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In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension.

Making use of the recent theory of noncommutative motives, we prove that Schur-finiteness in the setting of Voevodsky’s mixed motives is invariant under homological projective duality. As an application, we show that the mixed motives of smooth linear sections of certain (Lagrangian) Grassmannians, spinor varieties, and determinantal varieties, are Schur-finite. Finally, we upgrade our applications from Schur-finiteness to Kimura-finiteness.

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 of commutative separable algebras CSep(k). Making use of these models, we then explain how the category Sep(k) can be described as a "fibered Z-order" over CSep(k). This viewpoint leads to several computations and structural properties of the category Sep(k). For example, we obtain a complete dictionary between directs sums of noncommutative motives of central simple algebras (= CSA) and sequences of elements in the Brauer group of k. As a first application, we establish two families of motivic relations between CSA which hold for every additive invariant (e.g. algebraic K-theory, cyclic homology, and topological Horhschild homology). As a second application, we compute the additive invariants of twisted flag varieties using solely the Brauer classes of the corresponding CSA. Along the way, we categorify the cyclic sieving phenomenon and compute the (rational) noncommutative motives of purely inseparable field extensions and of dg Azumaya algebras. Keywords: Noncommutative motives; Separable algebra; Brauer group; Twisted flag variety; Hecke algebra; Convolution; Cyclic sieving phenomenon;dg Azumaya algebra

C. Weibel, and Thomason and Trobaugh, proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. We generalize this result from schemes to the broad setting of dg categories. Along the way, we extend the Bass–Quillen fundamental theorem as well as Stienstra’s foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the du Val singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain vanishing and divisibility properties of algebraic K-theory (without coefficients). Keywords: A¹-homotopy, algebraic K-theory, Witt vectors, sheaf of dg algebras, dg orbit category, cluster category, du Val singularities, noncommutative algebraic geometry

In this article we prove that the numerical Grothendieck group of every smooth proper dg category is invariant under primary field extensions, and also that the mod-n algebraic K-theory of every dg category is invariant under extensions of separably closed fields. As a byproduct, we obtain an extension of Suslin’s rigidity theorem, as well as of Yagunov-Østvær’s equivariant rigidity theorem, to singular varieties. Among other applications, we show that base-change along primary field extensions yields a faithfully flat morphism between noncommutative motivic Galois groups. Finally, along the way, we introduce the category of n-adic noncommutative mixed motives. Keywords: Algebraic cycles, K-theory, noncommutative algebraic geometry, National Science Foundation (U.S.) (CAREER Award 1350472), Portuguese Science and Technology Foundation (Grant PEst-OE/MAT/UI0297/2014)

To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, étale, Hodge, etc.) as well as its ring of periods. In this note, following an insight of Kontsevich, we prove that, after suitable modifications, these classical constructions can be extended from schemes to the broad setting of differential graded (dg) categories. This leads to new additive invariants of dg categories, which we compute in the case of differential operators, as well as to a theory of periods of dg categories. Among other applications, we prove that the ring of periods of a scheme is invariant under projective homological duality. Along the way, we explicitly describe the modified mixed realizations using the Tannakian formalism.

In this note we prove some structural properties of all the A1-homotopy invariants of corner skew Laurent polynomial algebras. As an application, we compute the mod-l algebraic K-theory of Leavitt path algebras using solely the kernel/cokernel of the incidence matrix. This leads naturally to some vanishing and divisibility properties of the K-theory of these algebras., National Science Foundation (U.S.). Faculty Early Career Development Program (Award #1350472), Portuguese Foundation for Science and Technology (project UID/MAT/00297/2013 (Centro de Matemática e Aplicações))

Making use of the recent theory of noncommutative mixed motives, we prove that the Voevodsky's mixed motive of a quadric fibration over a smooth curve is Kimura-finite.

Grothendieck conjectured in the sixties that the even Künneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's original conjectures in the case of intersections of quadrics and linear sections of determinantal varieties. Along the way, we also prove the case of quadric fibrations and intersections of bilinear divisors., National Science Foundation (U.S.) (Award 1350472)

In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck's category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost-Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy -theory groups of a Noetherian scheme of Krull dimension vanish below . In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy -theory groups vanish below . Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy -theory group.

We prove that Toën’s secondary Grothendieck ring is isomorphic to the Grothendieck ring of smooth proper pretriangulated dg categories previously introduced by Bondal, Larsen, and Lunts. Along the way, we show that those short exact sequences of dg categories in which the first term is smooth proper and the second term is proper are necessarily split. As an application, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a commutative ring of characteristic zero, it distinguishes between dg Azumaya algebras associated to nontorsion cohomology classes and dg Azumaya algebras associated to torsion cohomology classes (= ordinary Azumaya algebras); in the case of a field of characteristic zero, it is injective; in the case of a field of positive characteristic p, it restricts to an injective map on the p-primary component of the Brauer group.

In this article we extend Voevodsky’s nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky’s original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds., National Science Foundation (U.S.) (Grant DMS-1201512), National Science Foundation (U.S.) (Grant PHY-1205440)

Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A[superscript 1]-homotopy invariant of dg categories E, we construct an associated Gysin triangle relating the value of E at the dg categories of perfect complexes of X, Z, and U. In the particular case where E is homotopy K-theory, this Gysin triangle yields a new proof of Quillen’s localization theorem, which avoids the use of devissage. As a first application, we prove that the value of E at a smooth scheme belongs to the smallest (thick) triangulated subcategory generated by the values of E at the smooth projective schemes. As a second application, we compute the additive invariants of relative cellular spaces in terms of the bases of the corresponding cells. Finally, as a third application, we construct explicit bridges relating motivic homotopy theory and mixed motives on the one side with noncommutative mixed motives on the other side. This leads to a comparison between different motivic Gysin triangles as well as to an etale descent result concerning noncommutative mixed motives with rational coefficients.

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, 2-cocycles, G-equivariant algebraic K-theory, etc. Among other results, we relate our theory with its commutative counterpart as well as with Panin’s theory. As a first application, we extend Panin’s computations, concerning twisted projective homogeneous varieties, to a large class of invariants. As a second application, we prove that whenever the category of perfect complexes of a G-scheme X admits a full exceptional collection of G-invariant (≠G-equivariant) objects, the G-equivariant Chow motive of X is of Lefschetz type. Finally, we construct a G-equivariant motivic measure with values in the Grothendieck ring of G-equivariant noncommutative Chow motives. Keywords: G-scheme; 2-cocycle; semidirect product algebra; twisted group algebra; equivariant algebraic K-theory; twisted projective homogeneous scheme; full exceptional collection; equivariant motivic measure; noncommutative algebraic geometry, National Science Foundation (U.S.) (Award 1350472)

Let A be a dg category, F : A →A be a dg functor inducing an equivalence of categories in degree-zero cohomology, and A /F be the associated dg orbit category. For every A¹-homotopy invariant E ( e.g. homotopy K -theory, K -theory with coefficients, étale K-theory, and periodic cyclic homology), we construct a distinguished triangle expressing E ( A /F )as the cone of the endomorphism E ( F ) − Id of E ( A ). In the particular case where F is the identity dg functor, this triangle splits and gives rise to the fundamental theorem. As a first application, we compute the A¹-homotopy invariants of cluster (dg) categories, and consequently of Kleinian singularities, using solely the Coxeter matrix. As a second application, we compute the A¹-homotopy invariants of the dg orbit categories associated with Fourier–Mukai autoequivalences.

In this note we relate the notions of Lefschetz type, decomposability, and isomorphism for Chow motives with the notions of trivial type, decomposability, and isomorphism for noncommutative motives. Some examples, counter-examples, and applications are also described.

In this note, by combining the work of Amiot–Iyama–Reiten and Thanhoffer de Völcsey–Van den Bergh on Cohen–Macaulay modules with the previous work of the author on orbit categories, we compute the algebraic K-theory with coefficients of cyclic quotient singularities.

I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev–Panin's computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneous varieties. As a second application, we show that the noncommutative motive of a twisted projective homogeneous variety is trivial if and only if the Brauer classes of the associated central simple algebras are trivial. Along the way we construct a fully-faithful ⊗-functor from Merkurjev–Panin's motivic category to Kontsevich's category of noncommutative Chow motives, which is of independent interest.

In this article we prove that Kontsevich’s category NC[subscript num](k)[subscript F] of noncommutative numerical motives is equivalent to the one constructed by the authors in [Marcolli and Tabuada, Noncommutative motives, numerical equivalence, and semisimplicity, Amer. J. Math., to appear, available at arXiv:1105.2950]. As a consequence, we conclude that NC[subscript num](k)[subscript F] is abelian semi-simple as conjectured by Kontsevich.

Let KK be a commutative ring. In this article we construct a well-behaved symmetric monoidal Quillen model structure on the category of small KK-categories which enhances classical Morita theory. Making use of it, we then obtain the usual categorification of the Brauer group and of its functoriality. Finally, we prove that the (contravariant) corestriction map for finite Galois extensions also lifts to this categorification., Fundação para a Ciência e a Tecnologia (Portugal) (PEst-OE/MAT/UI0297/2011), NEC Corporation (NEC Award 2742738)

V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemman-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts., NEC Corporation (Award 2742738)

In this article we further the study of noncommutative numerical motives, initiated in [30, 31]. By exploring the change-of-coefficients mechanism, we start by improving some of the main results of [30]. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor [over-bar HP∗]on the category of noncommutative Chow motives. This allows us to introduce the correct noncommutative analogues CNC and DNC of Grothendieck's standard conjectures C and D. Assuming C[subscript NC], we prove that NNum(k)F can be made into a Tannakian category NNum[superscript †](k)F by modifying its symmetry isomorphism constraints. By further assuming D[subscript NC], we neutralize the Tannakian category Num†(k)F using [over-bar HP∗]. Via the (super-)Tannakian formalism, we then obtain well-defined noncommutative motivic Galois (super-)groups. Finally, making use of Deligne-Milne's theory of Tate triples, we construct explicit morphisms relating these noncommutative motivic Galois (super-)groups with the classical ones as suggested by Kontsevich., National Science Foundation (U.S.) (NSF grant DMS-0901221), National Science Foundation (U.S.) (NSF grant DMS-1007207), National Science Foundation (U.S.) (NSF grant DMS-1201512), National Science Foundation (U.S.) (NSF grant PHY-1205440), National Science Foundation (U.S.) (CAREER Award #1350472), Fundação para a Ciência e a Tecnologia (Portugal) (project grant UID/MAT/00297/2013)

Conjectures of Beilinson-Bloch type predict that the low-degree rational Chow groups of intersections of quadrics are one-dimensional. This conjecture was proved by Otwinowska in [20]. By making use of homological projective duality and the recent theory of (Jacobians of) non-commutative motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an application, we make use of Vial's work [26], [27] to describe the rational Chow motives of these complete intersections and show that smooth fibrations into such complete intersections over bases S of small dimension satisfy Murre's conjecture (when dim(S) [less than or equal to] 1), Grothendieck's standard conjecture of Lefschetz type (when dim(S) [less than or equal to] 2), and Hodge's conjecture (when dim(S) [less than or equal to] 3)., National Science Foundation (U.S.) (CAREER Award #1350472), Fundação para a Ciência e a Tecnologia (Portugal) (project grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações))

We prove that the functor ring-of-rational-Witt-vectors W[subscript 0](−) becomes co-representable in the category of noncommutative motives. As an application, we obtain an immediate extension of W[subscript 0](−) from commutative rings to schemes. Then, making use of the theory of noncommutative motives, we classify all natural transformations of the functor K-theory-of-automorphisms., NEC Corporation (NEC Award-2742738), Fundação para a Ciência e a Tecnologia (Portugal) (PEst-OE/MAT/UI0297/2011 (CMA))