Uniqueness of uniform decompositions in exact categories
The authors extend uniqueness theorems on uniform decompositions of \textit{H. Krause} [Expo. Math. 33, No. 4, 535--549 (2015; Zbl 1353.18011)], \textit{L. Diracca} and \textit{A. Facchini} [J. Pure Appl. Algebra 172, No. 2--3, 183--191 (2002; Zbl 1006.18010)] (Krull-Remak Schmidt Theorem in abelian categories) to certain Quillen exact categories. Two objects \(A\) and \(B\) of an exact category \((\mathcal{A}, \mathcal{E})\) are (i) \(\mathcal{E}\)-essentially related if there exists \(\mathcal{E}\)-essential inflations \(A' \rightarrow A\) and \(B' \rightarrow B\) such that \(A' \cong B'\) and (ii) in the same \(\mathcal{E}\)-monogeny class if there exist inflations \(A \rightarrow B\) and \(B \rightarrow A.\) For an exact category \((\mathcal{A}, \mathcal{E}),\) let \(\mathcal{I}\) be the class of objects \(A\) such that \(A\) has an \(\mathcal{E}\)-uniform \(\mathcal{E}\)-injective envelope \(\mathcal{E}(A)\) with local endomorphism ring, and there exists an inflation \(A \rightarrow \mathcal{E}(A).\) Let \((\mathcal{A},\mathcal{E})\) be a weakly idempotent complete exact category, and let \(A = \bigoplus_{i=1}^m A_i\) and \(B = \bigoplus_{j=1}^n B_j\) be objects of \(\mathcal{A}\) such that \(A_i\)'s and \(B_j\)'s \(\in \mathcal{I}.\) If \((\mathcal{A}, \mathcal{E})\) had finite intersections and \(A,B\) are \(\mathcal {E}\)-essentially related, then the authors prove that \(m= n\) and there is a permutation \(\sigma\) of \(\{1,2,\dots, n\}\) such that \(A_i\) and \(B_{\sigma(i)}\) are \(\mathcal{E}\)-essentially related for every \(i \in \{1,2, \dots, n \}.\) If \(A\) and \(B\) are in the same \(\mathcal{E}\)-monogeny class, then they show that \(m = n\) and there is a permutation \(\sigma\) of \(\{1,2,\dots, n\}\) such that \(A_i\) and \(B_{\sigma(i)}\) are in the same \(\mathcal{E}\)-monogeny class for every \(i \in \{1,2, \dots, n \}.\) Moreover they generalize Krause's theorem to quasi abelian categories and give further applications to abelian categories.
scientific article; zbMATH DE number 7317124
scientific article