zbMATH — the first resource for mathematics

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
Full Text: DOI
[1] Auslander, M, Coherent functors, (), 189-231 · Zbl 0192.10902
[2] Auslander, M; Bridger, M, Stable module theory, Mem. amer. math. soc., 94, (1969) · Zbl 0204.36402
[3] Auslander, M; Buchweitz, R.O, Maximal Cohen-Macaulay approximations, Soc. math. France, 38, 5-37, (1989) · Zbl 0697.13005
[4] {\scM. Auslander and E. L. Green}, Modules over endomorphism rings, preprint. · Zbl 0804.16030
[5] Auslander, M; Reiten, I, Stable equivalence of Artin algebras, (), 8-71
[6] Auslander, M; Reiten, I, On a generalized version of the Nakayama conjecture, (), 69-74 · Zbl 0337.16004
[7] Auslander, M; Reiten, I, Representation theory of Artin algebras III, almost split sequences, Comm. algebra, 3, 239-294, (1975) · Zbl 0331.16027
[8] Auslander, M; Smalø, S.O, Preprojective modules over Artin algebras, J. algebra, 66, 61-122, (1980) · Zbl 0477.16013
[9] Auslander, M; Smalø, S.O, Almost split sequences in subcategories, J. algebra, Addendum, J. algebra, 71, 592-594, (1981) · Zbl 0474.16022
[10] Bongartz, K, Tilted algebras, (), 26-38
[11] Happel, D, On the derived category of a finite dimensional algebra, Comm. math. helv., 62, No. 3, 339-389, (1987) · Zbl 0626.16008
[12] Happel, D; Ringel, C.M, Tilted algebras, Trans. amer. math. soc., 274, 399-443, (1982) · Zbl 0503.16024
[13] {\scK. Igusa, S. O. Smalø, and G. Todorov}, Finite projectivity and contravariant finiteness, Proc. Amer. Math. Soc. · Zbl 0696.16024
[14] Jensen, C.U; Lenzing, H, Homological dimension and representation type of algebras under base field extension, Manuscripta math., 39, 1-13, (1982) · Zbl 0498.16023
[15] Miayshita, T, Tilting modules of finite projective dimension, Math. Z., 193, 113-146, (1986)
[16] {\scJ. Rickart and A. Scofield}, Cocovers and tilting modules, preprint.
[17] Sharp, R.Y, Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings, Proc. London math. soc., 25, 3, 303-328, (1972) · Zbl 0244.13015
[18] {\scS. O. Smalø}, Functorial finite subcategories over triangular matrix rings, Proc. Amer. Math. Soc. · Zbl 0724.16003
[19] Wakamatsu, T, On modules with trivial selfextensions, J. algebra, 114, 106-114, (1988) · Zbl 0646.16025
[20] {\scT. Wakamatsu}, Stable equivalence of selfinjective algebras and a generalization of tilting modules, preprint. · Zbl 0726.16009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.