Abstract
Let \(\mathcal {P}^{<\infty } ({\Lambda }\text {-mod})\) be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra Λ. We develop a widely applicable criterion that reduces the test for contravariant finiteness of \(\mathcal {P}^{<\infty } ({\Lambda }\text {-mod})\) in Λ-mod to corner algebras eΛe for suitable idempotents e ∈Λ. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of \(\mathcal {P}^{<\infty } ({\Lambda }\text {-mod})\). The consequences pursued here hinge on the fact that this finiteness condition is known to be equivalent to the existence of a strong tilting object in Λ-mod. We moreover characterize the situation in which the process of strongly tilting Λ-mod allows for unlimited iteration: This occurs precisely when, in the category \(\text {mod-}\widetilde {\Lambda }\) of right modules over the strongly tilted algebra \(\widetilde {\Lambda }\), the subcategory of modules of finite projective dimension is in turn contravariantly finite; the latter condition can, once again, be tested on suitable corners eΛe of the original algebra Λ. In the (frequently occurring) positive case, the sequence of consecutive strong tilts, \(\widetilde {\Lambda }\), \(\widetilde {\widetilde {\Lambda }}\), \(\widetilde {\widetilde {\widetilde {\Lambda }}}, \dots \), is shown to be periodic with period 2 (up to Morita equivalence); moreover, any two adjacent categories in the sequence \(\mathcal {P}^{<\infty } (\text {mod-}\widetilde {\Lambda })\), \(\mathcal {P}^{<\infty }(\widetilde {\widetilde {\Lambda }}\text {-mod})\), \(\mathcal {P}^{<\infty }(\text {mod-}\widetilde {\widetilde {\widetilde {\Lambda }}}), \dots \), alternating between right and left modules, are dual via contravariant Hom-functors induced by tilting bimodules which are strong on both sides. Our methods rely on comparisons of right \(\mathcal {P}^{<\infty }\)-approximations in the categories Λ-mod, eΛe-mod and the Giraud subcategory of Λ-mod determined by e; these interactions hold interest in their own right. In particular, they underlie our analysis of the indecomposable direct summands of strong tilting modules.
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Presented by: Christof Geiss
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The second named author would like to thank all members of Department of Mathematics at the University of Murcia for their warm hospitality during her stay as a post doc, supported by a grant of the research project 19880/GERM/15 of the Fundación Séneca of Murcia. The third named author has been supported by the Grant PID2020-113206GB-I00 funded by MCIN/AEI/10.13039/501100011033
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Huisgen-Zimmermann, B., Nazemian, Z. & Saorín, M. Contravariant Finiteness and Iterated Strong Tilting. Algebr Represent Theor 26, 2433–2465 (2023). https://doi.org/10.1007/s10468-022-10180-z
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DOI: https://doi.org/10.1007/s10468-022-10180-z
Keywords
- Projective dimension
- Contravariantly finite subcategory
- Finitistic dimension conjecture
- Strong tilting module
- Corner algebra
- Giraud subcategoy
- Morita context