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tau-tilting finite algebras, bricks, and g-vectors

Demonet, Laurent ; Iyama, Osamu and Jasso, Gustavo LU (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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author
; and
publishing date
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}},
}