Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T16:02:07.707Z Has data issue: false hasContentIssue false
  • English
  • Français

On Differential Torsion Theories and Rings with Several Objects

Part of: Rings and algebras arising under various constructions Abelian categories

Published online by Cambridge University Press:  08 April 2019

Abhishek Banerjee
Abhishek Banerjee*
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore-560012, India Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ${\mathcal{R}}$ be a small preadditive category, viewed as a “ring with several objects.” A right${\mathcal{R}}$-module is an additive functor from ${\mathcal{R}}^{\text{op}}$ to the category $Ab$ of abelian groups. We show that every hereditary torsion theory on the category $({\mathcal{R}}^{\text{op}},Ab)$ of right ${\mathcal{R}}$-modules must be differential.

Keywords

hereditary torsion theorydifferential torsion theoryring with several objects

MSC classification

Primary: 16S90: Torsion theories; radicals on module categories
Secondary: 18E05: Preadditive, additive categories 18E40: Torsion theories, radicals
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 703 - 714
Copyright
© Canadian Mathematical Society 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Banerjee, A., On Auslander’s formula and cohereditary torsion pairs . Commun. Contemp. Math. 20(2018), no. 6, 1750071. https://doi.org/10.1142/S0219199717500717 Google Scholar
Beligiannis, A. and Reiten, I., Homological and homotopical aspects of torsion theories . Mem. Amer. Math. Soc. 188(2007), no. 883. https://doi.org/10.1090/memo/0883 Google Scholar
Bland, P. E., Differential torsion theory . J. Pure Appl. Algebra 204(2006), 1–8. https://doi.org/10.1016/j.jpaa.2005.03.005 Google Scholar
Borceux, F., Handbook of categorical algebra. 2. Categories and structures . Encyclopedia of Mathematics and its Applications, 51, Cambridge University Press, Cambridge, 1994.Google Scholar
Cibils, C. and Solotar, A., Galois coverings, Morita equivalence and smash extensions of categories over a field . Doc. Math. 11(2006), 143–159.Google Scholar
Estrada, S. and Virili, S., Cartesian modules over representations of small categories . Adv. Math. 310(2017), 557–609. https://doi.org/10.1016/j.aim.2017.01.030 Google Scholar
Garkusha, G. A., Grothendieck categories. arxiv:math/9909030 Google Scholar
Golan, J. S., Extensions of derivations to modules of quotients . Comm. Algebra 9(1981), 275–281. https://doi.org/10.1080/00927878108822580 Google Scholar
Kaygun, A. and Khalkhali, M., Bivariant Hopf cyclic cohomology . Comm. Algebra 38(2010), no. 7, 2513–2537. https://doi.org/10.1080/00927870903417695 Google Scholar
Lomp, C. and van den Berg, J., All hereditary torsion theories are differential . J. Pure Appl. Algebra 213(2009), no. 4, 476–478. https://doi.org/10.1016/j.jpaa.2008.07.018 Google Scholar
Lowen, W. and van den Bergh, M., Hochschild cohomology of abelian categories and ringed spaces . Adv. Math. 198(2005), no. 1, 172–221. https://doi.org/10.1016/j.aim.2004.11.010 Google Scholar
Lowen, W. and van den Bergh, M., Deformation theory of abelian categories . Trans. Amer. Math. Soc. 358(2006), no. 12, 5441–5483. https://doi.org/10.1090/S0002-9947-06-03871-2 Google Scholar
Lowen, W., Hochschild cohomology with support . Int. Math. Res. Not. IMRN 2015 no. 13, 4741–4812. https://doi.org/10.1093/imrn/rnu079 Google Scholar
Mitchell, B., Rings with several objects . Adv. Math. 8(1972), 1–161. https://doi.org/10.1016/0001-8708(72)90002-3 Google Scholar
Papachristou, C. and Vas̆, L., A note on (𝛼, 𝛽)-higher derivations and their extensions to modules of quotients . In: Ring and module theory , Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010, pp. 165–174.Google Scholar
Tanaka, M., Locally nilpotent derivations on modules . J. Math. Kyoto Univ. 49(2009), no. 1, 131–159. https://doi.org/10.1215/kjm/1248983033 Google Scholar
Vas̆, L., Differentiability of torsion theories . J. Pure Appl. Algebra 210(2007), no. 3, 847–853. https://doi.org/10.1016/j.jpaa.2006.12.002 Google Scholar
Vas̆, L., Extending ring derivations to right and symmetric rings and modules of quotients . Comm. Algebra 37(2009), no. 3, 794–810. https://doi.org/10.1080/00927870802271664 Google Scholar
Xu, F., On the cohomology rings of small categories . J. Pure Appl. Algebra 212(2008), no. 11, 2555–2569. https://doi.org/10.1016/j.jpaa.2008.04.004 Google Scholar
Xu, F., Hochschild and ordinary cohomology rings of small categories . Adv. Math. 219(2008), no. 6, 1872–1893. https://doi.org/10.1016/j.aim.2008.07.014 Google Scholar