Biextensions and $1$-motives

C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (3), 2005 pp. 65–71
(Received: 2004-10-04 )

Cristiana Bertolin, D-Math HG G 33.4, ETH-Zentrum, CH-8092 Z ̈urich, Switzerland; email:


Let \(S\) be a scheme and let \(G_i\) (for \(i = 1,2,3\)) be an extension of an abelian \(S\)-scheme \(A_i\) by a \(S\)-torus \(Y_i (1)\). The first result of this note is that the category of biextensions of \((G_1,G_2)\) by \(G_3\) is equivalent to the category of biextensions of the underlying abelian \(S\)-schemes \((A_1,A_2)\) by the underlying \(S\)-torus \(Y_3(1)\). Using this theorem we define the notion of biextension of \(1\)-motives by \(1\)-motives. If \({\mathcal{M}}(S)\) denotes the conjectural Tannakian category generated by \(1\)-motives over \(S\) (in a geometrical sense), as a candidate for the morphisms of \({\mathcal{M}}(S)\) from the tensor product of two \(1\)-motives \(M_1 \otimes M_2\) to another \(1\)-motive \(M_3\), we propose the isomorphism classes of biextensions of \((M_1,M_2)\) by \(M_3\). This definition is compatible with the realizations of \(1\)-motives. Moreover, generalizing this definition we obtain, modulo isogeny, the geometrical notion of morphism of \({\mathcal{M}}(S)\) from a finite tensor product of \(1\)-motives to another \(1\)-motive.

Soit \(S\) un schéma. On définit la notion de biextension de \(1\)-motifs par des \(1\)-motifs. De plus, si \({\mathcal{M}}(S)\) désigne la catégorie Tannakienne engendrée par les \(1\)-motifs sur \(S\) (en un sense géométrique), on définit les morphismes de \({\mathcal{M}}(S)\) du produit tensoriel de deux \(1\)-motifs \(M_1 \otimes M_2\) vers un \(1\)-motif \(M_3,\) comme étant la classe d’isomorphismes des biextensions \((M_1,M_2)\) par \(M_3\). En généralisant cette définition, on obtient, modulo isogénies, la notion de morphisme de \({\mathcal{M}}(S)\) d’un produit tensoriel fini de \(1\)-motifs vers un autre \(1\)-motif.

Keywords: 1-motives, biextensions, morphisms, tensor products

AMS Subject Classification: Epimorphisms; monomorphisms; special classes of morphisms; null morphisms 18A20

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