Top-stable and layer-stable degenerations and hom-order
Volume 108 / 2007
Abstract
Using geometrical methods, Huisgen-Zimmermann showed that if $M$ is a module with simple top, then $M$ has no proper degeneration $M<_{\deg} N$ such that $\mathfrak{r} ^tM/\mathfrak{r} ^{t+1}M\simeq \mathfrak{r} ^tN/\mathfrak{r} ^{t+1}N$ for all $t$. Given a module $M$ with square-free top and a projective cover $P$, she showed that $\dim_k\mathop{\rm Hom} (M,M)=\dim_k\mathop{\rm Hom} (P,M)$ if and only if $M$ has no proper degeneration $M<_{\deg}N$ where $M/\mathfrak{r} M\simeq N/\mathfrak{r} N$. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from our results. In particular, we find that her second result holds not just for modules with square-free top, but also for indecomposable modules in general.
