Submodule-closed subcategories of finite type

Nguyen Viet Dung¹

¹ Department of Mathematics Ohio University–Zanesville Zanesville, OH 43701, U.S.A.

Colloquium Mathematicum, Tome 147 (2017) no. 1, pp. 125-144

Résumé

Let $R$ be a left pure semisimple ring, and $\mathcal {C}$ a full subcategory of finitely generated left $R$-modules such that $\mathcal {C}$ is closed under finite direct sums and submodules. It is shown that if $\mathcal {C}$ has an infinite number of non-isomorphic indecomposable modules, then $\mathcal {C}$ contains a submodule-closed subcategory of finite type $\mathcal {A}$ (i.e. $\mathcal {A}$ has only finitely many non-isomorphic indecomposable modules) which is maximal among all submodule-closed subcategories of finite type in $\mathcal {C}$, and moreover $\mathcal {A}$ contains an indecomposable module which is not the source of a left almost split morphism in $R$-mod. If $R$ is an indecomposable hereditary left pure semisimple ring, a maximal submodule-closed subcategory of finite type of $R$-mod always contains the preprojective component of $R$-mod, and if such a ring $R$ has only two simple modules, the unique maximal submodule-closed subcategory of finite type in $R$-mod can be described explicitly.

Détail

DOI : 10.4064/cm6819-3-2016

Keywords: pure semisimple ring mathcal full subcategory finitely generated r modules mathcal closed under finite direct sums submodules shown mathcal has infinite number non isomorphic indecomposable modules mathcal contains submodule closed subcategory finite type mathcal mathcal has only finitely many non isomorphic indecomposable modules which maximal among submodule closed subcategories finite type mathcal moreover mathcal contains indecomposable module which source almost split morphism r mod indecomposable hereditary pure semisimple ring maximal submodule closed subcategory finite type r mod always contains preprojective component r mod ring has only simple modules unique maximal submodule closed subcategory finite type r mod described explicitly