The Voevodsky motive of a rank one semiabelian variety (Q2844762)

Let \({\mathbf{DM}}^{\text{eff}}_{{-,}{\text{ét}}}(k,{\mathbb Q})\) be the category of étale motivic complexes with rational coefficients defined by V.Voevodsky. Let \(G\) be a semiabelian variety over a perfect field \(k.\) The main result of the work is that if the torus part of \(G\) is \({\mathbb G}_{m}\) then there exists an isomorphism: NEWLINE\[NEWLINEM(G)\cong \mathrm{Sym}(M_{1}(G)):={\bigoplus_{n\geq 0}}\mathrm{Sym}^{n}(M_{1}(G)).NEWLINE\]NEWLINE The proof is based on the classical decomposition for abelian veriety \(A\) in the category of effective Chow motives (cf. [\textit{C. Deninger} and \textit{J. Murre}, J. Reine Angew. Math. 422, 201--219 (1991; Zbl 0745.14003)], [\textit{K. Künnemann}, Proc. Symp. Pure Math. 55, Pt. 1, 189--205 (1994; Zbl 0823.14032)], [\textit{K. Künnemann}, Invent. Math. 113, No. 1, 85--102 (1993; Zbl 0806.14001)], [\textit{A. J. Scholl}, Proc. Symp. Pure Math. 55, Pt. 1, 163--187 (1994; Zbl 0814.14001)] and [\textit{A. M. Shermenev}, Usp. Mat. Nauk 26, No. 3(159), 215--216 (1971; Zbl 0217.04902)]): NEWLINE\[NEWLINEh(A)\cong {\bigoplus_{i\geq 0}}h^{i}(A):={\bigoplus_{i\geq 0}}\mathrm{Sym}^{i}(h^{1}(A)).NEWLINE\]NEWLINE \(M_{1}(G)\) denotes here the ordinary étale sheaf \({\tilde G}:={\text{Mor}}_{{\text{Sch}}/k}(_,G)\) viewed as complex concentrated in degree \(0.\) The proof is ineffective in a sense that the author cannot construct the isomorphism. The author has a candidate for this isomorphism \({\phi}_{G}: M(G)\rightarrow \mathrm{Sym}(M_{1}(G))\) but he is able to prove that \({\phi}_{G}\) is an isomorphism in the particular cases i.e. when \(G\) is an abelian variety or \(G\) is a torus.