Covers induced by Ext
Let \({\mathbf C}\subseteq\text{Mod-}R\) and \({\mathbf B}\subseteq R\text{-Mod}\). Then \(^\bot{\mathbf C}\) is defined to be the class \(\text{Ker Ext}(-,{\mathbf C})=\{D\mid\text{Ext}(D,C)=0\) for all \(C\in{\mathbf C}\}\) and \(\text{Ker Tor}(-,{\mathbf B})=\{A\mid\text{Tor}(A,B)=0\) for all \(B\in{\mathbf B}\}\). If \(M\in\text{Mod-}R\), then \(\phi\in\text{Hom}(A,M)\) (\(A\in{\mathbf C}\)) is called a \(\mathbf C\)-pre-cover of \(M\) if the induced map \(\text{Hom}(A',A)\to\text{Hom}(A',M)\) is surjective for all \(A'\in{\mathbf C}\). If each \(\psi\in\text{Hom}(A,A)\) satisfying \(\phi=\phi\psi\) is an automorphism of \(A\), then \(\phi\) is said to be a \(\mathbf C\)-cover. It is proved that each right \(R\)-module has a \(^\bot{\mathbf C}\)-cover if \(\mathbf C\) is any class of pure-injective modules. On the other hand, for any class \(\mathbf B\) each module has a \(\text{Ker Tor}(-,{\mathbf B})\)-cover. These results therefore provide a generalization of the flat cover conjecture. \(\mathbf C\)-pre-covers and \(\mathbf C\)-covers are studied intensively for modules over hereditary Dedekind domains. For Dedekind domains \(^\bot{\mathbf C}\) is described explicitly for any class of cotorsion modules \(\mathbf C\). The paper ends with three open problems.
scientific article; zbMATH DE number 1526429
scientific article