Wakamatsu tilting modules.
Let \(A\) be an Artin algebra. A self-orthogonal \(A\)-module, \(T\), is called a `Wakamatsu tilting module' if \(_AA\) admits a (possibly infinite) coresolution by modules from \(\text{add}(T)\). A full subcategory in \(A\text{-mod}\) is called `resolving' if it is closed under direct sums, direct summands, isomorphisms, extensions and kernels of epimorphisms, and contains all projective modules. A `cotorsion theory' is a pair \((\mathcal{C,D})\) of subcategories of \(A\)-modules such that \(\mathcal C\) is the left orthogonal complement of \(\mathcal D\) with respect to \(\text{Ext}^1\), and \(\mathcal D\) is the right orthogonal complement of \(\mathcal C\) with respect to \(\text{Ext}^1\). The class \({\mathcal C}\cap{\mathcal D}\) is called the `kernel' of \((\mathcal{C,D})\). In the paper under review the main results are the following two facts about the Wakamatsu tilting modules: Theorem. There is a natural bijection between the isomorphism classes of basic Wakamatsu tilting modules and coresolving subcategories in \(A\text{-mod}\) with an \(\text{Ext}\)-projective generator, maximal among those with the same \(\text{Ext}\)-projective generator. Theorem. There is a natural bijection between the isomorphism classes of basic Wakamatsu tilting modules and cotorsion theories, minimal among those with the same kernel and the first category having an \(\text{Ext}\)-injective cogenerator. Both results have natural dual versions, which are also included in the paper. The paper is finished with a discussion of properties of Wakamatsu tilting modules of finite projective dimension and complements of Wakamatsu tilting modules.
scientific article; zbMATH DE number 2105342
scientific article