On the motive of a commutative algebraic group (Q273529)

In this paper, the authors prove a formula for the motive of a commutative group scheme of finite type over a field, which is parallel to the classical formula for the Hopf algebra structure on the cohomology of a commutative Lie group.NEWLINENEWLINELet \(k\) be any field and \(G\) be a commutative group scheme of finite type over \(k\). The formula is stated in Voevodsky's category \(\mathrm{DM}^{\mathrm{eff}}(k,\mathbb{Q})\) of effective motives with rational coefficients defined using Nisnevich sheaves with transferts. The sheaf of abelian groups represented by \(G\) naturally admits transferts, hence defines an object \(M_1(G)\in\mathrm{DM}^{\mathrm{eff}}(k,\mathbb{Q})\), and there is a canonical morphism \(\alpha_G:M(G)\rightarrow M_1(G)\).NEWLINENEWLINEWe can now state the main theorem: the morphism \(\alpha_G\) determines a canonical isomorphism NEWLINE\[NEWLINE M(G)\simeq \left(\bigoplus_{n=0}^{\mathrm{kd}(G)} \mathrm{Sym}^n M_1(G)\right)\otimes M(\pi_0(G)) NEWLINE\]NEWLINE in \(\mathrm{DM}^{\mathrm{eff}}(k,\mathbb{Q})\) which is natural in \(G\) and an isomorphism of Hopf algebra objects. The \(i\)-th summand can be characterized as an \(n^i\)-eigenobject for the action of the multiplication by \(n\) on \(G\). The integer \(\mathrm{kd}(G)\) is explicit.NEWLINENEWLINEMany corollaries of this motivic decomposition are presented; in particular, the motive \(M(G)\) is Kimura-finite of dimension \(\mathrm{kd}(G)\).NEWLINENEWLINEIf \(G=A\) is an abelian variety, this isomorphism can be interpreted (via a theorem of Voevodsky) as providing an explicit Chow-Künneth decomposition for the motive of \(A\), and a closely related result has been proven in this language by \textit{A. M. Sermenev} [Funct. Anal. Appl. 8, 47--53 (1974); translation from Funkts. Anal. Prilozh. 8, No. 1, 55--61 (1974; Zbl 0294.14003)], \textit{C. Deninger} and \textit{J. Murre} [J. Reine Angew. Math. 422, 201--219 (1991; Zbl 0745.14003)] and \textit{K. Künnemann} [Proc. Symp. Pure Math. 55, 189--205 (1994; Zbl 0823.14032)].NEWLINENEWLINEThe proof of the main theorem proceeds by devissage using the structure theory of commutative group schemes. The cases of unipotent groups and tori are quite easy, and the case of abelian varieties is obtained by a careful identification of the decomposition of Künnemann mentioned above. The main difficulty is then to pass to the case of semi-abelian varieties; this is accomplished by induction on the rank of the torus, using a Kimura finiteness argument to reduce to checking some compatibilities after applying the \(\ell\)-adic realisation functor.NEWLINENEWLINEThe paper features three appendices; the second and the third one, studying symmetric (co)algebras, their universal properties and associated filtrations, are of independent interest.NEWLINENEWLINEThe paper is a continuation of the PhD thesis of \textit{S. Enright-Ward} [The Voevodsky motive of a rank one semiabelian variety. Freiburg im Breisgau: Univ. Freiburg, Fakultät für Mathematik und Physik (Diss. 2013) (2012; Zbl 1286.14001)].NEWLINENEWLINEThe motivic decomposition of this paper has been used to study (higher) Chow groups of semi-abelian varieties in [\textit{R. Sugiyama}, Doc. Math., J. DMV 19, 1061--1084 (2014; Zbl 1318.19004)] and has been extended to smooth commutative group schemes over a general base in [\textit{G. Ancona}, \textit{A. Huber} and the reviewer, Algebr. Geom. 3, No. 2, 150--178 (2016; Zbl 1354.14011)].