Jacobians of Noncommutative Motives

M. Marcolli, G. Tabuada
2014 Moscow Mathematical Journal  
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
more » ... rf dg (X)) (associated to the derived dg category perf 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.
doi:10.17323/1609-4514-2014-14-3-577-594 fatcat:omh2xgh5cnat3poaokla4fliwm