# Moscow Mathematical Journal

Volume 14, Issue 3, July–September 2014 pp. 577–594.

Jacobians of Noncommutative Motives

**Authors**:
M. Marcolli (1) and G. Tabuada (2)

**Author institution:**(1) Mathematics Department, Mail Code 253-37, Caltech, 1200 E. California Blvd. Pasadena, CA 91125, USA

(2) Department of Mathematics, MIT, Cambridge, MA 02139, USA and

Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516 Caparica, Portugal

**Summary: **

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

**(perf**

*J*_{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*.

2010 Mathematics Subject Classification. 14C15, 14H40, 14K02, 14K30, 18D20.

**Keywords:**Jacobians, abelian varieties, isogeny, noncommutative motives.

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