| Title:
|
On the structure of dg-categories of relative singularities (English) |
| Author:
|
Pippi, Massimo |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
6 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
375-402 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we show that every object in the dg category of relative singularities ${\bf Sing}(B,\underline f)$ associated to a pair $(B,\underline f)$, where $B$ is a ring and $\underline f \in B^n$, is equivalent to an homotopy retract of a $K(B,\underline f)$-dg module concentrated in $n + 1$ degrees, where $K(B,\underline f)$ denotes the Koszul algebra associated to $(B,\underline f)$. When $n = 1$, we show that Orlov’s comparison theorem, which relates the dg category of relative singularities and that of matrix factorizations of an LG-model, holds true without any regularity assumption on the potential. (English) |
| Keyword:
|
dg categories of relative singularities |
| Keyword:
|
matrix factorizations |
| Keyword:
|
non commutative algebraic geometry |
| MSC:
|
14B05 |
| MSC:
|
18G80 |
| idZBL:
|
Zbl 1505.14005 |
| idMR:
|
MR4456599 |
| DOI:
|
10.21136/HS.2022.08 |
| . |
| Date available:
|
2026-03-13T10:00:45Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153452 |
| . |
| Reference:
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[1] Amiot, C.: Cluster categories for algebras of global dimension 2 and quivers with potential..Ann. Inst. Fourier., 59:2525–2590 MR 2640929 |
| Reference:
|
[2] Auslander, M., Buchsbaum, D. A.: Homological dimension in noetherian rings..In Proc. Nat. Acad. Sci. U.S.A., volume 42, no. 1, pages 36––38 |
| Reference:
|
[3] Auslander, M., Buchsbaum, D. A.: Homological dimension in local rings..Trans. Amer. Math. Soc., 85:390–405 |
| Reference:
|
[4] Ben-Zvi, D., Francis, J., Nadler, D.: Integral transforms and drinfeld centers in derived algebraic geometry..J. Amer. Math. Soc., 23(4):909––966 MR 2669705 |
| Reference:
|
[5] Blanc, A., Robalo, M., Toën, B., Vezzosi, G.: Motivic realizations of singularity categories and vanishing cycles..Journal de l’Ecole Polytechnique, 5:651–747 MR 3877165 |
| Reference:
|
[6] Buchweitz, R. O.: Maximal cohen-macaulay modules and tate-cohomology over gorenstein rings..Unpublished manuscript |
| Reference:
|
[7] Burke, J., Walker, M.: Matrix factorizations over projective schemes..Homology, Homotopy and Applications, 14(2):37–61 MR 3007084 |
| Reference:
|
[8] Drinfeld, V.: Dg quotients of dg categories..J. Algebra, 272(2):643––691 MR 2028075 |
| Reference:
|
[9] Efimov, A. I.: Cyclic homology of categories of matrix factorizations..Internat. Math. Res. Notices, 2018(12):3834–3869 MR 3815168 |
| Reference:
|
[10] Efimov, A. I., Positselski, L.: Coherent analogues of matrix factorizations and relative singularity categories..Algebra and Number Theory, 9(5):1159––1292 MR 3366002 |
| Reference:
|
[11] Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representation..Transaction of the American Mathematical Society, 260(1):35–64 |
| Reference:
|
[12] Gaitsgory, D., Rozenblyum, N.: A study in derived algebraic geometry. Vol. 1. Correspondences and duality.Volume 221 of Mathematical Surveys and Monographs. American Mathematical Society, Provedence, RI MR 3701352 |
| Reference:
|
[13] Iyama, O., Yang, D.: Quotients of triangulated categories and equivalences of buchweitz, orlov and amiot-guo-keller..American Journal of Mathematics, 142(5):1641–1659 MR 4150654 |
| Reference:
|
[14] Keller, B.: On the cyclic homology of exact categories..Journal of Pure and Applied Algebra, 136(1):1–56 |
| Reference:
|
[15] Keller, B.: On differential graded categories..In Proceedings of the International Congress of Mathematicians, August 22-30, 2006, Madrid, volume 2, pages 151–190. Eur. Math. Soc., Zürich MR 2275593 |
| Reference:
|
[16] Orlov, D.: Triangulated categories of singularities and d-branes in landau-ginzburg models..Algebr. Geom. Metody, Svyazi i Prilozh. Trudy Mat. Inst. Steklov., 246:240––262 MR 2101296 |
| Reference:
|
[17] Orlov, D.: Matrix factorizations for nonaffine lg-models..Math. Ann., 353(1):95–108 MR 2910782 |
| Reference:
|
[18] Preygel, A.: Thom-Sebastiani and duality for Matrix factorizations..PhD thesis, Massachusetts Institute of Technology |
| Reference:
|
[19] Robalo, M.: Motivic homotopy theory of non-commutative spaces..PhD thesis, Université Montpellier 2 |
| Reference:
|
[20] Serre, J. P.: Sur la dimension homologique des anneaux et des modules noethériens..In Proceedings of the international symposium on algebraic number theory, Tokyo and Nikko,1955, Science Council of Japan, Tokyo, 1956, pages 175––189 |
| Reference:
|
[21] Tabuada, G.: Une structure de catégorie de modèles de quillen sur la catégorie des dg-catégories..Comptes Rendus Mathématique, 340(1):15–19 MR 2112034 |
| Reference:
|
[22] Toën, B.: The homotopy theory of dg categories and derived morita theory..Invent. Math., 167(3):615–667 MR 2276263 |
| Reference:
|
[23] Toën, B.: Lectures on dg categories..In Topics in algebraic and topological K-theory, volume 2008 of Lect. Notes in Math., pages 243–302. Springer, Berlin MR 2762557 |
| Reference:
|
[24] Toën, B.: Proper local complete intersection morphisms preserve perfect complexes..arXiv:1210.2827 |
| Reference:
|
[25] Toën, B., Vaquie, M.: Moduli of objects in dg categories..Ann. Sci. École Norm. Sup., 40(3):387–444 MR 2493386 |
| Reference:
|
[26] Toën, B., Vezzosi, G.: Trace and künneth formulas for singularity categories and applications..arXiv:1710.05902 MR 4434529 |
| . |
