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Injective homogeneity and the Auslander-Gorenstein property. (English) Zbl 0830.16010
For a finitely generated right module $$M$$ over a ring $$R$$, the nonnegative integers $$\text{u.gr}(M)=\text{sup}\{n\mid\text{Ext}^n_R (M,R)\neq 0\}$$ and $$j(M)=\inf\{n\mid\text{Ext}^n_R (M,R)\neq 0\}$$ are called the upper grade and the grade of $$M$$, respectively. A noetherian ring $$R$$ with finite injective dimension is called an Auslander-Gorenstein ring, if, given any finitely generated $$R$$-module $$M$$ and any integer $$i$$, $$j(N)\geq i$$ for every submodule $$N$$ of $$\text{Ext}^i_R (M,R)$$, and $$R$$ is called Macaulay, if furthermore $$j(M) + \text{K.dim}(M)=\text{K.dim}(R)$$ for every finitely generated module $$M$$. The Auslander-Gorenstein and Macaulay properties are closely related to some other homological properties. For example, J. T. Stafford and J. Zhang [J. Algebra 168, 988-1026 (1994; Zbl 0812.16046)] have shown that for a noetherian P.I. ring $$R$$ to be Auslander-Gorenstein and Macaulay, it suffices to be injectively smooth, that is, to satisfy $$\text{u.gr}(R/P)=\text{inj.dim}(R)$$ for every maximal ideal $$P$$. In the paper under review, the author shows that this latter condition is necessary as well. He also shows that $$R$$ is Auslander-Gorenstein when it is injectively homogeneous, that is, when $$\text{u.gr}(R/P)=\text{u.gr}(R/Q)$$ for any two maximal ideals $$P$$ and $$Q$$ that belong to the same clique. The transfer of injective homogeneity and of injective smoothness between strongly group-graded rings and their coefficient rings is studied, and it is applied to establish the following result. Let $$G$$ be a finite group and let $$S=R(G)$$ be a strongly $$G$$-graded ring with noetherian coefficient ring $$R$$. Then $$R$$ is Auslander-Gorenstein (Auslander-Gorenstein and Macaulay) if and only if so is $$S$$.

##### MSC:
 16E10 Homological dimension in associative algebras 16P40 Noetherian rings and modules (associative rings and algebras) 16W50 Graded rings and modules (associative rings and algebras) 16R40 Identities other than those of matrices over commutative rings
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