Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-08T01:37:05.649Z Has data issue: false hasContentIssue false
  • English
  • Français

FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

Part of: Rings and algebras arising under various constructions Rings and algebras with additional structure Homological methods Abelian categories Modules, bimodules and ideals

Published online by Cambridge University Press:  07 May 2019

LEONID POSITSELSKI Open the ORCID record for LEONID POSITSELSKI [Opens in a new window]
LEONID POSITSELSKI*
Affiliation:
Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow119048, Russia Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow127051, Russia and Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa31905, Israel e-mail: positselski@yandex.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

Let R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

MSC classification

Primary: 16D40: Free, projective, and flat modules and ideals 16E10: Homological dimension 16S90: Torsion theories; radicals on module categories 16W60: Valuations, completions, formal power series and related constructions 18E10: Exact categories, abelian categories
Secondary: 16E30: Homological functors on modules (Tor, Ext, etc.) 18E35: Localization of categories
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 2 , May 2020 , pp. 383 - 439
Copyright
Copyright © Glasgow Mathematical Journal Trust 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

Hügel, L. Angeleri and Hrbek, M., Silting modules over commutative rings, Int. Math. Res. Not. 2017(13) (2017), 4131–4151. arXiv:1602.04321 [math.RT].Google Scholar
Hügel, L. Angeleri, Marks, F., Št’ovček, J., Takahashi, R. and Vitória, J., Flat ring epimorphisms and universal localizations of commutative rings, Electronic preprint. arXiv:1807.01982 [math.RT].Google Scholar
Hügel, L. Angeleri and Sánchez, J., Tilting modules arising from ring epimorphisms, Algebras Represent. Theory. 14(2) (2011), 217–246. arXiv:0804.1313 [math.RT].CrossRefGoogle Scholar
Bazzoni, S. and Positselski, L., S-almost perfect commutative rings, Electronic preprint. arXiv:1801.04820 [math.AC].Google Scholar
Bazzoni, S. and Positselski, L., Contramodules over pro-perfect topological rings, the covering property in categorical tilting theory, and homological ring epimorphisms, Electronic preprint arXiv:1807.10671 [math.CT].Google Scholar
Bazzoni, S. and Salce, L., Strongly flat covers, J. Lond. Math. Soc. 66(2) (2002), 276–294.CrossRefGoogle Scholar
Bazzoni, S. and Salce, L., On strongly flat modules over integral domains, Rocky Mt. J. Math. 34(2) (2004), 417–439.10.1216/rmjm/1181069861CrossRefGoogle Scholar
Bourbaki, N., Algèbre Commutative, Chapitres 1 à 4. Masson, Paris, 1985 (Springer-Verlag, Berlin–Heidelberg–New York, 2006).CrossRefGoogle Scholar
Eklof, P. C. and Trlifaj, J., How to make Ext vanish, Bull. Lond. Math. Soc. 33(1) (2001), 41–51.CrossRefGoogle Scholar
Facchini, A. and Nazemian, Z., Covering classes, strongly flat modules, and completions, Electronic preprint. arXiv:1808.02397 [math.RA].Google Scholar
Faith, C., Algebra: rings, modules and categories I (Springer-Verlag, Berlin–Heidelberg–New York, 1973).CrossRefGoogle Scholar
Gabriel, P., Des catégories abéliennes, Bull. de la Soc. Math. de France 90 (1962), 323–448.Google Scholar
Geigle, W. and Lenzing, H., Perpendicular categories with applications to representations and sheaves. J. Algebra 144(2) (1991), 273–343.CrossRefGoogle Scholar
Göbel, R. and Trlifaj, J., Approximations and endomorphism algebras of modules, Second revised and extended edition. (De Gruyter, Berlin–Boston, 2012).Google Scholar
Hrbek, M., One-tilting classes and modules over commutative rings. J. Algebra 462 (2016), 1–22. arXiv:1507.02811 [math.AC].Google Scholar
Marks, F. and Št’ovček, J., Universal localizations via silting. Electronic preprint arXiv:1605.04222 [math.RT], in Proceedings of the Royal Society of Edinburgh, Sect. A (to appear)Google Scholar
Matlis, E., Cotorsion modules, Memoirs Am. Math. Soc. 49 (1964), 66 pp.Google Scholar
Nunke, R. J., Modules of extensions over Dedekind rings, Illinois J. Math. 3(2) (1959), 222–241.CrossRefGoogle Scholar
Popesco, N. and Gabriel, P., Caractérisation des catégoies abéliennes avec générateurs and limites inductives exactes, Comptes Rendus Acad. Sci. Paris 258 (1964), 4188–4190.Google Scholar
Positselski, L., Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures. Appendix C in collaboration with D. Rumynin; Appendix D in collaboration with S. Arkhipov. Monografie Matematyczne, vol. 70, (Birkhäuser/Springer, Basel, 2010). xxiv+349 pp. arXiv:0708.3398 [math.CT].CrossRefGoogle Scholar
Positselski, L., Weakly curved A∞-algebras over a topological local ring, in Mémoires de la Soc. Math. de France (to appear), Electronic preprint. arXiv:1202.2697 [math.CT].Google Scholar
Positselski, L., Contramodules, Electronic preprint. arXiv:1503.00991 [math.CT].Google Scholar
Positselski, L., Triangulated Matlis equivalence, J. Algebra Appl. 17(4) (2018), article ID 1850067, 44 pp. arXiv:1605.08018 [math.CT].10.1142/S0219498818500676CrossRefGoogle Scholar
Positselski, L., Smooth duality and co-contra correspondence, Electronic preprint. arXiv:1609.04597 [math.CT].Google Scholar
Positselski, L., Abelian right perpendicular subcategories in module categories, Electronic preprint. arXiv:1705.04960 [math.CT].Google Scholar
Positselski, L. and Rosický, J., Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories, J. Algebra 483 (2017), 83–128. arXiv:1512.08119 [math.CT].CrossRefGoogle Scholar
Positselski, L. and Slávik, A., Flat morphisms of finite presentation are very flat, Electronic preprint. arXiv:1708.00846 [math.AC].Google Scholar
Positselski, L. and Slávik, A., On strongly flat and weakly cotorsion modules. Math. Zeitschrift 291 (2019), 831–875. DOI: 10.1007/s00209-018-2116-z, arXiv:1708.06833 [math.AC].CrossRefGoogle Scholar
Positselski, L. and Št’ovček, J., The tilting-cotilting correspondence, Electronic preprint. arXiv:1710.02230 [math.CT]. = 4150Google Scholar
Stenström, B., Rings of quotients. An introduction to methods of ring theory (Springer-Verlag, Berlin–Heidelberg–New York, 1975).Google Scholar
Trlifaj, J., Cotorsion theories induced by tilting and cotilting modules, in Abelian groups, rings and modules (Perth, 2000), Contemp. Math., vol. 273 (AMS, Providence, 2001), 285–300.Google Scholar
Trlifaj, J., Covers, envelopes, and cotorsion theories, Lecture notes for the workshop “Homological methods in module theory”, Cortona, September 2000, 39 pp. Available at http://matematika.cuni.cz/dl/trlifaj/NALG077cortona.pdfGoogle Scholar
Xu, J., Flat covers of modules, Lecture Notes in Math., vol. 1634 (Springer-Verlag, Berlin, 1996).CrossRefGoogle Scholar