On compactly generated torsion pairs and the classification of co-𝑡-structures for commutative noetherian rings

Šťovíček, Jan; Pospíšil, David

doi:10.1090/tran/6561

On compactly generated torsion pairs and the classification of co-$t$-structures for commutative noetherian rings
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Abstract:

We classify compactly generated co-$t$-structures on the derived category of a commutative noetherian ring. In order to accomplish this, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the literature) in triangulated categories that resembles Bousfield localization theory. Finally, we show that the category of perfect complexes over a connected commutative noetherian ring admits only the trivial co-$t$-structures and (de)suspensions of the canonical co-$t$-structure and use this to describe all silting objects in the category.

References

Takuma Aihara and Osamu Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668. MR 2927802, DOI 10.1112/jlms/jdr055
Leovigildo Alonso Tarrío, Ana Jeremías López, and Manuel Saorín, Compactly generated $t$-structures on the derived category of a Noetherian ring, J. Algebra 324 (2010), no. 3, 313–346. MR 2651339, DOI 10.1016/j.jalgebra.2010.04.023
Lidia Angeleri Hügel, David Pospíšil, Jan Šťovíček, and Jan Trlifaj, Tilting, cotilting, and spectra of commutative Noetherian rings, Trans. Amer. Math. Soc. 366 (2014), no. 7, 3487–3517. MR 3192604, DOI 10.1090/S0002-9947-2014-05904-7
Lidia Angeleri Hügel and Manuel Saorín, t-Structures and cotilting modules over commutative noetherian rings, Math. Z. 277 (2014), no. 3-4, 847–866. MR 3229968, DOI 10.1007/s00209-014-1281-y
Luchezar Avramov and Stephen Halperin, Through the looking glass: a dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983) Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 1–27. MR 846435, DOI 10.1007/BFb0075446
Paul Balmer and Giordano Favi, Generalized tensor idempotents and the telescope conjecture, Proc. Lond. Math. Soc. (3) 102 (2011), no. 6, 1161–1185. MR 2806103, DOI 10.1112/plms/pdq050
Hanno Becker, Models for singularity categories, Adv. Math. 254 (2014), 187–232. MR 3161097, DOI 10.1016/j.aim.2013.11.016
Alexander Beilinson and Vadim Vologodsky, A DG guide to Voevodsky’s motives, Geom. Funct. Anal. 17 (2008), no. 6, 1709–1787. MR 2399083, DOI 10.1007/s00039-007-0644-5
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Apostolos Beligiannis, Relative homological algebra and purity in triangulated categories, J. Algebra 227 (2000), no. 1, 268–361. MR 1754234, DOI 10.1006/jabr.1999.8237
Apostolos Beligiannis and Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207. MR 2327478, DOI 10.1090/memo/0883
A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258 (English, with English and Russian summaries). MR 1996800, DOI 10.17323/1609-4514-2003-3-1-1-36
A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 519–541. MR 1039961, DOI 10.1070/IM1990v035n03ABEH000716
A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669–683 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 93–107. MR 1055981, DOI 10.1070/SM1991v070n01ABEH001253
A. I. Bondal and D. O. Orlov, Semiorthogonal decomposition for algebraic varieties, Preprint, arXiv:alg-geom/9506012, 1995.
M. V. Bondarko, Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences, and mixed complexes of sheaves: a survey, Preprint, arXiv:0903.0091v4, 2010.
M. V. Bondarko, Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), no. 3, 387–504. MR 2746283, DOI 10.1017/is010012005jkt083
A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. MR 551009, DOI 10.1016/0040-9383(79)90018-1
Theo Bühler, Exact categories, Expo. Math. 28 (2010), no. 1, 1–69. MR 2606234, DOI 10.1016/j.exmath.2009.04.004
Denis-Charles Cisinski, Images directes cohomologiques dans les catégories de modèles, Ann. Math. Blaise Pascal 10 (2003), no. 2, 195–244 (French, with French summary). MR 2031269
Denis-Charles Cisinski and Amnon Neeman, Additivity for derivator $K$-theory, Adv. Math. 217 (2008), no. 4, 1381–1475. MR 2382732, DOI 10.1016/j.aim.2007.10.003
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
J. Franke, Uniqueness theorems for certain triangulated categories with an Adams spectral sequence, 1996, Preprint, available at http://www.math.uiuc.edu/K-theory/0139/Adams.pdf.
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
Moritz Groth, Derivators, pointed derivators and stable derivators, Algebr. Geom. Topol. 13 (2013), no. 1, 313–374. MR 3031644, DOI 10.2140/agt.2013.13.313
Moritz Groth, Kate Ponto, and Michael Shulman, Mayer-Vietoris sequences in stable derivators, Homology Homotopy Appl. 16 (2014), no. 1, 265–294. MR 3211746, DOI 10.4310/HHA.2014.v16.n1.a15
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
A. Grothendieck, Les dérivateurs, Available at http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html, 1991.
Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
Alex Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (1988), no. 383, vi+78. MR 920963, DOI 10.1090/memo/0383
Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041, DOI 10.1090/surv/099
Mitsuo Hoshino, Yoshiaki Kato, and Jun-Ichi Miyachi, On $t$-structures and torsion theories induced by compact objects, J. Pure Appl. Algebra 167 (2002), no. 1, 15–35. MR 1868115, DOI 10.1016/S0022-4049(01)00012-3
Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
Mark Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553–592. MR 1938704, DOI 10.1007/s00209-002-0431-9
Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895, DOI 10.1090/memo/0610
Osamu Iyama and Yuji Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168. MR 2385669, DOI 10.1007/s00222-007-0096-4
C. U. Jensen, Les foncteurs dérivés de $\underleftarrow {\mmlToken {mi}{lim}}$ et leurs applications en théorie des modules, Lecture Notes in Mathematics, Vol. 254, Springer-Verlag, Berlin-New York, 1972. MR 0407091
Peter Jørgensen and David Pauksztello, The co-stability manifold of a triangulated category, Glasg. Math. J. 55 (2013), no. 1, 161–175. MR 3001338, DOI 10.1017/S0017089512000420
Bernhard Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), no. 4, 379–417. MR 1052551, DOI 10.1007/BF02568439
Bernhard Keller, Derived categories and universal problems, Comm. Algebra 19 (1991), no. 3, 699–747. MR 1102982, DOI 10.1080/00927879108824166
Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406
Bernhard Keller, A remark on the generalized smashing conjecture, Manuscripta Math. 84 (1994), no. 2, 193–198. MR 1285956, DOI 10.1007/BF02567453
Bernhard Keller and Pedro Nicolás, Weight structures and simple dg modules for positive dg algebras, Int. Math. Res. Not. IMRN 5 (2013), 1028–1078. MR 3031826, DOI 10.1093/imrn/rns009
B. Keller and D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 239–253. Deuxième Contact Franco-Belge en Algèbre (Faulx-les-Tombes, 1987). MR 976638
G. M. Kelly, Chain maps inducing zero homology maps, Proc. Cambridge Philos. Soc. 61 (1965), 847–854. MR 188273, DOI 10.1017/s0305004100039207
Steffen Koenig and Dong Yang, Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438. MR 3178243
Henning Krause, Derived categories, resolutions, and Brown representability, Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 101–139. MR 2355771, DOI 10.1090/conm/436/08405
Henning Krause, Localization theory for triangulated categories, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 161–235. MR 2681709, DOI 10.1017/CBO9781139107075.005
Henning Krause and Jan Šťovíček, The telescope conjecture for hereditary rings via Ext-orthogonal pairs, Adv. Math. 225 (2010), no. 5, 2341–2364. MR 2680168, DOI 10.1016/j.aim.2010.04.027
L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482, DOI 10.1007/BFb0075778
Georges Maltsiniotis, La $K$-théorie d’un dérivateur triangulé, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 341–368 (French, with French summary). MR 2342836, DOI 10.1090/conm/431/08280
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
Octavio Mendoza Hernández, Edith Corina Sáenz Valadez, Valente Santiago Vargas, and María José Souto Salorio, Auslander-Buchweitz context and co-$t$-structures, Appl. Categ. Structures 21 (2013), no. 5, 417–440. MR 3097052, DOI 10.1007/s10485-011-9271-2
Amnon Neeman, The chromatic tower for $D(R)$, Topology 31 (1992), no. 3, 519–532. With an appendix by Marcel Bökstedt. MR 1174255, DOI 10.1016/0040-9383(92)90047-L
Amnon Neeman, The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547–566. MR 1191736
Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
David Pauksztello, Compact corigid objects in triangulated categories and co-$t$-structures, Cent. Eur. J. Math. 6 (2008), no. 1, 25–42. MR 2379950, DOI 10.2478/s11533-008-0003-2
David Pauksztello, A note on compactly generated co-$t$-structures, Comm. Algebra 40 (2012), no. 2, 386–394. MR 2889469, DOI 10.1080/00927872.2010.528714
Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR 737778, DOI 10.2307/2374308
Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
Manuel Saorín and Jan Šťovíček, On exact categories and applications to triangulated adjoints and model structures, Adv. Math. 228 (2011), no. 2, 968–1007. MR 2822215, DOI 10.1016/j.aim.2011.05.025
N. Spaltenstein, Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154. MR 932640
Jan Št’ovíček, Deconstructibility and the Hill lemma in Grothendieck categories, Forum Math. 25 (2013), no. 1, 193–219. MR 3010854, DOI 10.1515/form.2011.113