On the Theory of 1-Motives

Barbieri-Viale, Luca

doi:10.1017/cbo9780511721496.003

Cited by 32 publications

(105 citation statements)

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“…Section 3 is a brief summary of the facts that we need on 1-motives (readers wishing to learn more about the theory of 1-motives are advised to read [1]). In Section 2 we prove some elementary results needed in the sequel.…”

Section: Introductionmentioning

confidence: 99%

Arithmetic duality theorems for 1-motives over function fields

González-Avilés¹

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this paper we obtain a Poitou-Tate exact sequence for finite and flat group schemes over a global function field. In addition, we extend the duality theorems for 1-motives over number fields obtained by D. Harari and T. Szamuely to the function field case. Theorem 1.2. Let M be a 1-motive over K with dual 1-motive M Ã . Then there exists a canonical pairing [ 1 ðK; MÞðpÞ Â [ 1 ðK; M Ã ÞðpÞ ! Q=Z whose left and right kernels are the maximal divisible subgroups of each group. The author was partially supported by Fondecyt grant 1061209. Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 6:10 AM 206 Gonzá lez-Avilé s, Duality for 1-motives over function fields Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 6:10 AM Gonzá lez-Avilé s, Duality for 1-motives over function fields Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 6:10 AM 208 Gonzá lez-Avilé s, Duality for 1-motives over function fields Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 6:10 AM

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Section: Introductionmentioning

confidence: 99%

Arithmetic duality theorems for 1-motives over function fields

González-Avilés¹

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…is the lattice Ker{Z tr (Z) → Z tr (π 0 (C))} and Alb 0 (C) is the connected component of the identity of the Serre-Albanese scheme Alb(C) of C. (Note that A(C, Z) = Alb − (C, Z) with the notation adopted in [6] and A(C, Z) ∼ = L 1 Alb(C, Z) according to [4].) On the other hand, we have a functor…”

Section: We Have a Representationmentioning

confidence: 99%

Nori $$1$$ 1 -motives

Ayoub

Barbieri-Viale

2014

Math. Ann.

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Abstract. Let EHM be Nori's category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM 1 ⊂ EHM generated by the i-th relative homology of pairs of varieties for i ∈ {0, 1}. We show that EHM 1 is naturally equivalent to the abelian category t M 1 of 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize t M 1 as the universal abelian category obtained, using Nori's formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on extensions of 1-motives in the category of mixed realizations for those extensions that are effective in Nori's sense.

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“…We also have "Cartier duals" (up to isogeny) Alb − (X, i) such that Alb − (X, 0) = Alb(X) is the classical Albanese variety for X smooth. ) which is left adjoint to the embedding Tot above (see [2]). The aim is that this operation will be rendering the 1-motives predicted by the Deligne conjecture.…”

Section: For Higher Dimensional Varieties Delignementioning

confidence: 99%

“…In order to deal with cohomological motives of singular schemes we need a "motivic" description of the Picard functor (see [2] For a pair X, Y ∈ Sm k we let c(X, Y ) ⊆ Z * (X × Y ) denote the sub-group of finite correspondences: it is generated by sub-schemes Z ⊆ X × Y which are finite over X and surjective on a connected component of X. A finite correspondence X Y is somewhat a "finite multivalued" function from X to Y .…”

Section: Below) a Motivic Cohomologymentioning

confidence: 99%

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A Pamphlet on Motivic Cohomology

Barbieri-Viale

2005

Milan j. math.

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Abstract. This is an "elementary" introduction to the conjectural theory of motives along the lines indicated by Grothendieck. We further quote recent developments, also presenting some advances due to Voevodsky, and applications to the study of algebraic cycles and differential forms. Ad astra per asperaIn order to fix a general "philosophical" framework of what a theory of motives looks like let's draw the following picture Motives ր ց Spaces −→ Structures with the following explanation:• Spaces ⇒ Structures: we usually go from spaces to structures associating several kind of invariants of the shape we are investigating 1 and call them cohomology theories of our spaces along with the structures they carry on; • Spaces ⇒ Motives: when a concept of space is fixed and a concept of cohomology theory is involved we then seek for a universal cohomology theory along with its structure of motive and call it motivic cohomology; • Motives ⇒ Structures: such a motive of the shape will be the finest invariant or structure associated to a space and would have several realizations yielding the various cohomology theories. 1 The most elementary is the dimension but we here imagine quite many other structures corresponding to homotopical, topological, differential or algebraic invariants which are linked to several visions of our spaces embodied in topological, differential and algebraic varieties or manifolds.2 Exactly as in the common sense we here just think of a musical or visual motive being further realized with several different instruments.

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On the Theory of 1-Motives

Cited by 32 publications

References 40 publications

Arithmetic duality theorems for 1-motives over function fields

Arithmetic duality theorems for 1-motives over function fields

Nori $$1$$ 1 -motives

A Pamphlet on Motivic Cohomology

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