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

Canonaco, Alberto; Stellari, Paolo

doi:10.4171/jems/820

Cited by 33 publications

(76 citation statements)

References 40 publications

(73 reference statements)

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“…By Theorem 2.5, the Verdier quotient T/L is α-compactly generated. Hence, the fact that (G1) and (G3) hold true follows from [17,Proposition 1.7]. By the discussion in Section 2.1, the set Q(S) satisfies (G2) as well, if α = ℵ 0 .…”

Section: 1mentioning

confidence: 93%

“…It is explained in the proof of [35, Theorem 5.10] that D(C) can be realized as a Verdier quotient Q : D(Mod(C α )) → D(C) such that the kernel of Q is generated by a small set of αcompact objects, for all sufficiently large regular cardinals α. Moreover, it is an easy exercise to show that D(Mod(C α )) is compactly generated by the set S of objects in the image of [17,Corollary 5.3]). Hence Q(S) can be identified with C α .…”

Section: 1mentioning

confidence: 99%

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A tour about existence and uniqueness of dg enhancements and lifts

Canonaco

Stellari

2017

Journal of Geometry and Physics

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This paper surveys the recent advances concerning the relations between triangulated (or derived) categories and their dg enhancements. We explain when some interesting triangulated categories arising in algebraic geometry have a unique dg enhancement. This is the case, for example, for the unbounded derived category of quasi-coherent sheaves on an algebraic stack or for its full triangulated subcategory of perfect complexes. Moreover we give an account of the recent results about the possibility to lift exact functors between the bounded derived categories of coherent sheaves on smooth schemes to dg (quasi-)functors.

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Section: 1mentioning

confidence: 93%

Section: 1mentioning

confidence: 99%

A tour about existence and uniqueness of dg enhancements and lifts

Canonaco

Stellari

2017

Journal of Geometry and Physics

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“…We know that the category of perfect complexes perf(X) has a unique dg enhancement perf dg (X) (cf. [19] or [8]), which is smooth and proper as a dg category. Moreover, suppose that the derived category of perfect complexes on X has a semiorthogonal decomposition of the form perf(X) = A 1 , ..., A n .…”

Section: Pure Motives Vs Noncommutative Motivesmentioning

confidence: 99%

Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds via noncommutative K3 surfaces

Ornaghi

Pertusi

2019

J. Noncommut. Geom.

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In the first part of this paper we will prove the Voevodsky's nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky's nilpotence conjecture for generic Gushel-Mukai fourfolds containing a τ -plane Gr(2, 3) and for ordinary Gushel-Mukai fourfolds containing a quintic del Pezzo surface.Theorem (BMT). Let X be a smooth projective k-scheme. The conjecture V(X) is equivalent to the conjecture V nc (perf dg (X)).Here, perf dg (X) denotes the unique enhancement of the derived category of perfect complexes on X. Making use of noncommutative motives, Voevodsky's conjecture was proven for example for quadric fibrations, intersection of quadrics, linear sections of Grassmannians, etc. (see [6] and [7] for details).The first result of this paper is the proof of Voevodsky's conjecture for cubic fourfolds and ordinary generic Gushel-Mukai fourfolds. We recall that a cubic fourfold is a smooth complex hypersurface of degree 3 in P 5 , while a Gushel-Mukai fourfold is a smooth and transverse intersection of the form Cone(Gr(2, V 5 )) ∩ Q, where Q is a quadric hypersurface in P 8 .Theorem (A). Let X be a cubic fourfold or an ordinary generic Gushel-Mukai fourfold; then the conjecture V (X) holds. 3

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“…If we consider triangulated categories of algebraic or geometric origin, e. g. derived categories of modules over an algebra or of sheaves on some space, it is natural to ask what properties their dg enhancements have and how these properties do or do not depend on properties of the algebra or the space. The focus of this article is on the smoothness of dg enhancements (see Definition 2.2), where we always work over a field k. Since dg enhancements are often essentially unique (see [LO10], [CS18]), we are a bit sloppy in this introduction and just say that a triangulated category is smooth when we mean that a certain natural dg enhancement has this property (cf. Definition 2.3 and Remark 2.5 for the choices used in this article).…”

Section: Introductionmentioning

confidence: 99%

Smoothness of Derived Categories of Algebras

Elagin¹,

Lunts²,

Schnürer³

2020

MMJ

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We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, thereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.

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

Cited by 33 publications

References 40 publications

A tour about existence and uniqueness of dg enhancements and lifts

A tour about existence and uniqueness of dg enhancements and lifts

Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds via noncommutative K3 surfaces

Smoothness of Derived Categories of Algebras

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