Hostname: page-component-7f64f4797f-7vqc4 Total loading time: 0 Render date: 2025-11-04T12:08:03.199Z 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: abhishekbanerjee1313@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 703 - 714
Copyright
© Canadian Mathematical Society 2019 

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