Abstract
We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri, Pospíšil, ŠÅ¥ovíček and Trlifaj (2014). We show that the n-tilting classes can equivalently be expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal:
Funding source: Czech Science Foundation
Award Identifier / Grant number: 14-15479S
Funding source: Czech Academy of Sciences
Award Identifier / Grant number: RVO: 6798584
Funding source: Charles University in Prague
Award Identifier / Grant number: SVV-2016-260336
Funding statement: Both authors were partially supported by the Czech Science Foundation under the grant no. 14-15479S. The first named author was also partially supported by RVO: 6798584 of the Czech Academy of Sciences, and by the project SVV-2016-260336 of the Charles University in Prague.
References
[1] L. Alonso Tarrío, A. Jeremías López and J. Lipman, Local homology and cohomology on schemes, Ann. Sc. École Norm. Supér. (4) 30 (1997), no. 1, 1–39. 10.1016/S0012-9593(97)89914-4Search in Google Scholar
[2] L. Alonso Tarrío, A. Jeremías López and M. Saorín, Compactly generated t-structures on the derived category of a Noetherian ring, J. Algebra 324 (2010), no. 3, 313–346. 10.1016/j.jalgebra.2010.04.023Search in Google Scholar
[3] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed., Grad. Texts in Math. 13, Springer, New York, 1992. 10.1007/978-1-4612-4418-9Search in Google Scholar
[4] L. Angeleri Hügel and F. U. Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math. 13 (2001), no. 2, 239–250. 10.1515/form.2001.006Search in Google Scholar
[5] L. Angeleri Hügel and M. Hrbek, Silting modules over commutative rings, Int. Math. Res. Not. IMRN 2017 (2017), no. 13, 4131–4151. Search in Google Scholar
[6] L. Angeleri Hügel, D. Pospíšil, J. Šťovíček and J. Trlifaj, Tilting, cotilting, and spectra of commutative Noetherian rings, Trans. Amer. Math. Soc. 366 (2014), no. 7, 3487–3517. 10.1090/S0002-9947-2014-05904-7Search in Google Scholar
[7] L. Angeleri Hügel and M. Saorín, t-Structures and cotilting modules over commutative noetherian rings, Math. Z. 277 (2014), no. 3–4, 847–866. 10.1007/s00209-014-1281-ySearch in Google Scholar
[8] S. Bazzoni, A characterization of n-cotilting and n-tilting modules, J. Algebra 273 (2004), no. 1, 359–372. 10.1016/S0021-8693(03)00432-0Search in Google Scholar
[9] S. Bazzoni, Cotilting and tilting modules over Prüfer domains, Forum Math. 19 (2007), no. 6, 1005–1027. 10.1515/FORUM.2007.039Search in Google Scholar
[10] S. Bazzoni, Equivalences induced by infinitely generated tilting modules, Proc. Amer. Math. Soc. 138 (2010), no. 2, 533–544. 10.1090/S0002-9939-09-10120-XSearch in Google Scholar
[11] S. Bazzoni, F. Mantese and A. Tonolo, Derived equivalence induced by infinitely generated n-tilting modules, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4225–4234. 10.1090/S0002-9939-2011-10900-6Search in Google Scholar
[12] S. Bazzoni and J. Šťovíček, All tilting modules are of finite type, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3771–3781. 10.1090/S0002-9939-07-08911-3Search in Google Scholar
[13] A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and Topology on Singular Spaces. I (Luminy 1981), Astérisque 100, Société Mathématique de France, Paris (1982), 5–171. Search in Google Scholar
[14] W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University, Cambridge, 1993. Search in Google Scholar
[15] R. Colpi and J. Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), no. 2, 614–634. 10.1006/jabr.1995.1368Search in Google Scholar
[16] W. Crawley-Boevey, Infinite-dimensional modules in the representation theory of finite-dimensional algebras, Algebras and Modules. I (Trondheim 1996), CMS Conf. Proc. 23, American Mathematical Society, Providence (1998), 29–54. Search in Google Scholar
[17] H. Dao and R. Takahashi, Classification of resolving subcategories and grade consistent functions, Int. Math. Res. Not. IMRN 2015 (2015), no. 1, 119–149. 10.1093/imrn/rnt141Search in Google Scholar
[18] W. G. Dwyer and J. P. C. Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002), no. 1, 199–220. 10.1353/ajm.2002.0001Search in Google Scholar
[19] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Exp. Math. 30, Walter de Gruyter, Berlin, 2000. 10.1515/9783110803662Search in Google Scholar
[20] L. Fiorot, F. Mattiello and M. Saorín, Derived equivalences induced by nonclassical tilting objects, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1505–1514. 10.1090/proc/13368Search in Google Scholar
[21] G. Garkusha and M. Prest, Torsion classes of finite type and spectra, K-theory and Noncommutative Geometry, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2008), 393–412. 10.4171/060-1/12Search in Google Scholar
[22] R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules. Volume 1. Approximations, Second revised and extended edition, De Gruyter Exp. Math. 41, Walter de Gruyter, Berlin, 2012. 10.1515/9783110218114Search in Google Scholar
[23] J. P. C. Greenlees, First steps in brave new commutative algebra, Interactions Between Homotopy Theory and Algebra, Contemp. Math. 436, American Mathematical Society, Providence (2007), 239–275. 10.1090/conm/436/08412Search in Google Scholar
[24] J. P. C. Greenlees and J. P. May, Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), no. 2, 438–453. 10.1016/0021-8693(92)90026-ISearch in Google Scholar
[25] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. 10.1090/S0002-9947-1969-0251026-XSearch in Google Scholar
[26] M. Hrbek, One-tilting classes and modules over commutative rings, J. Algebra 462 (2016), 1–22. 10.1016/j.jalgebra.2016.05.014Search in Google Scholar
[27] P. T. Johnstone, Stone Spaces, Cambridge Stud. Adv. Math. 3, Cambridge University, Cambridge, 1982. Search in Google Scholar
[28] M. Kashiwara and P. Schapira, Categories and Sheaves, Grundlehren Math. Wiss. 332, Springer, Berlin, 2006. 10.1007/3-540-27950-4Search in Google Scholar
[29] J. Kock and W. Pitsch, Hochster duality in derived categories and point-free reconstruction of schemes, Trans. Amer. Math. Soc. 369 (2017), no. 1, 223–261. 10.1090/tran/6773Search in Google Scholar
[30] H. Krause, Localization theory for triangulated categories, Triangulated Categories, London Math. Soc. Lecture Note Ser. 375, Cambridge University, Cambridge (2010), 161–235. 10.1017/CBO9781139107075.005Search in Google Scholar
[31] T. Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer, New York, 1991. 10.1007/978-1-4684-0406-7Search in Google Scholar
[32] L. G. Lewis, Jr., J. P. May, M. Steinberger and J. E. McClure, Equivariant Stable Homotopy Theory, Lecture Notes in Math. 1213, Springer, Berlin, 1986. 10.1007/BFb0075778Search in Google Scholar
[33] Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146. 10.1007/BF01163359Search in Google Scholar
[34] P. Nicolás and M. Saorín, Parametrizing recollement data for triangulated categories, J. Algebra 322 (2009), no. 4, 1220–1250. 10.1016/j.jalgebra.2009.04.035Search in Google Scholar
[35] D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge University, London, 1968. 10.1017/CBO9780511565922Search in Google Scholar
[36] M. Porta, L. Shaul and A. Yekutieli, On the homology of completion and torsion, Algebr. Represent. Theory 17 (2014), no. 1, 31–67. 10.1007/s10468-012-9385-8Search in Google Scholar
[37] L. Positselski, Dedualizing complexes and MGM duality, J. Pure Appl. Algebra 220 (2016), no. 12, 3866–3909. 10.1016/j.jpaa.2016.05.019Search in Google Scholar
[38] L. Positselski and J. Šťovíček, The Tilting–Cotilting Correspondence, Int. Math. Res. Not. IMRN (2019), 10.1093/imrn/rnz116. 10.1093/imrn/rnz116Search in Google Scholar
[39] M. Prest, Purity, Spectra and Localisation, Encyclopedia Math. Appl. 121, Cambridge University, Cambridge, 2009. 10.1017/CBO9781139644242Search in Google Scholar
[40] P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), no. 2, 161–180. 10.7146/math.scand.a-14399Search in Google Scholar
[41] B. Stenström, Rings of Quotients. An Introduction to Methods of Ring Theory, Grundlehren Math. Wiss. 217, Springer, New York, 1975. 10.1007/978-3-642-66066-5_1Search in Google Scholar
[42] R. W. Thomason, The classification of triangulated subcategories, Compos. Math. 105 (1997), no. 1, 1–27. 10.1023/A:1017932514274Search in Google Scholar
[43] J. Šaroch and J. Šťovíček, The countable telescope conjecture for module categories, Adv. Math. 219 (2008), no. 3, 1002–1036. 10.1016/j.aim.2008.05.012Search in Google Scholar
[44] J. Šťovíček, Derived equivalences induced by big cotilting modules, Adv. Math. 263 (2014), 45–87. 10.1016/j.aim.2014.06.007Search in Google Scholar
[45] J. Šťovíček, J. Trlifaj and D. Herbera, Cotilting modules over commutative Noetherian rings, J. Pure Appl. Algebra 218 (2014), no. 9, 1696–1711. 10.1016/j.jpaa.2014.01.008Search in Google Scholar
[46] C. A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge University, Cambridge, 1994. 10.1017/CBO9781139644136Search in Google Scholar
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