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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Wide subcategories of $d$-cluster tilting subcategories


Authors: Martin Herschend, Peter Jørgensen and Laertis Vaso
Journal: Trans. Amer. Math. Soc. 373 (2020), 2281-2309
MSC (2010): Primary 16G10, 18A20, 18E10
DOI: https://doi.org/10.1090/tran/8051
Published electronically: January 23, 2020
MathSciNet review: 4069219
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Abstract:

A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics.

If $\Phi$ is a finite dimensional algebra, then each functorially finite wide subcategory of $\operatorname {mod}( \Phi )$ is of the form $\phi _{ {\textstyle *}}\big ( \operatorname {mod}( \Gamma ) \big )$ in an essentially unique way, where $\Gamma$ is a finite dimensional algebra and $\Phi \stackrel { \phi }{ \longrightarrow } \Gamma$ is an algebra epimorphism satisfying $\operatorname {Tor}^{ \Phi }_1( \Gamma ,\Gamma ) = 0$.

Let $\mathscr {F} \subseteq \operatorname {mod}( \Phi )$ be a $d$-cluster tilting subcategory as defined by Iyama. Then $\mathscr {F}$ is a $d$-abelian category as defined by Jasso, and we call a subcategory of $\mathscr {F}$ wide if it is closed under sums, summands, $d$-kernels, $d$-cokernels, and $d$-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of $\mathscr {F}$ is of the form $\phi _{ {\textstyle *}}( \mathscr {G} )$ in an essentially unique way, where $\Phi \stackrel { \phi }{ \longrightarrow } \Gamma$ is an algebra epimorphism satisfying $\operatorname {Tor}^{ \Phi }_d( \Gamma ,\Gamma ) = 0$, and $\mathscr {G} \subseteq \operatorname {mod}( \Gamma )$ is a $d$-cluster tilting subcategory.

We illustrate the theory by computing the wide subcategories of some $d$-cluster tilting subcategories $\mathscr {F} \subseteq \operatorname {mod}( \Phi )$ over algebras of the form $\Phi = kA_m / (\operatorname {rad} kA_m )^{ \ell }$.


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Additional Information

Martin Herschend
Affiliation: Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
MR Author ID: 771009
Email: martin.herschend@math.uu.se

Peter Jørgensen
Affiliation: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
Email: peter.jorgensen@ncl.ac.uk

Laertis Vaso
Affiliation: Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
MR Author ID: 1309760
Email: laertis.vaso@math.uu.se

Keywords: Algebra epimorphism, $d$-abelian category, $d$-cluster tilting subcategory, $d$-homological pair, $d$-pseudoflat morphism, functorially finite subcategory, higher homological algebra, wide subcategory
Received by editor(s): August 2, 2017
Received by editor(s) in revised form: March 7, 2019
Published electronically: January 23, 2020
Additional Notes: This work was supported by EPSRC grant EP/P016014/1 “Higher Dimensional Homological Algebra”.
Dedicated: Dedicated to Idun Reiten on the occasion of her 75th birthday
Article copyright: © Copyright 2020 American Mathematical Society