Top-stable and layer-stable degenerations and hom-order

S. O. Smalø ¹   ; A. Valenta ¹

¹ Department of Mathematical Sciences University of Science and Technology N-7491 Trondheim, Norway

Colloquium Mathematicum, Tome 108 (2007) no. 1, pp. 63-71

Résumé

Using geometrical methods, Huisgen-Zimmermann showed that if $M$ is a module with simple top, then $M$ has no proper degeneration $M$ 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 $M$ 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.

Détail

DOI : 10.4064/cm108-1-6

Keywords: using geometrical methods huisgen zimmermann showed module simple top has proper degeneration deg mathfrak mathfrak simeq mathfrak mathfrak given module square free top projective cover she showed dim mathop hom dim mathop hom only has proper degeneration deg where mathfrak simeq mathfrak prove here these results general form hom order instead degeneration order prove algebraically results huisgen zimmermann follow consequences results particular her second result holds just modules square free top indecomposable modules general