Recollements induced by good (co)silting dg-modules

Rongmin Zhu; Jiaqun Wei

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 2, page 453-473
  • ISSN: 0011-4642

Abstract

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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 𝐃 ( C , d ) of C , 𝐃 ( B , d ) of B and 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.

How to cite

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Zhu, Rongmin, and Wei, Jiaqun. "Recollements induced by good (co)silting dg-modules." Czechoslovak Mathematical Journal 73.2 (2023): 453-473. <http://eudml.org/doc/299555>.

@article{Zhu2023,
abstract = {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.},
author = {Zhu, Rongmin, Wei, Jiaqun},
journal = {Czechoslovak Mathematical Journal},
keywords = {silting object; dg-algebra; cosilting dg-module; recollement},
language = {eng},
number = {2},
pages = {453-473},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Recollements induced by good (co)silting dg-modules},
url = {http://eudml.org/doc/299555},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Zhu, Rongmin
AU - Wei, Jiaqun
TI - Recollements induced by good (co)silting dg-modules
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 453
EP - 473
AB - 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.
LA - eng
KW - silting object; dg-algebra; cosilting dg-module; recollement
UR - http://eudml.org/doc/299555
ER -

References

top
  1. Hügel, L. Angeleri, 10.1112/blms.12264, Bull. Lond. Math. Soc. 51 (2019), 658-690. (2019) Zbl07118830MR3990384DOI10.1112/blms.12264
  2. Hügel, L. Angeleri, Herbera, D., 10.1512/iumj.2008.57.3325, Indiana Univ. Math. J. 57 (2008), 2459-2517. (2008) Zbl1206.16002MR2463975DOI10.1512/iumj.2008.57.3325
  3. Hügel, L. Angeleri, Marks, F., Vitória, J., 10.1093/imrn/rnv191, Int. Math. Res. Not. 2016 (2016), 1251-1284. (2016) Zbl1367.16005MR3493448DOI10.1093/imrn/rnv191
  4. Bazzoni, S., 10.1090/S0002-9939-09-10120-X, Proc. Am. Math. Soc. 138 (2010), 533-544. (2010) Zbl1233.16008MR2557170DOI10.1090/S0002-9939-09-10120-X
  5. Bazzoni, S., Mantese, F., Tonolo, A., 10.1090/S0002-9939-2011-10900-6, Proc. Am. Math. Soc. 139 (2011), 4225-4234. (2011) Zbl1232.16004MR2823068DOI10.1090/S0002-9939-2011-10900-6
  6. Bazzoni, S., Pavarin, A., 10.1016/j.jalgebra.2013.03.037, J. Algebra 388 (2013), 338-363. (2013) Zbl1303.16013MR3061693DOI10.1016/j.jalgebra.2013.03.037
  7. Bazzoni, S., Šťovíček, J., 10.1016/j.aim.2016.09.028, Adv. Math. 305 (2017), 351-401. (2017) Zbl1386.13043MR3570139DOI10.1016/j.aim.2016.09.028
  8. Becker, H., 10.1016/j.aim.2013.11.016, Adv. Math. 254 (2014), 187-232. (2014) Zbl1348.16009MR3161097DOI10.1016/j.aim.2013.11.016
  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. (1982) Zbl0536.14011MR0751966
  10. Beligiannis, A., Reiten, I., 10.1090/memo/0883, Memoirs of the American Mathematical Society 883. AMS, Providence (2007). (2007) Zbl1124.18005MR2327478DOI10.1090/memo/0883
  11. Breaz, S., Modoi, G. C., Derived equivalences induced by good silting complexes, Available at https://arxiv.org/abs/1707.07353 (2017), 14 pages. (2017) 
  12. Breaz, S., Modoi, G. C., 10.1007/s10468-019-09930-3, Algebr. Represent. Theory 23 (2020), 2113-2129. (2020) Zbl1471.16011MR4165012DOI10.1007/s10468-019-09930-3
  13. Chen, H., Xi, C., 10.1112/plms/pdr056, Proc. Lond. Math. Soc. (3) 104 (2012), 959-996. (2012) Zbl1255.18013MR2928333DOI10.1112/plms/pdr056
  14. Chen, H., Xi, C., 10.2969/jmsj/78477847, J. Math. Soc. Japan 71 (2019), 515-554. (2019) Zbl1433.18007MR3943449DOI10.2969/jmsj/78477847
  15. Hovey, M., Palmieri, J. H., Strickland, N. P., 10.1090/memo/0610, Memoirs or the American Mathematical Society 610. AMS, Providence (1997). (1997) Zbl0881.55001MR1388895DOI10.1090/memo/0610
  16. Keller, B., 10.24033/asens.1689, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 63-102. (1994) Zbl0799.18007MR1258406DOI10.24033/asens.1689
  17. Keller, B., 10.1017/CBO9780511735134.005, Handbook of Tilting Theory London Mathematical Society Lecture Note Series 332. Cambridge University Press, London (2007), 49-104. (2007) MR2384608DOI10.1017/CBO9780511735134.005
  18. Marks, F., Vitória, J., 10.1016/j.jalgebra.2017.12.031, J. Algebra 501 (2018), 526-544. (2018) Zbl1441.16011MR3768148DOI10.1016/j.jalgebra.2017.12.031
  19. Nicolás, P., On Torsion Torsionfree Triples: Ph.D. Thesis, Available at https://arxiv.org/abs/0801.0507 (2008), 184 pages. (2008) 
  20. Nicolás, P., Saorín, M., 10.1016/j.jalgebra.2009.04.035, J. Algebra 322 (2009), 1220-1250. (2009) Zbl1187.18008MR2537682DOI10.1016/j.jalgebra.2009.04.035
  21. Nicolás, P., Saorín, M., 10.1007/s10485-017-9495-x, Appl. Categ. Struct. 26 (2018), 309-368. (2018) Zbl1396.18009MR3770911DOI10.1007/s10485-017-9495-x
  22. Nicolás, P., Saorín, M., Zvonareva, A., 10.1016/j.jpaa.2018.07.016, J. Pure Appl. Algebra 223 (2019), 2273-2319. (2019) Zbl1436.18013MR3906550DOI10.1016/j.jpaa.2018.07.016
  23. Pauksztello, D., 10.1080/00927870802623344, Commun. Algebra 37 (2009), 2337-2350. (2009) Zbl1189.18005MR2536923DOI10.1080/00927870802623344
  24. Schwede, S., Shipley, B. E., 10.1112/S002461150001220X, Proc. Lond. Math. Soc., III. Ser. 80 (2000), 491-511. (2000) Zbl1026.18004MR1734325DOI10.1112/S002461150001220X
  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., 10.1090/conm/480, Rings, Modules and Representations Contemporary Mathematics 480. AMS, Providence (2009), 319-334. (2009) Zbl1180.13034MR2508160DOI10.1090/conm/480
  28. Wei, J., 10.1007/s11856-012-0093-1, Isr. J. Math. 194 (2013), 871-893. (2013) Zbl1286.16011MR3047094DOI10.1007/s11856-012-0093-1
  29. Yang, D., 10.1090/S0002-9939-2011-10898-0, Proc. Am. Math. Soc. 140 (2012), 83-91. (2012) Zbl1244.18013MR2833519DOI10.1090/S0002-9939-2011-10898-0
  30. Yekutieli, A., 10.1017/9781108292825, Cambridge Studies in Advanced Mathematics 183. Cambridge University Press, Cambridge (2020). (2020) Zbl1444.18001MR3971537DOI10.1017/9781108292825

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