Generators and representability of functors in commutative and noncommutative geometry

By A. Bondal, Michel van den Bergh. 2002

768 citations recorded by the source corpus.

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/69eb91fbcdf5fff41031df03bd124a602ed0c888

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Citations in the corpus

Annihilation of cohomology and strong generation of module categories (2014, methodology): “The part of this result dealing with rings essentially of finite type over a field was proved by Rouquier [29, Theorem 7.38] when the field is perfect, and extended to the general case by Keller and Van den Bergh [19, Proposition 5.1.2].”
SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES (2014, methodology): “at is idempotent complete is a full subcategory in a proper triangulated category T1. Then T is admissible in T1. 20 The proof of the Theorem 3.16 works for DG categories without any corrections (see [BVB]). Moreover, the DG version can be deduced from Theorem 3.16. Theorem 3.18. Let A be a small DG”
Homological projective duality (2005, background): “i(A T) → i(A S). Since HomT (i(A T),i(A S)) ∼= HomA(A T,A S) the correspondence T → A T is a functor T → A, left adjoint to i: A → T . Similarly one can construct a right adjoint functor. Lemma 2.12 ([BV]). If Xis a smooth projective variety then Db(X) is saturated. Corollary 2.13. If Xis a smooth p”
Hodge theoretic aspects of mirror symmetry (2008, background): “ters in practice are aﬃne. For instance usual quasicompact quasi-separated schemes of ﬁnite type over C are aﬃne when viewed as nc-spaces. This follows from a deep theorem of Bondal and van den Bergh [BvdB03] which asserts that for such a scheme Xthe category C X = D(Qcoh(X)) has a compact generator”
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (2008, neutral): “These ideas were combined with homotopical techniques by B¨okstedt and Neeman [BoN], and further developed and enhanced by Neeman [N1, N2] and many others [K e, BV, To1].”
Derived equivalences and finite dimensional algebras (2006, neutral): “The notion of finite-dimensionality corresponds to Bondal–Van den Bergh’s property of being strongly finitely generated [10].”
Derived Categories (2016, neutral): “The paper [26] by A.”
Derived algebraic geometry (2014, background): “heme Xpossesses a quasi-coherent derived category D qcoh(X), which for us will be the derived category of (unbounded) complexes of O X-modules with quasi-coherent cohomology sheaves (see for instance [Bond-Vand]). In the same way, a derived scheme X possesses a quasi-coherent derived 1-category L qc”