Derived functors and Hilbert polynomials over Gorenstein rings

Abstract

Let $(A,\mathfrak{m},k)$ be a Gorenstein ring of dimension $d\ge 1$, $N$ a perfect module of dimension $t\ge 1$ and $I$ an ideal of definition of $N$. For a non-free maximal Cohen-Macaulay (=MCM) $A$-module $M$ and an integer $i\ge 1$, it is well known that the functions $n\mapsto \ell (To{r}_i^A(M,N/I^{n+1}N))$ and $n\mapsto \ell (Ex{t}_A^i(M,N/I^{n+1}N))$ are of polynomial types of degrees $r_i^{I,N}(M)$ and $s_{I,N}^i(M)$, respectively. We prove that $r_i^{I,N}(M)\le t-1$ and $s_{I,N}^i(M)\le t-1$ and when $I$ is the maximal ideal $\mathfrak{m}$, both the inequalities become equalities. We also show that $r_i^{I,N}(M)\le r_1^{I,N}(\hat{I}{\text{c◯}}^{d}k)$, $s_{I,N}^i(M)\le s_{I,N}^1(\hat{I}{\text{c◯}}^{d}k)$ and $r_i^{I,N}(\hat{I}{\text{c◯}}^{d}k)=r_1^{I,N}(\hat{I}{\text{c◯}}^{d}k)=s_{I,N}^1(\hat{I}{\text{c◯}}^{d}k)=s_{I,N}^i(\hat{I}{\text{c◯}}^{d}k)$. \end