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Homologically finite-dimensional objects in triangulated categories

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arxiv 2211.09418 v4 pith:NAHD6PVR submitted 2022-11-17 math.AG math.KT

Alexander Kuznetsov , Evgeny Shinder This is my paper · ORCID
classification math.AGmath.KT
keywords categorycategoriesderivedtriangulatedfinite-dimensionalgeometrichomologicallyobjects
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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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Notes on the Kuznetsov component of cubic sevenfolds

    math.AG 2026-07 accept novelty 7.0 of 10

    For smooth cubic sevenfolds, the Kuznetsov component embeds into the derived category of Clifford modules on P^4; for a general nodal-line cubic, its Kuznetsov component admits an explicit weakly crepant categorical r...

  2. Categorical resolutions of cuspidal singularities

    math.AG 2024-11 conditional novelty 6.0 of 10

    A crepant categorical resolution of an isolated A2 singularity is shown to be a Verdier localization, with kernel generated by two 2-spherical objects (even dimension) or one non-spherical object (odd), and for cuspid...

  3. Approximable Triangulated Categories and Reflexive DG-categories

    math.AG 2024-11 accept novelty 6.0 of 10

    A locally finite approximable DG-category with a strong generator inside its finitely valued modules is finite-reflexive; this yields reflexivity for proper schemes, Azumaya algebras, and proper connective DG-algebras.

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