A cohomological study of local rings of embedding codepth 3

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

No reviews yet.

No comments yet.