Related papers: Nori 1-motives

In the paper M. Somekawa, {\it{On Milnor $K$-groups attached at semi-Abelian varieties}}, K-theory, \textbf{4} (1990) p.105, Somekawa conjectures that his Milnor K-group $K(k,G_1,...,G_r)$ attached to semi-abelian varieties $G_1$,...,$G_r$…

We define a theory of etale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of…

For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient…

These lecture notes cover four topics. There is a proof of the fact that the functors represented by the motivic Eilenberg-Maclane spaces on the motivic homotopy category coincide with the motivic cohomology defined in terms of the motivic…

In this paper we suggest a new general formalism for studying the invariants of polyhedra and manifolds comming from the theory of von Neumann algebras. First, we examine generality in which one may apply the construction of the extended…

We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of…

The goal of this paper is to introduce Hodge 1-motives of algebraic varieties and to state a corresponding cohomological Grothendieck-Hodge conjecture, generalizing the classical Hodge conjecture to arbitrarily singular proper schemes.

As a higher genus version of universal mixed elliptic motives by Hain and Matsumoto, we consider mixed Teichm\"uller motives as certain motivic local systems on the moduli space of pointed curves. We show that the category of mixed…

In this note we prove that Kontsevich's category NCnum of noncommutative numerical motives is equivalent to the one constructed by the authors. As a consequence, we conclude that NCnum is abelian semi-simple as conjectured by Kontsevich.

In this paper we construct a Q-linear tannakian category MEM_1 of universal mixed elliptic motives over the moduli space M_{1,1} of elliptic curves. It contains MTM, the category of mixed Tate motives unramified over the integers. Each…

Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of…

We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular…

We show that in the case of primary field extensions, the extension of scalars of Deligne $1$-motives admits a left adjoint, called Chow image, and a right adjoint, called Chow trace. This generalizes Chow's results on abelian varieties.…

In this paper, we prove an $\mathbb{A}^1$-homology version of the Whitehead theorem with dimension bound. We also prove an excision theorem for $\mathbb{A}^1$-homology, Suslin homology and $\mathbb{A}^1$-homotopy sheaves. In order to prove…

Let $X$ be a smooth compact projective variety over $\mathbb C$. Let $H^2(\pi_1(X),\mathbb R)^{1,1}$ be the intersection of $H^{1,1}(X,{\mathbb R})$ with the image of the map $H^2(\pi_1(X),{\mathbb R})\to H^2(X)$ induced by the classifying…

Following Eilenberg-Steenrod axiomatic approach we construct the universal ordinary homology theory for any homological structure on a given category by representing ordinary theories with values in abelian categories. For a convenient…

We prove that arbitrary pullbacks, as well as Betti and \'etale realisation functors, are t-exact for the constructible motivic t-structure on the category of cohomological 1-motives over a base scheme.

Given a torsion theory (Y,X) in an abelian category C, the reflector I from C to the torsion-free subcategory X induces a reflective factorisation system (E, M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e that…

We extend Tate duality for Galois cohomology of abelian varieties to $1$-motives over a $p$-adic field, improving a result of Harari and Szamuely. To do this, we replace Galois cohomology with the condensed cohomology of the Weil group.…

We construct a theory of motivic cohomology for quasi-compact, quasi-separated schemes of equal characteristic, which is related to non-connective algebraic $K$-theory via an Atiyah--Hirzebruch spectral sequence, and to \'etale cohomology…