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From subcategories to the entire module categories

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Published/Copyright: November 12, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 1

Abstract

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2×2 lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

Keywords: Relative stable Auslander algebra; triangular matrix algebra; Gorenstein projective module; almost split sequence; Auslander–Reiten quiver
MSC 2010: 16G10; 16G60; 16G50

Communicated by Freydoon Shahidi

Funding statement: This research was supported in part by a grant from School of Mathematics, Institute for Research in Fundamental Sciences (IPM).

Acknowledgements

The author gratefully thanks the referee for the constructive comments and recommendations which definitely helped to improve the readability and quality of the paper. He would like to thank Hideto Asashiba for some useful comments to improve the English of the manuscript.

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Received: 2019-10-07
Revised: 2020-10-26
Published Online: 2020-11-12
Published in Print: 2021-01-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
  3. On three-variable expanders over finite valuation rings
  4. Completely positive maps of order zero on pro-𝐶∗-algebras
  5. Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series
  6. Special value formula for the twisted triple product L-function and an application to the restricted L2-norm problem
  7. Generalised Iwasawa invariants and the growth of class numbers
  8. Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth
  9. The geometric sieve for quadrics
  10. On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness
  11. Commutative algebraic monoid structures on affine surfaces
  12. Representations induced from the Zelevinsky segment and discrete series in the half-integral case
  13. A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)
  14. From subcategories to the entire module categories
  15. Coherent state transform for Landau levels on quasi-tori
  16. On rational homotopy and minimal models
https://doi.org/10.1515/forum-2019-0276
Keywords for this article
Relative stable Auslander algebra; triangular matrix algebra; Gorenstein projective module; almost split sequence; Auslander–Reiten quiver

Articles in the same Issue

  1. Frontmatter
  2. Optimal sup norm bounds for newforms on GL2 with maximally ramified central character
  3. On three-variable expanders over finite valuation rings
  4. Completely positive maps of order zero on pro-𝐶∗-algebras
  5. Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series
  6. Special value formula for the twisted triple product L-function and an application to the restricted L2-norm problem
  7. Generalised Iwasawa invariants and the growth of class numbers
  8. Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth
  9. The geometric sieve for quadrics
  10. On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness
  11. Commutative algebraic monoid structures on affine surfaces
  12. Representations induced from the Zelevinsky segment and discrete series in the half-integral case
  13. A genuine analogue of the Wiener Tauberian theorem for some Lorentz spaces on SL(2,ℝ)
  14. From subcategories to the entire module categories
  15. Coherent state transform for Landau levels on quasi-tori
  16. On rational homotopy and minimal models
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