Exact Krull-Schmidt categories with finitely many indecomposables (Q2070952)

An Ext-category is an exact category with enough projectives and enough injectives such that every split epimorphism has a kernel. Notice that many naturally arising exact categories are, in fact, Ext-categories. Let \(\mathcal{A}\) be an Ext-category with the Krull-Schmidt property. It is of interest to understand when the set \(\mathrm{ind}(\mathcal{A})\) of (isomorphism classes of) indecomposable objects in \(\mathcal{A}\) is finite. A main result in the paper under review gives, under some technical assumptions on \(\mathcal{A}\), several interesting conditions which are all equivalent to finiteness of the set \(\mathrm{ind}(\mathcal{A})\). One of these equivalent conditions is that the homotopy category \(\mathrm{Ext}(\mathcal{A})\) of conflations in \(\mathcal{A}\) (which is proved to be abelian) is a length category, meaning that every object in this category has finite length. A second equivalent condition is that the category \(\mathrm{mod}(\mathcal{A})\) of finitely presented \(\mathcal{A}\)-modules is abelian and noetherian. A third equivalent condition is that \(\mathcal{A}\) has almost split sequences, and the homotopy category \(\mathsf{M}(\mathcal{A})\) of morphisms in \(\mathcal{A}\) is \(L\)-finite. This condition, of which the last part is somewhat technical (however, it is explained well in Section 4), is really the key condition in the author's proof. One application of this result yields several different characterizations of when a Cohen-Macaulay order \(\Lambda\) over a complete regular local ring \(R\) is representation finite. The last section of the paper contains an interesting characterization of Ext-categories \(\mathcal{A}\) of finite type, which means that one has \(\mathcal{A} = \mathrm{add}(A)\) for some object \(A \in \mathcal{A}\) (here the Krull-Schmidt property is not assumed).