Homologically finite-dimensional objects in triangulated categories

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.

[1]  Ana Cristina L'opez Mart'in,et al.  Indecomposability of derived categories for arbitrary schemes , 2023, 2301.06818.

[2]  E. Shinder,et al.  Categorical absorptions of singularities and degenerations , 2022, 2207.06477.

[3]  M. Bondarko.  Producing"new"semi-orthogonal decompositions in arithmetic geometry , 2022, 2203.07315.

[4]  Dylan Spence,et al.  A note on semiorthogonal indecomposability of some Cohen-Macaulay varieties , 2021, Journal of Pure and Applied Algebra.

[5]  Greg Stevenson,et al.  Proper connective differential graded algebras and their geometric realizations , 2019, European Journal of Mathematics.

[6]  Nebojsa Pavic,et al.  K-theory and the singularity category of quotient singularities , 2018, Annals of K-Theory.

[7]  E. Shinder,et al.  Derived categories of singular surfaces , 2018, Journal of the European Mathematical Society.

[8]  Andrea T. Ricolfi,et al.  Moduli spaces of semiorthogonal decompositions in families , 2020, 2002.03303.

[9]  D. Orlov.  Finite-dimensional differential graded algebras and their geometric realizations , 2019, Advances in Mathematics.

[10]  A. Efimov.  Homotopy finiteness of some DG categories from algebraic geometry , 2013, Journal of the European Mathematical Society.

[11]  E. Shinder,et al.  Obstructions to Semiorthogonal Decompositions for Singular Threefolds I: K-Theory , 2019, 1910.09531.

[12]  Haibo Jin.  Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations , 2018, 1812.03737.

[13]  A. Neeman.  Approximable triangulated categories , 2018, 1806.06995.

[14]  A. Neeman.  Triangulated categories with a single compact generator and a Brown representability theorem , 2018, 1804.02240.

[15]  D. Orlov.  SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES , 2014, 1402.7364.

[16]  Shinnosuke Okawa,et al.  Nonexistence of semiorthogonal decompositions and sections of the canonical bundle , 2015, 1508.00682.

[17]  A. Kuznetsov,et al.  Categorical resolutions of irrational singularities , 2012, 1212.6170.

[18]  H. Krause.  Krull-Schmidt categories and projective covers , 2014, 1410.2822.

[19]  Shinnosuke Okawa.  Semi-orthogonal decomposability of the derived category of a curve , 2011, 1104.4902.

[20]  V. Lunts.  Categorical resolution of singularities , 2009, 0905.4566.

[21]  D. Orlov.  Remarks on generators and dimensions of triangulated categories , 2008, 0804.1163.

[22]  M. Ballard.  Derived categories of sheaves on singular schemes with an application to reconstruction , 2008, 0801.2599.

[23]  Xiao-Wu Chen.  Generalized Serre duality , 2006, math/0610258.

[24]  A. Kuznetsov.  Lefschetz decompositions and categorical resolutions of singularities , 2006, math/0609240.

[25]  Bernhard Keller,et al.  On differential graded categories , 2006, math/0601185.

[26]  Dmitri Orlov,et al.  Triangulated categories of singularities and equivalences between Landau-Ginzburg models , 2005, math/0503630.

[27]  Raphael Rouquier,et al.  Dimensions of triangulated categories , 2003, math/0310134.

[28]  M. Bergh,et al.  Generators and representability of functors in commutative and noncommutative geometry , 2002, math/0204218.

[29]  M. Hoshino,et al.  On t-structures and torsion theories induced by compact objects , 2000, math/0005172.

[30]  R. Thomason.  The classification of triangulated subcategories , 1997, Compositio Mathematica.

[31]  Amnon Neeman,et al.  The Grothendieck duality theorem via Bousfield’s techniques and Brown representability , 1996 .

[32]  Bernhard Keller,et al.  Deriving DG categories , 1994 .

[33]  F. Catanese.  Pluricanonical — Gorenstein — Curves , 1982 .