Iterated Extensions and Uniserial Length Categories

Abstract

In this paper, we study length categories using iterated extensions. We fix a field k, and for any family S of orthogonal k-rational points in an Abelian k-category \(\mathcal {A}\), we consider the category Ext(S) of iterated extensions of S in \(\mathcal {A}\), equipped with the natural forgetful functor \(\mathbf {Ext}(\mathsf {S}) \to \mathbf {\mathcal {A}}(\mathsf {S})\) into the length category \(\mathbf {\mathcal {A}}(\mathsf {S})\). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel’s criterion, we give a complete classification of the indecomposable objects in \(\mathbf {\mathcal {A}}(\mathsf {S})\) when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family S in \(\mathcal {A}\). As an application, we classify all graded holonomic D-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when D is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish.

References

  1. 1.

    Amdal, I.K., Ringdal, F.: Catégories unisérielles. C. R. Acad. Sci. Paris Sér A-B 267, A247–A249 (1968)

    MATH  Google Scholar 

  2. 2.

    Boutet de Monvel, L.: \({\mathcal D}\)-modules holonômes réguliers en une variable. Mathematics and Physics (Paris, 1979/1982), Progr. Math., vol. 37, pp 313–321. Birkhäuser, Boston (1983). MR 728427

    Google Scholar 

  3. 3.

    Chen, X.-W., Krause, H.: Introduction to coherent sheaves on weighted projective lines. arXiv:0911.4473 (2009)

  4. 4.

    Coutinho, S.C.: A Primer of Algebraic D-Modules London Mathematical Society Student Texts, vol. 33. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  5. 5.

    Eriksen, E.: Differential operators on monomial curves. J. Algebra 264(1), 186–198 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Eriksen, E: Graded Holonomic D-modules on Monomial Curves. arXiv:1803.04367 (2018)

  7. 7.

    Eriksen, E., Laudal, O.A., Siqveland, A.: Noncommutative Deformation Theory, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2017)

    Book  Google Scholar 

  8. 8.

    Eriksen, E: Computing noncommutative deformations of presheaves and sheaves of modules. Canad. J. Math. 62(3), 520–542 (2010). MR 2666387

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gabriel, P.: Indecomposable Representations. II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), pp 81–104. Academic Press, London (1973)

    Google Scholar 

  10. 10.

    Laudal, O.A.: Noncommutative deformations of modules. Homology Homotopy Appl. 4(2), part 2, 357–396 (2002). The Roos Festschrift volume, 2

    MathSciNet  Article  Google Scholar 

  11. 11.

    May, J.P.: Matric Massey products. J. Algebra 12, 533–568 (1969). MR 238929

    MathSciNet  Article  Google Scholar 

  12. 12.

    Quillen, D.: On the endomorphism ring of a simple module over an enveloping algebra. Proc. Amer. Math. Soc. 21, 171–172 (1969)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Ringel, C.M.: Representations of K-species and bimodules. J. Algebra 41(2), 269–302 (1976)

    MathSciNet  Article  Google Scholar 

Download references

Funding

Open Access funding provided by Norwegian Business School.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Eivind Eriksen.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Presented by: Michel Brion

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Eriksen, E. Iterated Extensions and Uniserial Length Categories. Algebr Represent Theor 24, 273–286 (2021). https://doi.org/10.1007/s10468-020-09946-0

Download citation

Keywords

  • Finite length categories
  • Uniserial categories
  • Iterated extensions
  • Noncommutative deformations

Mathematics Subject Classification (2010)

  • Primary 18E10
  • Secondary 16G99
  • 14F10