We define a linear structure on Grothendieck’s arithmetic fundamental group ${\pi }_1(X,x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group $Gal(\overline{k}/k)$ to ${\pi }_1(X,x)$ with the existence of a neutral fiber functor on the category which linearizes it. We apply the construction to affine curves and neutral fiber functors coming from a tangent vector at a rational point at infinity, in order to follow this rational point in the universal covering of the affine...

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic algebras). This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $$Jacobian set– the set of all algebra endomorphisms of a polynomial algebra with the Jacobian $1$ – the Jacobian conjecture claims that the Jacobian...

In this paper we study the structure and the degeneracies of the Mumford-Tate group $MT\left(M\right)$ of a 1-motive $M$ defined over $ℂ$. This group is an algebraic $\mathbb{Q}$- group acting on the Hodge realization of $M$ and endowed with an increasing filtration $W_{•}$. We prove that the unipotent radical of $MT\left(M\right)$, which is $W_{-1}\left(MT\left(M\right)\right)$, injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1....