## 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.

### 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
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### References:

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