We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally free sheaves has a unique dg enhancement.

Uniqueness of dg enhancements for the derived category of a Grothendieck category / A. Canonaco, P. Stellari. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - 20:11(2018), pp. 2607-2641.

Uniqueness of dg enhancements for the derived category of a Grothendieck category

P. Stellari
2018

Abstract

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally free sheaves has a unique dg enhancement.
Dg categories; dg enhancements; triangulated categories
Settore MAT/03 - Geometria
   Spazi di moduli e applicazioni.
   MINISTERO DELL'ISTRUZIONE E DEL MERITO
   RBFR12DZRV_001

   Geometria delle Varietà Algebriche
   MINISTERO DELL'ISTRUZIONE E DEL MERITO
   2010S47ARA_006
2018
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/595674
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