# Recent developments on noncommutative motives

@article{Tabuada2016RecentDO, title={Recent developments on noncommutative motives}, author={Gonçalo Tabuada}, journal={arXiv: Algebraic Geometry}, year={2016} }

This survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive invariants of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity conjecture; prove a far-reaching noncommutative generalization of the Weil conjectures; prove Grothendieck's standard conjectures of type C+ and D, Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases; embed the (cohomological) Brauer… Expand

#### Paper Mentions

#### 12 Citations

Noncommutative motives in positive characteristic and their applications

- Mathematics
- Advances in Mathematics
- 2019

Let k be a base field of positive characteristic. Making use of topological periodic cyclic homology, we start by proving that the category of noncommutative numerical motives over k is abelian… Expand

A note on Grothendieck’s standard conjectures of type 𝐶⁺ and 𝐷 in positive characteristic

- Mathematics
- Proceedings of the American Mathematical Society
- 2019

Making use of topological periodic cyclic homology, we extend Grothendieck’s standard conjectures of type
C
+
\mathrm {C}^+
and
D
\mathrm {D}
(with respect to… Expand

Noncommutative Weil conjecture

- Mathematics
- 2018

In this article, following an insight of Kontsevich, we extend the famous Weil conjecture (as well as the strong form of the Tate conjecture) from the realm of algebraic geometry to the broad setting… Expand

Finite generation of the numerical Grothendieck group

- Mathematics
- 2017

Let k be a finite base field. In this note, making use of topological periodic cyclic homology and of the theory of noncommutative motives, we prove that the numerical Grothendieck group of every… Expand

Grothendieck's standard conjecture of type D in positive characteristic for linear sections of determinantal varieties

- Mathematics
- 2017

Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjecture of type D (with respect to crystalline cohomology theory) from smooth projective schemes to smooth… Expand

A note on secondary K-theory II

- Mathematics
- 2015

This note is the sequel to [A note on secondary K-theory. Algebra and Number Theory 10 (2016), no. 4, 887-906]. Making use of the recent theory of noncommutative motives, we prove that the canonical… Expand

Schur-finiteness (and Bass-finiteness) conjecture for quadric fibrations and for families of sextic du Val del Pezzo surfaces.

- Mathematics
- 2017

Let Q -> B be a quadric fibration and T -> B a family of sextic du Val del Pezzo surfaces. Making use of the recent theory of noncommutative mixed motives, we establish a precise relation between the… Expand

HPD-invariance of the Tate, Beilinson and Parshin conjectures

- Mathematics
- 2017

We prove that the Tate, Beilinson and Parshin conjectures are invariant under Homological Projective Duality (=HPD). As an application, we obtain a proof of these celebrated conjectures (as well as… Expand

HPD-invariance of the Tate conjecture

- Mathematics
- 2017

We prove that the Tate conjecture is invariant under Homological Projective Duality (=HPD). As an application, we prove the Tate conjecture in the new cases of linear sections of determinantal… Expand

Noncommutative counterparts of celebrated
conjectures

- Mathematics
- 2018

In this survey, written for the proceedings of the conference K-theory in algebra, analysis and topology, Buenos-Aires, Argentina (satellite event of the ICM 2018), we give a rigorous overview of the… Expand

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