The counting formula for indecomposable modules over string algebra

The counting formula for indecomposable modules over string algebra

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Let $ A = kQ/I $ be a string algebra.We show that, if for any vertex $ v Agitator Cam Dogs $ 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), Spin Mop where $ P(v) $ (resp.$ E(v) $) is the indecomposable projective (resp.injective) module corresponded by the vertex $ v $.