots ,n_r)\in {\mathbb{N}}^r$, set $G_{\underline{n}}:={\mathcal{M}}_{\underline{n}}$ or $G_{\underline{n}}:=M/{\mathbf{I}}^{\underline{n}}N$, where ${\mathbf{I}}^{\underline{n}}=I_1^{n_1}\cdots I_r^{n_r}$. Suppose $F$ is a coherent functor on the category of finitely generated $R$-modules. We prove that the set ${\mathrm{Ass}}_{\mathrm{R}}(\mathrm{F}({\mathrm{G}}_{\underline{\mathrm{n}}}))$ of associate primes and $\mathrm{grade}(\mathrm{J},\mathrm{F}({\mathrm{G}}_{\underline{\mathrm{n}}}))$ stabilize for all $\underline{n}\gg 0$, where $J$ is a non-zero ideal of $R$. Furthermore, if the length ${\lambda }_R(F(G_{\underline{n}}))$ is finite for all $\underline{n}\gg 0$, then there exists a polynomial $P$ in $r$ variables over $\mathbb{Q}$ such that ${\lambda }_R(F(G_{\underline{n}}))=P(\underline{n})$ for all $\underline{n}\gg 0$. When $R$ is a local ring, and $G_{\underline{n}}=M/{\mathbf{I}}^{\underline{n}}N$, we give a sharp upper bound of the total degree of $P$. As applications, when $R$ is a local ring, we show that for each fixed $i\ge 0$, the $i$th Betti number ${\beta }_i^R(F(G_{\underline{n}}))$ and Bass number ${\mu }_R^i(F(G_{\underline{n}}))$ are given by polynomials in $\underline{n}$ for all $\underline{n}\gg 0$. Thus, in particular, the projective dimension ${\mathrm{pd}}_{\mathrm{R}}(\mathrm{F}({\mathrm{G}}_{\underline{\mathrm{n}}}))$ (resp., injective dimension ${\mathrm{id}}_{\mathrm{R}}(\mathrm{F}({\mathrm{G}}_{\underline{\mathrm{n}}}))$) is constant for all $\underline{n}\gg 0$.

arXiv comment9 pages, comments and suggestions are welcome!