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

Noncommutative motives in positive characteristic and their applications
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 abelianExpand
A note on Grothendieck’s standard conjectures of type 𝐶⁺ and 𝐷 in positive characteristic
Making use of topological periodic cyclic homology, we extend Grothendieck’s standard conjectures of type C + \mathrm {C}^+ and D \mathrm {D} (with respect toExpand
Noncommutative Weil conjecture
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 settingExpand
Finite generation of the numerical Grothendieck group
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 everyExpand
Grothendieck's standard conjecture of type D in positive characteristic for linear sections of determinantal varieties
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 smoothExpand
A note on secondary K-theory II
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 canonicalExpand
Schur-finiteness (and Bass-finiteness) conjecture for quadric fibrations and for families of sextic du Val del Pezzo surfaces.
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 theExpand
HPD-invariance of the Tate, Beilinson and Parshin conjectures
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 asExpand
HPD-invariance of the Tate conjecture
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 determinantalExpand
Noncommutative counterparts of celebrated conjectures
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 theExpand
...
1
2
...

References

SHOWING 1-10 OF 104 REFERENCES
Equivariant noncommutative motives
Given a finite group G, we develop a theory of G-equivariant noncommutative motives. This theory provides a well-adapted framework for the study of G-schemes, Picard groups of schemes, G-algebras,Expand
A note on Grothendieck's (noncommutative) standard conjecture of type D
Grothendieck conjectured in the sixties that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend this celebratedExpand
Noncommutative motives of separable algebras
In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory ofExpand
Levine's motivic comparison theorem revisited
Abstract For a field of characteristic zero Levine has proved in [M. Levine, Mixed motives, Math. Surv. Monogr. 57, American Mathematical Society, 1998.], Part I, Ch. VI, 2.5.5, that the triangulatedExpand
Equivariant intersection theory
In this paper we develop an equivariant intersection theory for actions of algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They areExpand
Finite dimensional motives and the Conjectures of Beilinson and Murre
We relate the notion of finite dimensionality of the Chow motive M(X) of a smooth projective variety X (as defined by S. Kimura) with the Conjectures of Beilinson, Bloch and Murre on the existence ofExpand
Higher Algebraic K-Theory of Schemes and of Derived Categories
In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of thoseExpand
The Gysin triangle via localization and A1-homotopy invariance
Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A1-homotopy invariant of dg categories E, we construct an associated Gysin triangle relatingExpand
Zeta Functions and Motives
We study properties of rigid K-linear ⊗-categories A, where K is a field of characteristic 0. When A is semi-simple, we introduce a notion of multiplicities for an object of A: they are rationalExpand
A note on secondary K-theory
We prove that Toen's secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen and Lunts. Along theExpand
...
1
2
3
4
5
...