Finitely Presented Functors and Degenerations# |
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| Authors: | S. O. Smal⊘ A. Valenta |
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| Affiliation: | 1. Department of Mathematics , Norwegian University of Science and Technology , Trondheim, Norway sverresm@math.ntnu.no;3. Department of Mathematics , Norwegian University of Science and Technology , Trondheim, Norway |
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| Abstract: | Given two d-dimensional Λ-modules M and N, then M degenerates to N if and only if there exists an exact sequence of the form 0→ U→ U ⊕ M→ N→ 0 for some U ? mod Λ (Zwara, 1998 Zwara , G. ( 1998 ). A degeneration-like order for modules . Arch. Math. 71 : 437 – 444 . [CSA] [CROSSREF] [Google Scholar]). Having this as a starting point, in this article we give a characterization of degenerations by the existence of a certain finitely presented functor. This gives new information about U in the sequence above. We show how this new information can be used to prove that M deg N even when M ⊕ X ≤ deg N ⊕ X for some X for modules over Λ q = k〈 x,y〉/〈 x 2,y 2,xy + qyx〉, q ≠ 0 in k, where k is an algebraically closed field. |
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| Keywords: | Degenerations of representations Finitely presented functors Representations of algebras |
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