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

Tom 108 / 2007

S. O. Smalø, A. Valenta Colloquium Mathematicum 108 (2007), 63-71 MSC: 14L30, 16D10, 16E30, 16G10, 16G20. DOI: 10.4064/cm108-1-6

Streszczenie

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.

Autorzy

  • S. O. SmaløDepartment of Mathematical Sciences
    University of Science and Technology
    N-7491 Trondheim, Norway
    e-mail
  • A. ValentaDepartment of Mathematical Sciences
    University of Science and Technology
    N-7491 Trondheim, Norway
    e-mail

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