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Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
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Using Dold--Puppe category approach to the duality in topology, we prove general duality theorem for the category of motives. As one of the applications of this general result we obtain, in particular, a generalization of Friedlander--Voevodskys duality to the case of arbitrary base field characteristic.
In this paper we describe the category of motives for an elliptic curve in the sense of Voevodsky as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.
After introducing the Ogus realization of 1-motives we prove that it is a fully faithful functor. More precisely, following a framework introduced by Ogus, considering an enriched structure on the de Rham realization of 1-motives over a number field, we show that it yields a full functor by making use of an algebraicity theorem of Bost.
In a number of recent works [6, 7] the authors have introduced and studied a functor $mathcal{F}_k$ which associates to each loose graph $Gamma$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a $k$-scheme, such that $mathcal{F}_k(Gamma)$ is largely controlled by the combinatorics of $Gamma$. Here, $k$ is a field, and we allow $k$ to be $mathbb{F}_1$, the field with one element. For each finite prime field $mathbb{F}_p$, it is noted in [6] that any $mathcal{F}_k(Gamma)$ is polynomial-count, and the polynomial is independent of the choice of the field. In this note, we show that for each $k$, the class of $mathcal{F}_k(Gamma)$ in the Grothendieck ring $K_0(texttt{Sch}_k)$ is contained in $mathbb{Z}[mathbb{L}]$, the integral subring generated by the virtual Lefschetz motive.
We investigate maximal tori in the Hochschild cohomology Lie algebra $HH^1(A)$ of a finite dimensional algebra $A$, and their connection with the fundamental groups associated to presentations of $A$. We prove that every maximal torus in $HH^1(A)$ arises as the dual of some fundamental group of $A$, extending work of Farkas, Green and Marcos; de la Pe~na and Saorin; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.
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