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Universal localizations via silting

  • Frederik Marks (a1) and Jan Št'ovíček (a2)
Abstract

We show that silting modules are closely related with localizations of rings. More precisely, every partial silting module gives rise to a localization at a set of maps between countably generated projective modules and, conversely, every universal localization, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.

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1Adachi, T., Iyama, O. and Reiten, I.. τ-tilting theory. Compositio Math. 150 (2014), 415452.
2Angeleri Hügel, L. and Archetti, M.. Tilting modules and universal localization. Forum Math. 24 (2012), 709731.
3Angeleri Hügel, L. and Coelho, F.. Infinitely generated tilting modules of finite projective dimension. Forum Math. 13 (2001), 239250.
4Angeleri Hügel, L. and Hrbek, M.. Silting modules over commutative rings. Int. Math. Res. Not. IMRN 13 (2017), 41314151.
5Angeleri Hügel, L. and Sánchez, J.. Tilting modules arising from ring epimorphisms. Algebr. Represent. Theory 14 (2011), 217246.
6Angeleri Hügel, L. and Sánchez, J.. Tilting modules over tame hereditary algebras. J. Reine Angew. Math. 682 (2013), 148.
7Angeleri Hügel, L., Herbera, D. and Trlifaj, J.. Tilting modules and Gorenstein rings. Forum Math. 18 (2006), 211229.
8Angeleri Hügel, L., Marks, F. and Vitória, J.. Silting modules. Int. Math. Res. Not. IMRN 4 (2016a), 12511284.
9Angeleri Hügel, L., Marks, F. and Vitória, J.. Silting modules and ring epimorphisms. Adv. Math. 303 (2016b), 10441076.
10Auslander, M.. Representation dimension of artin algebras, Queen Mary College Mathematics Notes (1971).
11Auslander, M., Reiten, I. and Smalø, S. O.. Representation theory of Artin algebras, Corrected reprint of the 1995 original. Cambridge Studies in Advanced Mathematics, vol. 36 (Cambridge: Cambridge University Press, 1997).
12Bazzoni, S. and Herbera, D.. One dimensional tilting modules are of finite type. Algebr. Represent. Theory 11 (2008), 4361.
13Bazzoni, S. and Šťovíček, J.. All tilting modules are of finite type. Proc. Amer. Math. Soc. 135 (2007), 37713781.
14Bazzoni, S. and Šťovíček, J.. Smashing localizations of rings of weak global dimension at most one. Adv. Math. 305 (2017), 351401.
15Bazzoni, S., Eklof, P. C. and Trlifaj, J.. Tilting cotorsion pairs. Bull. London Math. Soc. 37 (2005), 683696.
16Brodmann, M. P. and Sharp, R. Y.. Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics, vol. 60 (Cambridge: Cambridge University Press, 1998).
17Bruns, W. and Herzog, J.. Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge: Cambridge University Press, 1993).
18Colpi, R. and Trlifaj, J.. Tilting modules and tilting torsion theories. J. Algebra 178 (1995), 614634.
19Colpi, R., Tonolo, A. and Trlifaj, J.. Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes. J. Pure Appl. Algebra 211 (2007), 223234.
20Dwyer, W. G. and Greenlees, J. P. C.. Complete modules and torsion modules. Amer. J. Math. 124 (2002), 199220.
21Eklof, P. C. and Trlifaj, J.. How to make Ext vanish. Bull. London Math. Soc. 33 (2001), 4151.
22Gabriel, P. and de la Penã, J. A.. Quotients of representation-finite algebras. Comm. Algebra 15 (1987), 279307.
23Geigle, W. and Lenzing, H.. Perpendicular categories with applications to representations and sheaves. J. Algebra 144 (1991), 273343.
24Göbel, R. and Trlifaj, J.. Approximations and endomorphism algebras of modules, volume 41 of de Gruyter Expositions in Mathematics (Berlin: Walter de Gruyter GmbH & Co. KG, 2006).
25Hartshorne, R.. Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (New York-Heidelberg: Springer-Verlag, 1977).
26Hellus, M. and Stückrad, J.. Local cohomology and Matlis duality. Univ. Iagel. Acta Math 45 (2007), 6370.
27Huneke, C.. Lectures on local cohomology, with Appendix 1 by Amelia Taylor, Contemp. Math. 436, Interactions between homotopy theory and algebra,pp. 5199, Amer. Math. Soc., Providence, RI (2007).
28Jasso, G.. Reduction of τ-tilting modules and torsion pairs. Int. Math. Res. Not. IMRN 16 (2015), 71907237.
29Krause, H. and Šťovíček, J.. The telescope conjecture for hereditary rings via Ext-orthogonal pairs. Adv. Math. 225 (2010), 23412364.
30Marks, F. and Šťovíček, J.. Torsion classes, wide subcategories and localisations. Bull. London Math. Soc. 49 (2017), 405416.
31Neeman, A. and Ranicki, A.. Noncommutative localisation in algebraic K-theory I. Geom. Topol. 8 (2004), 13851425.
32Ranicki, A.. Noncommutative localization in topology. In Proc. 2002 ICMS Edinburgh meeting on Noncommutative Localization in Algebra and Topology, LMS Lecture Notes Series, vol. 330,pp. 81102 (Cambridge: Cambridge University Press, 2006).
33Schofield, A.. Representations of rings over skew fields. LMS Lecture Note Series, vol. 92 (Cambridge: Cambridge University Press, 1985).
34Schofield, A.. Severe right Ore sets and universal localisation, preprint, arXiv:0708.0246.
35Schofield, A.. Universal localisations of hereditary rings, preprint, arXiv:0708.0257.
36Šťovíček, J. and Trlifaj, J.. All tilting modules are of countable type. Bull. London Math. Soc. 39 (2007), 121132.
37Whitehead, J. M.. Projective modules and their trace ideals. Comm. Algebra 8 (1980), 18731901.
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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