Jacobians of Noncommutative Motives
Author(s)Trigo Neri Tabuada, Goncalo Jo; Marcolli, Matilde
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In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a Q-linear additive Jacobian functor N → J(N) from the category of noncommutative Chow motives to the category of abelian varieties up to isogeny, with the following properties: (i) the first de Rham cohomology group of J(N) agrees with the subspace of the odd periodic cyclic homology of N which is generated by algebraic curves; (ii) the abelian variety J(perf[subscript dg](X)) (associated to the derived dg category perf[subscript dg](X) of a smooth projective k-scheme X) identifies with the product of all the intermediate algebraic Jacobians of X. As an application, every semi-orthogonal decomposition of the derived category perf(X) gives rise to a decomposition of the intermediate algebraic Jacobians of X.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Moscow Mathematical Journal
Independent University of Moscow
Marcolli, Matilde, and Goncalo Tabuada. "Jacobians of Noncommutative Motives." Moscow Mathematical Journal, Volume 14, Number 3 (July-September 2014), 577-594.