Let $ A = kQ/I $ be a string algebra. We show that, if for any vertex $ v $ of its bound quiver $ (Q, I) $, there exists at most one arrow (resp. at most two arrows) ending with $ v $ and there exist at most two arrows (resp. at most one arrow) starting with $ v $, then the number of indecomposable modules over $ A $ is $ \dim_{k}A+\Sigma $, where $ \Sigma $ is induced by $ rad P(v) $ (resp. $ E(v)/\mathrm{soc} E(v) $) with decomposable socle (resp. top), where $ P(v) $ (resp. $ E(v) $) is the indecomposable projective (resp. injective) module corresponded by the vertex $ v $.
Citation: Haicun Wen, Mian-Tao Liu, Yu-Zhe Liu. The counting formula for indecomposable modules over string algebra[J]. AIMS Mathematics, 2024, 9(9): 24977-24988. doi: 10.3934/math.20241217
Let $ A = kQ/I $ be a string algebra. We show that, if for any vertex $ v $ of its bound quiver $ (Q, I) $, there exists at most one arrow (resp. at most two arrows) ending with $ v $ and there exist at most two arrows (resp. at most one arrow) starting with $ v $, then the number of indecomposable modules over $ A $ is $ \dim_{k}A+\Sigma $, where $ \Sigma $ is induced by $ rad P(v) $ (resp. $ E(v)/\mathrm{soc} E(v) $) with decomposable socle (resp. top), where $ P(v) $ (resp. $ E(v) $) is the indecomposable projective (resp. injective) module corresponded by the vertex $ v $.
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Haicun Wen, Mian-Tao Liu, Yu-Zhe Liu. The counting formula for indecomposable modules over string algebra[J]. AIMS Mathematics, 2024, 9(9): 24977-24988. doi: 10.3934/math.20241217
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