Dlab’s theorem and tilting modules for stratified algebras. (English) Zbl 1127.16009
In the first main result of the paper under review the author obtains a module-theoretic characterization of algebras, which are standardly stratified in the sense of E. Cline, B. Parshall and L. Scott [“Stratifying endomorphism algebras”, Mem. Am. Math. Soc. 591 (1996; Zbl 0888.16006)], in terms of proper costandard filtrations for injective modules. This result is a generalization of V. Dlab’s characterization of quasi-hereditary algebras [from An. Ştiinţ. Univ. “Ovidius” Constanţa, Ser. Mat. 4, No. 2, 43-54 (1996; Zbl 0873.16008)].
The proper costandard modules are then used to develop tilting theory for standardly stratified algebras. The author proves the existence of tilting modules (i.e. those modules which have both standard and proper costandard filtrations) and gives a classification of indecomposable tilting modules. This is used to define the Ringel dual of a standardly stratified algebra, and it is shown that the opposite of this Ringel dual is standardly stratified. This unifies and generalizes the results of C. M. Ringel [Math. Z. 208, No. 2, 209-224 (1991; Zbl 0725.16011)] and I. Ágoston, D. Happel, E. Lukács and L. Unger [J. Algebra 226, No. 1, 144-160 (2000; Zbl 0949.16012)].
In the last part of the paper, the author obtains some estimates for the finitistic dimension of a standardly stratified algebra in terms of finitistic dimension of the endomorphism algebras of standard modules. For the so-called weakly properly stratified algebras some stronger estimates in terms of projective dimensions of tilting modules are obtained.
The proper costandard modules are then used to develop tilting theory for standardly stratified algebras. The author proves the existence of tilting modules (i.e. those modules which have both standard and proper costandard filtrations) and gives a classification of indecomposable tilting modules. This is used to define the Ringel dual of a standardly stratified algebra, and it is shown that the opposite of this Ringel dual is standardly stratified. This unifies and generalizes the results of C. M. Ringel [Math. Z. 208, No. 2, 209-224 (1991; Zbl 0725.16011)] and I. Ágoston, D. Happel, E. Lukács and L. Unger [J. Algebra 226, No. 1, 144-160 (2000; Zbl 0949.16012)].
In the last part of the paper, the author obtains some estimates for the finitistic dimension of a standardly stratified algebra in terms of finitistic dimension of the endomorphism algebras of standard modules. For the so-called weakly properly stratified algebras some stronger estimates in terms of projective dimensions of tilting modules are obtained.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16G30 | Representations of orders, lattices, algebras over commutative rings |
| 16E10 | Homological dimension in associative algebras |
| 16G10 | Representations of associative Artinian rings |
Keywords:
stratified algebras; tilting modules; Ringel duals; finitistic dimension; quasi-hereditary algebras; costandard modules; filtrationsReferences:
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