Tilting, cotilting, and spectra of commutative noetherian rings
HTML articles powered by AMS MathViewer
- by Lidia Angeleri Hügel, David Pospíšil, Jan Šťovíček and Jan Trlifaj PDF
- Trans. Amer. Math. Soc. 366 (2014), 3487-3517 Request permission
Abstract:
We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated modules of bounded projective dimension are classified. We also relate our results to Hochster’s Conjecture on the existence of finitely generated maximal Cohen-Macaulay modules.References
- Lidia Angeleri Hügel and Flávio Ulhoa Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math. 13 (2001), no. 2, 239–250. MR 1813669, DOI 10.1515/form.2001.006
- Lidia Angeleri Hügel, Dolors Herbera, and Jan Trlifaj, Tilting modules and Gorenstein rings, Forum Math. 18 (2006), no. 2, 211–229. MR 2218418, DOI 10.1515/FORUM.2006.013
- Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
- S. Bazzoni, Cotilting modules are pure-injective, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3665–3672. MR 1998172, DOI 10.1090/S0002-9939-03-06938-7
- Silvana Bazzoni, A characterization of $n$-cotilting and $n$-tilting modules, J. Algebra 273 (2004), no. 1, 359–372. MR 2032465, DOI 10.1016/S0021-8693(03)00432-0
- Silvana Bazzoni, Cotilting and tilting modules over Prüfer domains, Forum Math. 19 (2007), no. 6, 1005–1027. MR 2367952, DOI 10.1515/FORUM.2007.039
- Silvana Bazzoni and Dolors Herbera, One dimensional tilting modules are of finite type, Algebr. Represent. Theory 11 (2008), no. 1, 43–61. MR 2369100, DOI 10.1007/s10468-007-9064-3
- Silvana Bazzoni and Jan Šťovíček, All tilting modules are of finite type, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3771–3781. MR 2341926, DOI 10.1090/S0002-9939-07-08911-3
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Aslak Bakke Buan and Henning Krause, Cotilting modules over tame hereditary algebras, Pacific J. Math. 211 (2003), no. 1, 41–59. MR 2016589, DOI 10.2140/pjm.2003.211.41
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR 1731415
- Riccardo Colpi and Claudia Menini, On the structure of $*$-modules, J. Algebra 158 (1993), no. 2, 400–419. MR 1226797, DOI 10.1006/jabr.1993.1138
- Riccardo Colpi and Jan Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), no. 2, 614–634. MR 1359905, DOI 10.1006/jabr.1995.1368
- H. Dao and R. Takahashi. Classification of resolving subcategories and grade consistent functions. Preprint, available at http://arxiv.org/pdf/1202.5605v1.pdf, 2012.
- Gabriella D’Este, Reflexive modules are not closed under submodules, Representations of algebras (São Paulo, 1999) Lecture Notes in Pure and Appl. Math., vol. 224, Dekker, New York, 2002, pp. 53–64. MR 1884806
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Edgar E. Enochs and Overtoun M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. MR 1753146, DOI 10.1515/9783110803662
- Rüdiger Göbel and Jan Trlifaj, Approximations and endomorphism algebras of modules, De Gruyter Expositions in Mathematics, vol. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006. MR 2251271, DOI 10.1515/9783110199727
- Ulrich Görtz and Torsten Wedhorn, Algebraic geometry I, Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden, 2010. Schemes with examples and exercises. MR 2675155, DOI 10.1007/978-3-8348-9722-0
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Melvin Hochster, Cohen-Macaulay modules, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 120–152. MR 0340251
- Henning Krause, The spectrum of a locally coherent category, J. Pure Appl. Algebra 114 (1997), no. 3, 259–271. MR 1426488, DOI 10.1016/S0022-4049(95)00172-7
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- H. A. Nielsen, Elementary commutative algebra. Lecture Notes, Department of Mathematical Sciences, University of Aarhus. Available at http://home.imf.au.dk/holger/eca05.pdf, 2005.
- David Pospíšil and Jan Trlifaj, Tilting for regular rings of Krull dimension two, J. Algebra 336 (2011), 184–199. MR 2802536, DOI 10.1016/j.jalgebra.2011.02.047
- Mike Prest, Purity, spectra and localisation, Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009. MR 2530988, DOI 10.1017/CBO9781139644242
- Paul Roberts, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 127–130. MR 799793, DOI 10.1090/S0273-0979-1985-15394-7
- Peter Schenzel, On birational Macaulayfications and Cohen-Macaulay canonical modules, J. Algebra 275 (2004), no. 2, 751–770. MR 2052635, DOI 10.1016/j.jalgebra.2003.12.016
- Jean-Pierre Serre, Local algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. Translated from the French by CheeWhye Chin and revised by the author. MR 1771925, DOI 10.1007/978-3-662-04203-8
- Donald Stanley and Binbin Wang, Classifying subcategories of finitely generated modules over a Noetherian ring, J. Pure Appl. Algebra 215 (2011), no. 11, 2684–2693. MR 2802159, DOI 10.1016/j.jpaa.2011.03.013
- Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953
- Ryo Takahashi, Classifying subcategories of modules over a commutative Noetherian ring, J. Lond. Math. Soc. (2) 78 (2008), no. 3, 767–782. MR 2456904, DOI 10.1112/jlms/jdn056
- Ryo Takahashi, Contravariantly finite resolving subcategories over commutative rings, Amer. J. Math. 133 (2011), no. 2, 417–436. MR 2797352, DOI 10.1353/ajm.2011.0011
- Jan Trlifaj and David Pospíšil, Tilting and cotilting classes over Gorenstein rings, Rings, modules and representations, Contemp. Math., vol. 480, Amer. Math. Soc., Providence, RI, 2009, pp. 319–334. MR 2508160, DOI 10.1090/conm/480/09383
- Jin Zhong Xu, Minimal injective and flat resolutions of modules over Gorenstein rings, J. Algebra 175 (1995), no. 2, 451–477. MR 1339651, DOI 10.1006/jabr.1995.1196
- Helmut Zöschinger, Linear-kompakte Moduln über noetherschen Ringen, Arch. Math. (Basel) 41 (1983), no. 2, 121–130 (German). MR 719414, DOI 10.1007/BF01196867
Additional Information
- Lidia Angeleri Hügel
- Affiliation: Dipartimento di Informatica, Settore di Matematica, Università degli Studi di Verona, Strada le Grazie 15 - Ca’ Vignal, 37134 Verona, Italy
- MR Author ID: 358523
- Email: lidia.angeleri@univr.it
- David Pospíšil
- Affiliation: Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
- Email: pospisil.david@gmail.com
- Jan Šťovíček
- Affiliation: Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
- Email: stovicek@karlin.mff.cuni.cz
- Jan Trlifaj
- Affiliation: Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
- MR Author ID: 174420
- ORCID: 0000-0001-5773-8661
- Email: trlifaj@karlin.mff.cuni.cz
- Received by editor(s): March 2, 2012
- Received by editor(s) in revised form: June 25, 2012
- Published electronically: February 6, 2014
- Additional Notes: This research was supported by GAČR 201/09/0816, GAČR 201/09/H012, GAČR P201/10/P084, as well as by MEC-DGESIC (Spain) through Project MTM2008-06201-C02-01, and by the Comissionat Per Universitats i Recerca de la Generalitat de Catalunya through Project 2009 SGR 1389
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 3487-3517
- MSC (2010): Primary 13C05, 13E05, 16D90; Secondary 13C14, 13C60, 13D07, 16E30
- DOI: https://doi.org/10.1090/S0002-9947-2014-05904-7
- MathSciNet review: 3192604