Degenerations and stable homomorphisms. (English) Zbl 1113.16024
Let \(\Lambda\) be a unital associative algebra over an algebraically closed field \(k\). Two relations are defined on finitely generated (f.g.) left \(\Lambda\)-modules: \(M\leq_{\hom}N\) iff \(\dim(X,M)\leq\dim(X,N)\) for all f.g. left \(\Lambda\)-modules; \(M\leq_{\text{stab}}N\) iff \(\dim\underline{(X,M)}\leq\dim\underline{(X,N)}\) for all f.g. left \(\Lambda\)-modules. Here \((X,Y)\) denotes the \(k\)-space of \(\Lambda\)-homomorphisms, and \(\underline{(X,Y)}\) is the \(k\)-space of \(\Lambda\)-stable homomorphisms.
The main results of the paper are: Let \(M,N\) be f.g. left \(\Lambda\)-modules with the same dimension vector. Then \(M\leq_{\hom}N\) implies \(M\leq_{\text{stab}}N\) (Proposition 1); The converse is true if \(\Lambda\) is hereditary (Proposition 2); There exists \(\Lambda\) such that the converse is not true (Proposition 3).
The main results of the paper are: Let \(M,N\) be f.g. left \(\Lambda\)-modules with the same dimension vector. Then \(M\leq_{\hom}N\) implies \(M\leq_{\text{stab}}N\) (Proposition 1); The converse is true if \(\Lambda\) is hereditary (Proposition 2); There exists \(\Lambda\) such that the converse is not true (Proposition 3).
Reviewer: Simion Sorin Breaz (Cluj-Napoca)
MSC:
16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
16G20 | Representations of quivers and partially ordered sets |
Keywords:
coverings; degenerations; representations of finite-dimensional algebras; stable module categories; finitely generated modulesReferences:
[1] | Auslander M., Illinois J. Math. 29 pp 280– (1985) |
[2] | DOI: 10.1017/CBO9780511623608 · doi:10.1017/CBO9780511623608 |
[3] | DOI: 10.1006/aima.1996.0053 · Zbl 0862.16007 · doi:10.1006/aima.1996.0053 |
[4] | DOI: 10.1007/BF01396624 · Zbl 0482.16026 · doi:10.1007/BF01396624 |
[5] | Gabriel P., Lecture Notes in Mathematics 831 |
[6] | Macdonald , I. D. ( 1968 ).The Theory of Groups. Oxford : Oxford University Press , p. 167 . · Zbl 0181.03501 |
[7] | Riedtmann C., Ann. Sci. Ecole Norm. Sup. 19 pp 275– (1986) |
[8] | DOI: 10.1090/S0002-9939-99-04714-0 · Zbl 0927.16008 · doi:10.1090/S0002-9939-99-04714-0 |
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