Homologically finite-dimensional objects in triangulated categories

• Alexander Kuznetsov ^[1] ; Evgeny Shinder ^[2]
  1. [1] Higher School of Economics, National Research University

     Higher School of Economics, National Research University

     Rusia

     
  2. [2] University of Bonn

     University of Bonn

     Kreisfreie Stadt Bonn, Alemania

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 2, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-024-01004-7
• Enlaces
  • Texto completo
• Resumen

  • In this paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for triangulated categories, and discuss crepant categorical contractions. We illustrate the introduced notions on examples of categories of geometric and algebraic origin and provide geometric applications. In particular, we apply our results to prove a bijection between semiorthogonal decompositions of the derived category of a singular variety and the derived category of its smoothing with support on the central fiber.

• Referencias bibliográficas
  • Ballard, M.: Derived categories of sheaves on singular schemes with an application to reconstruction. Adv. Math. 227(2), 895–919 (2011)
  • Ben-Zvi, D., Nadler, D., Preygel, A.: Integral transforms for coherent sheaves. J. Eur. Math. Soc. 19(12), 3763–3812 (2017)
  • Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1183–1205...
  • Bondarko, M.V.: Producing new semi-orthogonal decompositions in arithmetic geometry. Sbornik Math. 215(4), 511–542 (2024)
  • Belmans, P., Okawa, S., Ricolfi, A.T.: Moduli spaces of semiorthogonal decompositions in families. Preprint arXiv:2002.03303 (2020)
  • Bondal, A., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1),...
  • Catanese, F.: Pluricanonical-Gorenstein-curves. In Enumerative geometry and classical algebraic geometry (Nice, 1981), volume 24 of Progr....
  • Chen, X.-W.: Generalized Serre duality. J. Algebra 328(1), 268–286 (2011)
  • Efimov, A.I.: Homotopy finiteness of some DG categories from algebraic geometry. J. Eur. Math. Soc. (JEMS) 22(9), 2879–2942 (2020)
  • Hoshino, M., Kato, Y., Miyachi, J.-I.: On t-structures and torsion theories induced by compact objects. J. Pure Appl. Algebra 167(1), 15–35...
  • Cattani, W., Giovenzana, F., Liu, S., Magni, P., Martinelli, L., Pertusi, L., Song, J.: Kernel of categorical resolutions of nodal singularities....
  • Jin, H.: Cohen–Macaulay differential graded modules and negative Calabi–Yau configurations. Adv. Math., 374:59, 2020. Id/No 107338
  • Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup.(4) 27(1), 63–102 (1994)
  • Keller, B.: On differential graded categories. In International Congress of Mathematicians. Vol. II, pp. 151–190. Eur. Math. Soc., Zürich...
  • Karmazyn, J., Kuznetsov, A., Shinder, E.: Derived categories of singular surfaces. J. Eur. Math. Soc. (JEMS) 24(2), 461–526 (2022)
  • Kuznetsov, A., Lunts, V.A.: Categorical resolutions of irrational singularities. Int. Math. Res. Not. IMRN 13, 4536–4625 (2015)
  • Kawatani, K., Okawa, S.: Nonexistence of semiorthogonal decompositions and sections of the canonical bundle. Preprint arXiv:1508.00682 (2015)
  • Kalck, M., Pavic, N., Shinder, E.: Obstructions to semiorthogonal decompositions for singular threefolds I: K-theory. Mosc. Math. J. 21(3),...
  • Krause, H.: Krull-Schmidt categories and projective covers. Expo. Math. 33(4), 535–549 (2015)
  • Kuznetsov, A., Shinder, E.: Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique, Special volume in honour...
  • Kuznetsov, A.: Lefschetz decompositions and categorical resolutions of singularities. Selecta Math. (N.S.) 13(4), 661–696 (2008)
  • Lekili, Y., Polishchuk, A.: Auslander orders over nodal stacky curves and partially wrapped Fukaya categories. J. Topol. 11(3), 615–644 (2018)
  • López-Martín, A.C., Sancho de Salas, F.: Indecomposability of derived categories for arbitrary schemes. Collect. Math. 75(3), 639–658 (2024)
  • Lunts, V.A.: Categorical resolution of singularities. J. Algebra 323(10), 2977–3003 (2010)
  • Neeman, A.: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9(1), 205–236 (1996)
  • Neeman, A.: The category Tc⨁op as functors on Tb c. Preprint arXiv:1806.05777 (2018)
  • Neeman, A.: Triangulated categories with a single compact generator and a Brown representability theorem. Preprint arXiv:1804.02240 (2021)
  • Neeman, A.: Approximable triangulated categories. In Representations of algebras, geometry and physics, volume 769 of Contemp. Math., pp....
  • Okawa, S.: Semi-orthogonal decomposability of the derived category of a curve. Adv. Math. 228(5), 2869–2873 (2011)
  • Orlov, D.O.: Triangulated categories of singularities, and equivalences between Landau-Ginzburg models. Mat. Sb. 197(12), 117–132 (2006)
  • Orlov, D.: Remarks on generators and dimensions of triangulated categories. Mosc. Math. J. 9(1), 153–159 (2009)
  • Orlov, D.: Smooth and proper noncommutative schemes and gluing of DG categories. Adv. Math. 302, 59–105 (2016)
  • Orlov, D.: Finite-dimensional differential graded algebras and their geometric realizations. Adv. Math. 366, 107096 (2020)
  • Pavic, N., Shinder, E.: K-theory and the singularity category of quotient singularities. Ann. K-Theory 6(3), 381–424 (2021)
  • Rouquier, R.: Dimensions of triangulated categories. J. K-Theory 1(2), 193–256 (2008)
  • Raedschelders, T., Stevenson, G.: Proper connective differential graded algebras and their geometric realizations. Eur. J. Math. 8(suppl....
  • Spence, D.: A note on semiorthogonal indecomposability for some Cohen-Macaulay varieties. J. Pure Appl. Algebra 226(10), 107076 (2022)
  • Thomason, R.W.: The classification of triangulated subcategories. Compositio Math. 105(1), 1–27 (1997)