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This article is cited in 5 scientific papers (total in 5 papers)
Jacobians of noncommutative motives
Matilde Marcollia, Gonçalo Tabuadabc a Mathematics Department, Mail Code 253-37, Caltech, 1200 E. California Blvd. Pasadena, CA 91125, USA
b Departamento de Matemática e CMA, FCT-UNL, Quinta da Torre, 2829-516 Caparica, Portugal
c Department of Mathematics, MIT, Cambridge, MA 02139, USA
Abstract:
In this article one extends the classical theory of (intermediate) Jacobians to the “noncommutative world”. Concretely, one constructs a $\mathbb Q$-linear additive Jacobian functor $N\mapsto\boldsymbol J(N)$ 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 $\boldsymbol J(N)$ agrees with the subspace of the odd periodic cyclic homology of $N$ which is generated by algebraic curves; (ii) the abelian variety $\boldsymbol J(\mathrm{perf}_\mathrm{dg}(X))$ (associated to the derived dg category $\mathrm{perf}_\mathrm{dg}(X)$ 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 $\mathrm{perf}(X)$ gives rise to a decomposition of the intermediate algebraic Jacobians of $X$.
Key words and phrases:
Jacobians, abelian varieties, isogeny, noncommutative motives.
Received: February 7, 2013; in revised form January 15, 2014
Citation:
Matilde Marcolli, Gonçalo Tabuada, “Jacobians of noncommutative motives”, Mosc. Math. J., 14:3 (2014), 577–594
Linking options:
https://www.mathnet.ru/eng/mmj533 https://www.mathnet.ru/eng/mmj/v14/i3/p577
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