Recollements induced by good (co)silting dg-modules

Zhu, Rongmin ; Wei, Jiaqun

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 453-473

Abstract (VO)

Let $U$ be a dg-$A$-module, $B$ the endomorphism dg-algebra of $U$. We know that if $U$ is a good silting object, then there exist a dg-algebra $C$ and a recollement among the derived categories ${\mathbf D}(C,d)$ of $C$, ${\mathbf D}(B,d)$ of $B$ and ${\mathbf D}(A,d)$ of $A$. We investigate the condition under which the induced dg-algebra $C$ is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained. Finally, some applications are given related to good 2-term silting complexes, good tilting complexes and modules.\looseness -1

Détail

DOI : 10.21136/CMJ.2023.0372-21

Classification : 16D90, 16E45, 18G80
Keywords: silting object; dg-algebra; cosilting dg-module; recollement

Bibliographie

[1] Hügel, L. Angeleri: Silting objects. Bull. Lond. Math. Soc. 51 (2019), 658-690. | DOI | MR | JFM

[2] Hügel, L. Angeleri, Herbera, D.: Mittag-Leffler conditions on modules. Indiana Univ. Math. J. 57 (2008), 2459-2517. | DOI | MR | JFM

[3] Hügel, L. Angeleri, Marks, F., Vitória, J.: Silting modules. Int. Math. Res. Not. 2016 (2016), 1251-1284. | DOI | MR | JFM

[4] Bazzoni, S.: Equivalences induced by infinitely generated tilting modules. Proc. Am. Math. Soc. 138 (2010), 533-544. | DOI | MR | JFM

[5] Bazzoni, S., Mantese, F., Tonolo, A.: Derived equivalence induced by infinitely generated $n$-tilting modules. Proc. Am. Math. Soc. 139 (2011), 4225-4234. | DOI | MR | JFM

[6] Bazzoni, S., Pavarin, A.: Recollements from partial tilting complexes. J. Algebra 388 (2013), 338-363. | DOI | MR | JFM

[7] Bazzoni, S., Šťovíček, J.: Smashing localizations of rings of weak global dimension at most one. Adv. Math. 305 (2017), 351-401. | DOI | MR | JFM

[8] Becker, H.: Models for singularity categories. Adv. Math. 254 (2014), 187-232. | DOI | MR | JFM

[9] Beilinson, A. A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and Topology on Singular Spaces. I Astérisque 100. Société Mathématique de France, Paris (1982), 5-171 French. | MR | JFM

[10] Beligiannis, A., Reiten, I.: Homological and Homotopical Aspects of Torsion Theories. Memoirs of the American Mathematical Society 883. AMS, Providence (2007). | DOI | MR | JFM

[11] Breaz, S., Modoi, G. C.: Derived equivalences induced by good silting complexes. Available at , 14 pages. | arXiv

[12] Breaz, S., Modoi, G. C.: Equivalences induced by infinitely generated silting modules. Algebr. Represent. Theory 23 (2020), 2113-2129. | DOI | MR | JFM

[13] Chen, H., Xi, C.: Good tilting modules and recollements of derived module categories. Proc. Lond. Math. Soc. (3) 104 (2012), 959-996. | DOI | MR | JFM

[14] Chen, H., Xi, C.: Good tilting modules and recollements of derived module categories. II. J. Math. Soc. Japan 71 (2019), 515-554. | DOI | MR | JFM

[15] Hovey, M., Palmieri, J. H., Strickland, N. P.: Axiomatic Stable Homotopy Theory. Memoirs or the American Mathematical Society 610. AMS, Providence (1997). | DOI | MR | JFM

[16] Keller, B.: Deriving DG categories. Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 63-102. | DOI | MR | JFM

[17] Keller, B.: Derived categories and tilting. Handbook of Tilting Theory London Mathematical Society Lecture Note Series 332. Cambridge University Press, London (2007), 49-104. | DOI | MR

[18] Marks, F., Vitória, J.: Silting and cosilting classes in derived categories. J. Algebra 501 (2018), 526-544. | DOI | MR | JFM

[19] Nicolás, P.: On Torsion Torsionfree Triples: Ph.D. Thesis. Available at , 184 pages. | arXiv

[20] Nicolás, P., Saorín, M.: Parametrizing recollement data for triangulated categories. J. Algebra 322 (2009), 1220-1250. | DOI | MR | JFM

[21] Nicolás, P., Saorín, M.: Generalized tilting theory. Appl. Categ. Struct. 26 (2018), 309-368. | DOI | MR | JFM

[22] Nicolás, P., Saorín, M., Zvonareva, A.: Silting theory in triangulated categories with coproducts. J. Pure Appl. Algebra 223 (2019), 2273-2319. | DOI | MR | JFM

[23] Pauksztello, D.: Homological epimorphisms of differential graded algebras. Commun. Algebra 37 (2009), 2337-2350. | DOI | MR | JFM

[24] Schwede, S., Shipley, B. E.: Algebras and modules in monoidal model categories. Proc. Lond. Math. Soc., III. Ser. 80 (2000), 491-511. | DOI | MR | JFM

[25] Authors, The Stacks Project: Stacks Project, an open source textbook and reference work on algebraic geometry. Available at https://stacks.math.columbia.edu/tag/04jd

[26] Authors, The Stacks Project: Stacks Project, an open source textbook and reference work on algebraic geometry. Available at https://stacks.math.columbia.edu/tag/09ks

[27] Trlifaj, J., Pospíšil, D.: Tilting and cotilting classes over Gorenstein rings. Rings, Modules and Representations Contemporary Mathematics 480. AMS, Providence (2009), 319-334. | DOI | MR | JFM

[28] Wei, J.: Semi-tilting complexes. Isr. J. Math. 194 (2013), 871-893. | DOI | MR | JFM

[29] Yang, D.: Recollements from generalized tilting. Proc. Am. Math. Soc. 140 (2012), 83-91. | DOI | MR | JFM

[30] Yekutieli, A.: Derived Categories. Cambridge Studies in Advanced Mathematics 183. Cambridge University Press, Cambridge (2020). | DOI | MR | JFM