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A noncommutative generalization of Auslander’s last theorem. (English) Zbl 1103.16011

Let \(R\) be a ring, a left \(R\)-module \(M\) is said to be Gorenstein projective if there exists an exact sequence \(\cdots P_1\to P_0\to P^0\to P^1\to\cdots\) of projective left \(R\)-modules which remains exact whenever \(\operatorname{Hom}_R(-,P)\) is applied to it for every projective module \(P\) and such that \(M=\ker(P^0\to P^1)\). The finitely generated Gorenstein projective modules coincide with the modules of \(G\)-dimension zero.
These modules were introduced by M. Auslander [in Séminaire d’algèbre commutative 1966/67, École Normale Supérieure de Jeunes Filles, Paris (1967; Zbl 0157.08301)]. For Gorenstein commutative local rings, Auslander announced that every finitely generated module has a finitely generated Gorenstein projective cover (or equivalently, a minimal maximal Cohen-Macaulay approximation). The first two authors and J. Xu [in Algebr. Represent. Theory 2, No. 3, 259-268 (1999; Zbl 0938.13005)] showed the result for Cohen-Macaulay rings having a dualizing module.
In this paper, the authors study the noncommutative situation. A ring \(R\) is said to be left \(n\)-perfect if every left flat \(R\)-module has projective dimension less than or equal to \(n\). It is shown that every finitely generated left \(R\)-module in the Auslander class over an \(n\)-perfect ring \(R\) having a dualizing module and admitting a Matlis dualizing module has a finitely generated Gorenstein projective cover.

MSC:

16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16D40 Free, projective, and flat modules and ideals in associative algebras
16E10 Homological dimension in associative algebras
16D20 Bimodules in associative algebras
16L30 Noncommutative local and semilocal rings, perfect rings
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