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Universal localizations via silting

Published online by Cambridge University Press:  27 December 2018

Frederik Marks
Affiliation:
University of Stuttgart, Institute for Algebra and Number Theory, Pfaffenwaldring 57, 70569 Stuttgart, Germany ([email protected])
Jan Št'ovíček
Affiliation:
Faculty of Mathematics and Physics, Department of Algebra, Charles University in Prague, Sokolovská 83, 186 75 Praha, Czech Republic ([email protected])

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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References

1Adachi, T., Iyama, O. and Reiten, I.. τ-tilting theory. Compositio Math. 150 (2014), 415452.Google Scholar
2Angeleri Hügel, L. and Archetti, M.. Tilting modules and universal localization. Forum Math. 24 (2012), 709731.Google Scholar
3Angeleri Hügel, L. and Coelho, F.. Infinitely generated tilting modules of finite projective dimension. Forum Math. 13 (2001), 239250.Google Scholar
4Angeleri Hügel, L. and Hrbek, M.. Silting modules over commutative rings. Int. Math. Res. Not. IMRN 13 (2017), 41314151.Google Scholar
5Angeleri Hügel, L. and Sánchez, J.. Tilting modules arising from ring epimorphisms. Algebr. Represent. Theory 14 (2011), 217246.Google Scholar
6Angeleri Hügel, L. and Sánchez, J.. Tilting modules over tame hereditary algebras. J. Reine Angew. Math. 682 (2013), 148.Google Scholar
7Angeleri Hügel, L., Herbera, D. and Trlifaj, J.. Tilting modules and Gorenstein rings. Forum Math. 18 (2006), 211229.Google Scholar
8Angeleri Hügel, L., Marks, F. and Vitória, J.. Silting modules. Int. Math. Res. Not. IMRN 4 (2016a), 12511284.Google Scholar
9Angeleri Hügel, L., Marks, F. and Vitória, J.. Silting modules and ring epimorphisms. Adv. Math. 303 (2016b), 10441076.Google Scholar
10Auslander, M.. Representation dimension of artin algebras, Queen Mary College Mathematics Notes (1971).Google Scholar
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).Google Scholar
12Bazzoni, S. and Herbera, D.. One dimensional tilting modules are of finite type. Algebr. Represent. Theory 11 (2008), 4361.Google Scholar
13Bazzoni, S. and Šťovíček, J.. All tilting modules are of finite type. Proc. Amer. Math. Soc. 135 (2007), 37713781.Google Scholar
14Bazzoni, S. and Šťovíček, J.. Smashing localizations of rings of weak global dimension at most one. Adv. Math. 305 (2017), 351401.Google Scholar
15Bazzoni, S., Eklof, P. C. and Trlifaj, J.. Tilting cotorsion pairs. Bull. London Math. Soc. 37 (2005), 683696.Google Scholar
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).Google Scholar
17Bruns, W. and Herzog, J.. Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge: Cambridge University Press, 1993).Google Scholar
18Colpi, R. and Trlifaj, J.. Tilting modules and tilting torsion theories. J. Algebra 178 (1995), 614634.Google Scholar
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.Google Scholar
20Dwyer, W. G. and Greenlees, J. P. C.. Complete modules and torsion modules. Amer. J. Math. 124 (2002), 199220.Google Scholar
21Eklof, P. C. and Trlifaj, J.. How to make Ext vanish. Bull. London Math. Soc. 33 (2001), 4151.Google Scholar
22Gabriel, P. and de la Penã, J. A.. Quotients of representation-finite algebras. Comm. Algebra 15 (1987), 279307.Google Scholar
23Geigle, W. and Lenzing, H.. Perpendicular categories with applications to representations and sheaves. J. Algebra 144 (1991), 273343.Google Scholar
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).Google Scholar
25Hartshorne, R.. Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (New York-Heidelberg: Springer-Verlag, 1977).Google Scholar
26Hellus, M. and Stückrad, J.. Local cohomology and Matlis duality. Univ. Iagel. Acta Math 45 (2007), 6370.Google Scholar
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).Google Scholar
28Jasso, G.. Reduction of τ-tilting modules and torsion pairs. Int. Math. Res. Not. IMRN 16 (2015), 71907237.Google Scholar
29Krause, H. and Šťovíček, J.. The telescope conjecture for hereditary rings via Ext-orthogonal pairs. Adv. Math. 225 (2010), 23412364.Google Scholar
30Marks, F. and Šťovíček, J.. Torsion classes, wide subcategories and localisations. Bull. London Math. Soc. 49 (2017), 405416.Google Scholar
31Neeman, A. and Ranicki, A.. Noncommutative localisation in algebraic K-theory I. Geom. Topol. 8 (2004), 13851425.Google Scholar
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).Google Scholar
33Schofield, A.. Representations of rings over skew fields. LMS Lecture Note Series, vol. 92 (Cambridge: Cambridge University Press, 1985).Google Scholar
34Schofield, A.. Severe right Ore sets and universal localisation, preprint, arXiv:0708.0246.Google Scholar
35Schofield, A.. Universal localisations of hereditary rings, preprint, arXiv:0708.0257.Google Scholar
36Šťovíček, J. and Trlifaj, J.. All tilting modules are of countable type. Bull. London Math. Soc. 39 (2007), 121132.Google Scholar
37Whitehead, J. M.. Projective modules and their trace ideals. Comm. Algebra 8 (1980), 18731901.Google Scholar