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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Preprojective partitions and the determinant of the Hom matrix

Authors: K. Igusa and G. Todorov
Journal: Proc. Amer. Math. Soc. 94 (1985), 189-197
MSC: Primary 16A64; Secondary 16A46, 16A60
MathSciNet review: 784160
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Abstract: If $ \Lambda $ is an artin algebra and $ \Lambda $ is the set of isomorphism classes of indecomposable finitely generated $ \Lambda $-modules, then there is a partition $ \operatorname{ind}\Lambda = { \cup _{i \geqslant 0}}{\underline{\underline P} _i}$, called the preprojective partition. We give an algorithm for computing this partition, which is given only in terms of numerical properties of the Auslander-Reiten quiver of $ \Lambda $.

If $ \Lambda $ is of finite representation type, then there are two essentially different proofs that the matrix $ \operatorname{Hom} = ({\text{lengt}}{{\text{h}}_{{\text{End(}}N)/{\text{rEnd(}}N{\text{)}}}}\operatorname{Hom}_{\Lambda }(M,N))$, where $ M,n \in \operatorname{ind}\Lambda $ has determinant $ + 1[{\mathbf{IT1}},{\mathbf{W1}},{\mathbf{Z1}}]$. We show that the paths between the Hom matrix and the identity matrix in $ \operatorname{GL}_{n}(\mathbf{R})$ given by these two proofs are homotopic.

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PII: S 0002-9939(1985)0784160-0
Keywords: Indecomposable modules, preprojective partitions, almost split sequences, rings of global dimension 2, Auslander-Reiten quiver
Article copyright: © Copyright 1985 American Mathematical Society