Abstract
Let \(\varLambda\) be a right-peak algebra in the sense of Simson (J Algebra 92:532–571, 1985). Denote by \(\mathrm {mod}_\mathrm{{sp}}(\varLambda )\) the full subcategory of \(\mathrm {mod}\text {-}\varLambda\) consisting of those right \(\varLambda\)-modules M such that the socle of M is a simple projective right \(\varLambda\)-module. Suppose that for any non-simple indecomposable right projective \(\varLambda\)-module P, \(\mathrm {rad}P=T^{l}\), for some indecomposable T and some positive integer l. Here we show that the Auslander–Reiten component of \(\mathrm {mod}_\mathrm{{sp}}(\varLambda )\) containing the simple projective right \(\varLambda\)-module is a preprojective component. To do so, we describe an Auslander–Reiten component of a Krull–Schmidt category with four suitable properties. Then we relate \(\mathrm {mod}_\mathrm{{sp}}(\varLambda )\) with a subcategory of morphisms between projective \(\varLambda\)-modules and with some category of prinjective modules. We prove that the right-peak algebras associated with representations and corepresentations of p-equipped posets satisfy the mentioned condition for all prime number p.
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Bautista, R., Dorado, I. A preprojective Auslander-Reiten component for the socle projective modules of some right-peak algebras. Bol. Soc. Mat. Mex. 28, 15 (2022). https://doi.org/10.1007/s40590-021-00407-2
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DOI: https://doi.org/10.1007/s40590-021-00407-2
Keywords
- Right-peak algebras
- Pseudo-hereditary projective module
- Auslander–Reiten component
- Morphism category
- Equipped poset