Uniqueness of uniform decompositions in abelian categories
The author proves a Krull-Schmidt type theorem for finite direct sums of uniform objects in an abelian category \(\mathcal A\). If \(\mathcal A\) has injective envelopes, the decomposition theorem turns into the Krull-Schmidt theorem for injectives. Therefore, the proof is achieved by embedding \(\mathcal A\) into the Grothendieck category \(\text{Lex}(\mathcal A)\) of left exact functors from \({\mathcal A}^{\text{op}}\) to \({\mathcal A}b\). As a Grothendieck category, \(\text{Lex}(\mathcal A)\) has injective envelopes. The main point of the paper is the observation, that a recent unique decomposition theorem of Diracca and Facchini which replaces isomorphism between injective envelopes of objects \(A, B\) by the existence of monomorphisms \(A\hookrightarrow B\hookrightarrow A\), arises as a special case of the author's result.
scientific article; zbMATH DE number 2005424