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Wakamatsu tilting modules. (English) Zbl 1075.16006

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.

MSC:

16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16G10 Representations of associative Artinian rings
16D90 Module categories in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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