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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Wide subcategories of $d$-cluster tilting subcategories
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by Martin Herschend, Peter Jørgensen and Laertis Vaso PDF
Trans. Amer. Math. Soc. 373 (2020), 2281-2309 Request permission

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 $\Phi = kA_m / (\operatorname {rad} kA_m )^{ \ell }$.

References
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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
  • 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
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 2281-2309
  • MSC (2010): Primary 16G10, 18A20, 18E10
  • DOI: https://doi.org/10.1090/tran/8051
  • MathSciNet review: 4069219