Embeddings of algebras in derived categories of surfaces

Belmans, Pieter; Raedschelders, Theo

doi:10.1090/proc/13497

Cited by 3 publications

(1 citation statement)

References 22 publications

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“…Remark 2.4. At least in the case when the base field k is algebraically closed, Sasha Kuznetsov has mentioned to me the following geometric argument, which is directly inspired from [5,Theorem 4.7], showing that the existence of a numerically exceptional collection of maximal length implies the unimodularity of the Néron-Severi lattice. Choose a basis of N 1 (S) consisting of classes of smooth curves C i which intersect pairwise transversally.…”

Section: 2mentioning

confidence: 99%

Exceptional collections, and the Néron–Severi lattice for surfaces

Vial

2017

Advances in Mathematics

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We work out properties of smooth projective varieties X over a (not necessarily algebraically closed) field k that admit collections of objects in the bounded derived category of coherent sheaves D b (X) that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Néron-Severi lattice for a smooth projective surface S with χ(O S ) = 1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with pg = q = 0 admit a numerically exceptional collection of maximal length. Another consequence is that a minimal geometrically rational surface with a numerically exceptional collection of maximal length is rational.As is apparent, the class of varieties admitting full exceptional collections is rather restricted. Perhaps the simplest constraint for a smooth projective variety to admit a full exceptional collection is the following. If we have a semi-orthogonal decomposition T = A, B , then. , E r is a full exceptional collection, then K 0 (X) = Z r . As will be explained in the introduction of Section 2, this implies that the Chow motive of X with rational coefficients is a direct sum of Lefschetz motives. Such a constraint was originally obtained via the theory of non-commutative motives by Marcolli and Tabuada [34].Chow motives and Lefschetz motives. Let R be a ring. The category of Chow motives with R-coefficients over k is constructed as follows. First one linearizes the category of smooth projective varieties over k by declaring that Hom(X, Y ) = CH d (X × Y ) ⊗ Z R, that is, by declaring that the morphism between X and Y are given by correspondences with R-coefficients modulo rational equivalence. Here, X is assumed to be of pure dimension d (otherwise one works component-wise) and the composition law is given byHere, e is the dimension of Y , and p XZ , p XY , p Y Z are the projections from X × Y × Z onto X × Z, X × Y, Y × Z, respectively. This R-linear category is far from being abelian, so that one formally adds to this R-linear category the images of idempotents. This is called taking the pseudo-abelian, or Karoubi, envelope. This new category is called the category of effective Chow motives, and objects are pairs (X, p), where X is a smooth projective variety of dimension d and p ∈ CH d (X × X) ⊗ Z R is an idempotent. When p is the class of the diagonal ∆ X in CH d (X × X), we write h(X) for (X, ∆ X ). In general, the object (X, p) should be thought of as the image of p acting on the motive h(X) of X. For example, in the category of effective Chow motives, the motive h(P 1 ) of the projective line becomes isomorphic to (P 1 , p) ⊕ (P 1 , q), where p := {0} × P 1 and q := P 1 × {0} are idempotents in Hom(h(P 1 ), h(P 1 )) := CH 1 (P 1 × P 1 ). The object (P 1 , p) is isomorphic to 1 := h(Spec k), and the motive (P 1 , p) is called the Lefschetz motive and is written 1(−1). The fiber product of two smooth projective varieties induces a...

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Section: 2mentioning

confidence: 99%

Exceptional collections, and the Néron–Severi lattice for surfaces

Vial

2017

Advances in Mathematics

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Exceptional collections in surface-like categories

Кузнецов¹

2017

Sb. Math.

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Exceptional collections in surface-like categories

Кузнецов¹,

Kuznetsov²

2017

Математический сборник

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Предложен категорный подход к недавним результатам М. Перлинга о комбинаторике исключительных наборов на численно рациональных поверхностях. С его помощью упрощены и обобщены некоторые из результатов Перлинга, а также критерий Виала существования численного исключительного набора. Библиография: 18 названий.

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Embeddings of algebras in derived categories of surfaces

Cited by 3 publications

References 22 publications

Exceptional collections, and the Néron–Severi lattice for surfaces

Exceptional collections, and the Néron–Severi lattice for surfaces

Exceptional collections in surface-like categories

Exceptional collections in surface-like categories

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