Covers induced by Ext. (English) Zbl 0981.16010

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.


16D90 Module categories in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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