A cohomological study of local rings of embedding codepth 3

Abstract

The generating series of the Bass numbers ${\mu }_R^i={\mathrm{rank}}_k{\mathrm{Ext}}_R^i(k,R)$ 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 ${\mu }_R^{d+i}\ge \gamma {\mu }_R^{d+i-1}$ holds for all $i\ge 0$, except for $i=2$ in two explicitly described cases, where ${\mu }_R^{d+2}={\mu }_R^{d+1}=2$. New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings.

In version 2 numerous typos have been corrected, details have been added in a few places, and local rearrangements have been made. To appear in JPAA. 24 pages

References in corpus (1)

• Growth in the minimal injective resolution of a local ring