The acyclic closure of an exact category and its triangulation (Q2204846)

Let \(\mathcal M\) be an exact category and \(\mathcal A\) be a full subcategory of \(\mathcal M\). \(\mathcal A\) is called is biresolving if (a) For each object \(M\in\mathcal M\) there exists a deflation \(A_0\twoheadrightarrow M\) and an inflation \(M\rightarrowtail A_1\) with \(A_0, A_1\in \mathcal A\). (b) If two terms of a conflation \(L\rightarrowtail M\twoheadrightarrow N\) belong to \(\mathcal A\), the third term is isomorphic to an object of \(\mathcal A\). For any exact category \(\mathcal A\) with splitting idempotents, the authors construct a maximal exact category \textbf{T}\((\mathcal A)\) (which is the acyclic closure of \(\mathcal A\)) containing \(\mathcal A\) as a biresolving subcategory. Important types of exact categories, including \(n\)-tilting torsion classes, categories of Cohen-Macaulay modules over a Cohen-Macaulay order, or categories of Gorenstein projectives, are shown to be of the form \textbf{T}\((\mathcal A)\). They also show that the quotient category \(\textbf{T}(\mathcal A)/\mathcal A\) in the sense of Grothendieck always exists and carries a triangulated structure. More generally, it is proved that any biresolving subcategory \(\mathcal A\) of an exact category \(\mathcal M\) gives a triangulated localization \(\mathcal M/\mathcal A\). Some examples are given about these results. As applications, some recent developments related to Gorenstein projectivity, non-commutative crepant resolutions, singularity categories, and Cohen-Macaulay representations are extended and improved in the new framework. For example, the concept of non-commutative resolution of a noetherian Frobenius category is extended to arbitrary exact categories, which leads to an overarching connection with representation dimension of exact categories and \(n\)-tilting.