A class of infinite-rank modules over tame hereditary algebras
Let \(A\) be a finite dimensional hereditary connected algebra. A module \(M\in\text{Mod-}A\) is called a \({\mathcal P}^\infty\)-torsion module if \(M\) has no non-zero preprojective direct summand. The torsion class of \({\mathcal P}^\infty\)-torsion modules determines a torsion theory \(({\mathcal T},{\mathcal F})\), introduced by C. M. Ringel (1979). It is proved that the full subcategory \(\mathcal T\) of \({\mathcal P}^\infty\)-torsion modules has enough projective objects, the \({\mathcal P}^\infty\)-projective modules. More precisely, if \(A\) is a tame hereditary algebra and \(Y\) is a regular partial tilting module, then there exists a non zero \({\mathcal P}^\infty\)-projective module \(M\) with \(\text{Hom}(M,Y)=0\). Then \({\mathcal P}^\infty\)-projectives divide into finitely many classes whose number depends on the number of regular partial tilting modules. The application of these results leads to the following theorem. Let \(M\) be a module of countable rank over a tame hereditary algebra \(A\). Then: a) If \(M\) is \({\mathcal P}^\infty\)-torsionfree we have \(\text{Ext}(M,N)=0\) for every \({\mathcal P}^\infty\)-torsion module \(N\). b) The submodule \({\mathcal P}^\infty M\) is a direct summand of \(M\). c) If \(M\) is a Baer module, then \(M\) is a direct sum of a \({\mathcal P}^\infty\)-projective and a \({\mathcal P}^\infty\)-torsionfree module.
finite dimensional hereditary connected algebras
scientific article; zbMATH DE number 218567