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.
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.
Reviewer: W.H.Gustafson (Lubbock)
MSC:
| 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 |
Keywords:
category of finitely generated left modules;Artin algebras;finite projective dimension;direct summands;filtrations;injective cotilting modules;finite resolutions;contravariantly finite subcategoriesCite
Full Text:DOI
References:
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