On the rate of graded modules

TL;DR

This paper establishes upper bounds for the rate of graded modules over certain Cohen-Macaulay local rings, linking the growth of minimal free resolutions to algebraic invariants like multiplicity.

Contribution

It provides new upper bounds for the rate of graded modules in Cohen-Macaulay local rings with specific multiplicity conditions.

Findings

Rate of modules over Cohen-Macaulay rings is bounded by t-1.

The rate of the associated graded ring equals t-1 under given conditions.

Modules annihilated by minimal reductions have rate at most t-1.

Abstract

Let $K$ be a field, $R$ a standard graded $K$-algebra and $M$ be a finitely generated graded $R$-module. The rate of $M$, $rate_R(M)$, is a measure of the growth of the shifts in the minimal graded free resolution of $M$. In this paper, we find upper bounds for this invariant. More precisely, let $(A,\mathfrak{n})$ be a regular local ring and $I\subseteq \mathfrak{n} ^t$ be an ideal of $A$, where $t\geq 2$. We prove that if $(B=A/I, \mathfrak{m} =\mathfrak{n} /I)$ is a Cohen-Macaulay local ring with multiplicity $e(B)= \binom{h+t-1}{h}$, where $h=embdim(B)-dim B$, then $rat(gr_{\mathfrak{m}}(B))=t-1$ and for every $B$-module $N$, which annihilated by a minimal reduction of $\mathfrak{m}$, $rate_{gr_{\mathfrak{m}}(B)}(gr_{\mathfrak{m}}(N))\leq t-1$.