Weakly Stable Torsion Classes
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Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class T can be associated with a dg ring AT; in well behaved situations there is a homological epimorphism A → AT. We end by studying torsion and completion with respect to a single regular and normal element.
KeywordsNoncommutative ring theory Torsion theories Derived categories Derived functors
Mathematics Subject Classification (2010)(Primary) 16S90 (Secondary) 16E35 18E30 18G10
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The author would like to thank Amnon Yekutieli for his assistance and many suggestions regarding the material in this paper, and the anonymous referee for their comments.
- 5.Goodearl, K.R., Warfield Jr., R.B.: An Introduction to Noncommutative Noetherian Rings, 2nd edn., vol. 61. London Mathematical Society Student Texts, Cambridge University Press, Cambridge (2004)Google Scholar
- 7.Grothendieck, A., Raynaud, M.: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), Advanced Studies in Pure Mathematics 2, North-Holland Publishing Company. Updated version by Yves Laszlo available at https://arxiv.org/abs/math/0511279 (1968)
- 8.Hartshorne, R.: Residues and Duality, Lecture Notes in Mathematics, vol. 20. Springer, Berlin (1966)Google Scholar
- 10.Kashiwara, M., Schapira, P.: Categories and Sheaves, Grundlehren der mathematischen Wissenschaften, vol. 332. Springer, Berlin (2006)Google Scholar
- 11.Krause, H.: Localization Theory for Triangulated Categories, Triangulated Categories, London Mathematical Society Lecture Note Series, vol. 375, pp. 161–253. Cambridge University Press (2010)Google Scholar
- 18.Stenström, B.: Rings of quotients, Grundlehren der mathematischen Wissenschaften, vol. 217. Springer, Berlin (1975)Google Scholar
- 19.Vyas, R., Yekutieli, A.: Weak proregularity, weak stability, and the noncommutative MGM equivalence. In press, J. Algebra. 61 pages. https://doi.org/10.1016/j.jalgebra.2018.07.023
- 20.Vyas, R., Yekutieli, A.: Dualizing complexes in the noncommutative arithmetic context, in preparationGoogle Scholar
- 22.Yekutieli, A.: Derived categories of bimodules, in preparationGoogle Scholar