A note on preprojective partitions over hereditary Artin algebras

Abstract

If Λ \Lambda is an artin algebra there is a partition of ind ⁡ Λ \operatorname {ind} \Lambda , the category of indecomposable finitely generated Λ \Lambda -modules, ind ⁡ Λ = ∪ i ⩾ 0 P _ _ i \operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i} , called the preprojective partition. We show that P _ _ i \underline {\underline {P}}_i can be easily constructed for hereditary artin algebras, if P _ _ i − 1 \underline {\underline {P}}_{i - 1} is known: A A is in P _ _ i \underline {\underline {P}}_i if and only if A A is not in P _ _ i − 1 \underline {\underline {P}}_{i - 1} and there is an irreducible map B → A B \to A , where B B is in P _ _ i − 1 \underline {\underline {P}}_{i - 1} .