A cohomological study of local rings of embedding codepth 3

TL;DR

This paper computes the generating series of Bass numbers for certain local rings with embedding dimension and depth constraints, revealing growth patterns and restrictions on their algebraic structures.

Contribution

It provides explicit rational formulas for Bass number series and establishes new growth and structural restrictions for local rings of embedding codepth 3.

Findings

Bass number series are rational functions for rings with e-d ≤ 3.

Existence of a growth factor γ > 1 for Bass numbers in most cases.

Restrictions on multiplicative structures of minimal free resolutions of length 3.

Abstract

The generating series of the Bass numbers $\mu^i_R=\mathrm{rank}_k \mathrm{Ext}^i_R(k,R)$ of local rings $R$ with residue field $k$ are computed in closed rational form, in case the embedding dimension $e$ of $R$ and its depth $d$ satisfy $e-d\le 3$. For each such $R$ it is proved that there is a real number $\gamma>1$, such that $\mu^{d+i}_R\ge\gamma\mu^{d+i-1}_R$ holds for all $i\ge 0$, except for $i=2$ in two explicitly described cases, where $\mu^{d+2}_R=\mu^{d+1}_R=2$. New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings.