Jacobians of Noncommutative Motives

• Main Authors: Trigo Neri Tabuada, Goncalo Jo (Contributor), Marcolli, Matilde (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Independent University of Moscow, 2015-01-30T19:40:22Z.
• Subjects: Article
• Online Access: Get fulltext

 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.