tau-tilting finite algebras, bricks, and g-vectors
(2019) In International Mathematics Research Notices 2019(3). p.852-892- Abstract
- The class of support τ-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article, we study τ-tilting finite algebras, that is, finite dimensional algebras A with finitely many isomorphism classes of indecomposable τ-rigid modules. We show that A is τ-tilting finite if and only if every torsion class in modA is functorially finite. Moreover we give a bijection between indecomposable τ-rigid A-modules and bricks of A satisfying a certain finiteness condition, which is automatic for τ-tilting finite algebras. We observe that cones generated by g-vectors of indecomposable direct summands of each support τ-tilting module form a simplicial complex Δ(A). We show... (More)
- The class of support τ-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article, we study τ-tilting finite algebras, that is, finite dimensional algebras A with finitely many isomorphism classes of indecomposable τ-rigid modules. We show that A is τ-tilting finite if and only if every torsion class in modA is functorially finite. Moreover we give a bijection between indecomposable τ-rigid A-modules and bricks of A satisfying a certain finiteness condition, which is automatic for τ-tilting finite algebras. We observe that cones generated by g-vectors of indecomposable direct summands of each support τ-tilting module form a simplicial complex Δ(A). We show that if A is τ-tilting finite, then Δ(A) is homeomorphic to an (n−1)-dimensional sphere, and moreover the partial order on support τ-tilting modules can be recovered from the geometry of Δ(A). (Less)
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https://lup.lub.lu.se/record/f97a1380-6b4d-4f8e-a87d-5871f1daaa2f
- author
- Demonet, Laurent ; Iyama, Osamu and Jasso, Gustavo LU
- publishing date
- 2019-02
- type
- Contribution to journal
- publication status
- published
- subject
- in
- International Mathematics Research Notices
- volume
- 2019
- issue
- 3
- pages
- 41 pages
- publisher
- Oxford University Press
- external identifiers
-
- scopus:85065778184
- ISSN
- 1073-7928
- DOI
- 10.1093/imrn/rnx135
- language
- English
- LU publication?
- no
- id
- f97a1380-6b4d-4f8e-a87d-5871f1daaa2f
- date added to LUP
- 2022-03-09 15:09:00
- date last changed
- 2022-04-26 00:26:33
@article{f97a1380-6b4d-4f8e-a87d-5871f1daaa2f, abstract = {{The class of support τ-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article, we study τ-tilting finite algebras, that is, finite dimensional algebras A with finitely many isomorphism classes of indecomposable τ-rigid modules. We show that A is τ-tilting finite if and only if every torsion class in modA is functorially finite. Moreover we give a bijection between indecomposable τ-rigid A-modules and bricks of A satisfying a certain finiteness condition, which is automatic for τ-tilting finite algebras. We observe that cones generated by g-vectors of indecomposable direct summands of each support τ-tilting module form a simplicial complex Δ(A). We show that if A is τ-tilting finite, then Δ(A) is homeomorphic to an (n−1)-dimensional sphere, and moreover the partial order on support τ-tilting modules can be recovered from the geometry of Δ(A).}}, author = {{Demonet, Laurent and Iyama, Osamu and Jasso, Gustavo}}, issn = {{1073-7928}}, language = {{eng}}, number = {{3}}, pages = {{852--892}}, publisher = {{Oxford University Press}}, series = {{International Mathematics Research Notices}}, title = {{tau-tilting finite algebras, bricks, and g-vectors}}, url = {{http://dx.doi.org/10.1093/imrn/rnx135}}, doi = {{10.1093/imrn/rnx135}}, volume = {{2019}}, year = {{2019}}, }