Commutative group schemes

Oort, Frans

doi:10.1007/bfb0097479

Cited by 159 publications

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“…it is enough to show that any element E'eExt^S; H, G^) can be split by an fppf covering of S. Moreover E' may be interpreted as a short, exact sequence of group schemes over S ( [15], § 17, Chapter III). Given such an exact sequence we get a commutative exact diagram of group schemes over S since n == rank (H) annihilates H [16].…”

Section: Sectionmentioning

confidence: 99%

Brauer groups of abelian schemes

Hoobler¹

1972

Ann. Sci. École Norm. Sup.

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Section: Sectionmentioning

confidence: 99%

Brauer groups of abelian schemes

Hoobler¹

1972

Ann. Sci. École Norm. Sup.

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“…If k is algebraically closed of characteristic zero then the category G pr k has homological dimension 1 (see [46], Thm. 10.1, or [39]), Sec. II.14, so every object of…”

Section: 1mentioning

confidence: 99%

“…In the case of an algebraically closed field a nice exposition of the properties of this category can be found in Chap. II of [39]. Here is a simple way to realize the dual D.K/ to an orbi-abelian variety K concretely.…”

Section: 1mentioning

confidence: 99%

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Kernel algebras and generalized Fourier–Mukai transforms

Polishchuk¹

2011

J. Noncommut. Geom.

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Abstract. We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting categories, such as Dmodules, equivariant sheaves and their twisted versions, arise as categories of modules over kernel algebras. We develop the techniques of constructing derived equivalences between these module categories. As one application we generalize the results of [44] concerning modules over algebras of twisted differential operators on abelian varieties. As another application we recover and generalize the results of Laumon [33] concerning an analog of the Fourier transform for derived categories of quasi-coherent sheaves on a dual pair of generalized 1-motives.

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“…More generally, any finite commutative group scheme is contained in some abelian variety (see [26,Sec. 15.4…”

Section: Proof (I)⇒(ii) Is Obvious and (Ii)⇒(i) Follows From Theorementioning

confidence: 99%

Homogeneous projective bundles over abelian varieties

Brion

2013

Algebra Number Theory

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We consider projective bundles (or Brauer-Severi varieties) over an abelian variety which are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative group schemes; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties. IntroductionThe main objects of this article are the projective bundles (or Brauer-Severi varieties) over an abelian variety X which are homogeneous, i.e., isomorphic to their pull-backs under all translations. Among these bundles, the projectivizations of vector bundles are well understood by work of Mukai (see [22]). Indeed, the vector bundles with homogeneous projectivization are exactly the semi-homogeneous vector bundles of [loc. cit.]. Those that are simple (i.e., their global endomorphisms are just scalars) admit several remarkable characterizations; for example, they are all obtained as direct images of line bundles under isogenies. Moreover, every indecomposable semi-homogeneous vector bundle is the tensor product of a unipotent bundle and of a simple semi-homogeneous bundle.In this article, we obtain somewhat similar statements for the structure of homogeneous projective bundles. We build on the results of our earlier paper [9] about homogeneous principal bundles under an arbitrary algebraic group; here we consider of course the projective linear group PGL n . In loose terms, the approach of [loc. cit.] reduces the classification of homogeneous bundles to that of commutative subgroup schemes of PGL n . The latter, carried out in Section 2, is based on the classical construction of Heisenberg groups and their irreducible representations.2010 Mathematics Subject Classification: Primary 14K05; Secondary 14F22, 14J60, 14L30. 1In Section 3, we introduce a notion of irreducibility for homogeneous projective bundles, which is equivalent to the group scheme of bundle automorphisms being finite. (The projectivization of a semi-homogeneous vector bundle E is irreducible if and only if E is simple). We characterize those projective bundles that are homogeneous and irreducible, by the vanishing of all the cohomology groups of their adjoint vector bundle (Proposition 3.7). Also, we show that the homogeneous irreducible bundles are classified by the pairs (H, e), where H is a finite subgroup of the dual abelian variety, and e : H × H → G m a non-degenerate alternating bilinear pairing (Proposition 3.1). Finally, we obtain a characterization of those homogeneous projective bundles that are projectivizations of vector bundles, first in the irreducible case (Proposition 3.10; it states in loose terms that the pairing e originates from a line bundle on X) and then in the general case (Theorem 3.11).The irreducible homogeneous projective bundles over an elliptic curve are exactly the projectivizations of indecomposable vector bundles wi...

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Commutative group schemes

Abstract: All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photo mechanical means (photostat, microfilm and/or micro card) or by other procedure withoilt written permission from Springer Verlag.

Cited by 159 publications

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Brauer groups of abelian schemes

Brauer groups of abelian schemes

Kernel algebras and generalized Fourier–Mukai transforms

Homogeneous projective bundles over abelian varieties

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