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Covers and envelopes by \(V\)-Gorenstein modules. (English) Zbl 1087.16002
Let \(R\) and \(S\) denote right and left Noetherian rings respectively and \(V\) a dualizing module. \(A(R)\) denotes the Auslander class relative to \(V\). Let \(\mathcal U\) be the class of all left \(R\)-modules \(N\) such that \(N\cong\operatorname{Hom}_S(V,E)\) for some \(E\in S\)-Mod injective.
A left \(R\)-module \(M\) is said to be \(V\)-Gorenstein injective if there exists an exact resolution \(\cdots\to U_1\to U_0\to U^0\to U^1\to\cdots\) with every \(U_i\) and \(U^i\) in \(\mathcal U\) and \(M=\text{Ker}(U^0\to U^1)\) and such that the sequence remains exact whenever \(\operatorname{Hom}_R(U,-)\) and \(\operatorname{Hom}_R(-,U)\) are applied for every \(U\in\mathcal U\). \(V\)-Gorenstein projectivity is defined dually.
Various equivalences for \(N\in R\)-Mod to be \(V\)-Gorenstein injective are found in terms of \(N\in A(R)\) and some properties concerning \(\text{Ext}_R^1(-,N)\). As a corollary to this theorem, one sees that the class of \(V\)-Gorenstein injective modules is closed under finite direct sums and direct summands. If \(V\text{-Gor\,Inj}\) is the class of \(V\)-Gorenstein injective \(R\)-modules and \(L\) is the subclass of \(A(R)\) of finite right \(U\)-dimension, \((L, V\text{-Gor\,Inj})\) is proven to be a perfect hereditary complete cotorsion theory of \(A(R)\). The results on \(V\)-Gorenstein injectivity are subsequently dualized to \(V\)-Gorenstein projectivity and if \(R\) is left perfect, then \((V\text{-Gor\,Proj},L)\) is shown to be a hereditary complete cotorsion theory of \(B(S)\), the Bass class (relative to \(V\)). Sufficient conditions for every finitely generated left \(S\)-module \(M\in B(S)\) to have a \(V\)-Gorenstein projective cover are also derived.

MSC:
16D50 Injective modules, self-injective associative rings
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16E05 Syzygies, resolutions, complexes in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
13C11 Injective and flat modules and ideals in commutative rings
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