323 research outputs found
On the derived category of 1-motives
This is the final version of the 2007 preprint titled "On the derived
category of 1-motives, I". It has been substantially expanded to contain a
motivic proof of (two thirds of) Deligne's conjecture on 1-motives with
rational coefficients, hence the new title. Compared to the 2007 preprint, the
additions mainly concern an abstract theory of realisations with weight
filtrations; Deligne's conjecture is tackled though them by an adjunction game.Comment: 242 pages - revised before acceptatio
On the derived category of 1-motives, I
We consider the category of Deligne 1-motives over a perfect field k of
exponential characteristic p and its derived category for a suitable exact
structure after inverting p. As a first result, we provide a fully faithful
embedding into an etale version of Voevodsky's triangulated category of
geometric motives. Our second main result is that this full embedding "almost"
has a left adjoint, that we call \LAlb. Applied to the motive of a variety we
thus get a bounded complex of 1-motives, that we compute fully for smooth
varieties and partly for singular varieties. As an application we give motivic
proofs of Roitman type theorems (in characteristic 0)
A note on relative duality for Voevodsky motives
Let X be an n-dimensional smooth proper variety over a field admitting
resolution of singularities, and Y,Z two disjoint closed subsets of X. We
establish an isomorphism M(X-Z,Y) isomorphic to M(X-Y,Z)^*(n)[2n] in
Voevodsky's triangulated category of geometric motives. Here, M(X-Z,Y) is the
motive of X -Z relative to its closed subset Y
Sharp de Rham realization
We introduce the "sharp" (universal) extension of a 1-motive (with additive
factors and torsion) over a field of characteristic zero. We define the "sharp
de Rham realization" by passing to the Lie-algebra. Over the complex numbers we
then show a (sharp de Rham) comparison theorem in the category of formal Hodge
structures. For a free 1-motive along with its Cartier dual we get a canonical
connection on their sharp extensions yielding a perfect pairing on sharp
realizations. We thus provide "one-dimensional sharp de Rham cohomology" of
algebraic varieties.Comment: 30 page
Deligne's Conjecture on 1-Motives
We reformulate a conjecture of Deligne on 1-motives by using the integral
weight filtration of Gillet and Soul\'e on cohomology, and prove it. This
implies the original conjecture up to isogeny. If the degree of cohomology is
at most two, we can prove the conjecture for the Hodge realization without
isogeny, and even for 1-motives with torsion.Comment: 41 pages published versio
Tensor structure for Nori motives
We construct a tensor product on Freyd's universal abelian category attached
to an additive tensor category or a tensor quiver and establish a universal
property. This is used to give an alternative construction for the tensor
product on Nori motives.Comment: Revised & updated version, 23 page
Crystalline realizations of 1-motives
We consider the crystalline realization of Deligne's 1-motives in positive
characteristics and prove a comparison theorem with the De Rham realization of
liftings to zero characteristic. We then show that one dimensional crystalline
cohomology of an algebraic variety, defined by universal cohomological descent
via de Jong's alterations, coincide with the crystalline realization of the
(cohomological) Picard 1-motive, over perfect fields.Comment: 54 pages, exposition improved, references & appendix adde
Nori 1-motives
Let EHM be Nori's category of effective homological mixed motives. In this
paper, we consider the thick abelian subcategory EHM_1 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 M_1 of Deligne 1-motives with
torsion; this is our main theorem. Along the way, we obtain several interesting
results. Firstly, we realize 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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