Applications of contravariantly finite subcategories. (English) Zbl 0774.16006

Let \(\Lambda\) be an Artin algebra, and let \({\mathcal P}^ \infty(\Lambda)\) be the category of finitely generated left \(\Lambda\)-modules that are of finite projective dimension. The authors first study what it means for \({\mathcal P}^ \infty(\Lambda)\) to be contravariantly finite in the category of all finitely generated left \(\Lambda\)-modules. A rich supply of examples can be obtained where \({\mathcal P}^ \infty(\Lambda)\) is contravariantly finite, and there are also many examples where it is not. In the case where \({\mathcal P}^ \infty(\Lambda)\) is contravariantly finite, there is a bound on the projective dimensions of modules of finite projective dimension. Also, there are modules \(A_ 1,\dots,A_ n\) in \({\mathcal P}^ \infty(\Lambda)\) such that the objects of \({\mathcal P}^ \infty(\Lambda)\) are just the direct summands of modules having filtrations with the associated quotients among the \(A_ i\).
If \(T\) is a module with \(\text{Ext}^ i_ \Lambda(T,T)=0\) for all \(i>0\), one constructs the category \(^ \perp T\) of modules \(X\) with \(\text{Ext}^ i_ \Lambda(X,T)=0\) for all \(i>0\). It is shown that \(T\) is an injective cotilting module if and only if \(^ \perp T\) is contravariantly finite and every \(\Lambda\)-module has a finite resolution by modules in \(^ \perp T\). In fact, this construction exhausts the contravariantly finite subcategories with this resolution property. Finally, it is shown how analogs of the Gorenstein and Cohen-Macaulay properties of commuative rings can be defined for Artin algebras in terms of special types of cotilting modules.


16G10 Representations of associative Artinian rings
16D90 Module categories in associative algebras
16G50 Cohen-Macaulay modules in associative algebras
16E10 Homological dimension in associative algebras
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