Noncommutative Motives Research Articles

Parametrised Presentability Over Orbital Categories

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.

An improvement of the Farrell-Jones conjecture for localising invariants

The Farrell-Jones conjecture for lax monoidal finitary localising invariants was recently proved by Bunke-Kasprowski-Winges [On the Farrell-Jones conjecture for localising invariants, 2021, arXiv:2111.02490]. In this short note, making use of the theory of noncommutative motives, we prove that the lax monoidal assumption is not necessary.

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most$3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric$3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

Embedding of the derived Brauer group into the secondary $K$-theory ring

In this note, making use of the recent theory of noncommutative motives, 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 regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show that the aforementioned canonical map is injective in the case of affine cones over smooth projective plane curves of degree \geq 4 as well as in the case of Mumford’s (famous) singular surface.

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 abelian semi-simple, as conjectured by Kontsevich. Then, we establish a far-reaching noncommutative generalization of the Weil conjectures, originally proven by Dwork and Grothendieck. In the same vein, we establish a far-reaching noncommutative generalization of the cohomological interpretations of the Hasse-Weil zeta function, originally proven by Hesselholt. As a third main result, we prove that the numerical Grothendieck group of every smooth proper dg category is a finitely generated free abelian group, as claimed (without proof) by Kuznetsov. Then, we introduce the noncommutative motivic Galois (super-)groups and, following an insight of Kontsevich, relate them to their classical commutative counterparts. Finally, we explain how the motivic measure induced by Berthelot's rigid cohomology can be recovered from the theory of noncommutative motives.

Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds via noncommutative K3 surfaces

In the first part of this paper we will prove the Voevodsky’s nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel–Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky’s nilpotence conjecture for generic Gushel–Mukai fourfolds containing a \tau -plane Gr(2, 3) and for ordinary Gushel–Mukai fourfolds containing a quintic del Pezzo surface.

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, 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.

Motivic realizations of singularity categories and vanishing cycles

In this paper we establish a precise comparison between vanishing cycles and the singularity category of Landau-Ginzburg models over an excellent Henselian discrete valuation ring. By using noncommutative motives, we first construct a motivic-adic realization functor for dg-categories. Our main result, then asserts that, given a Landau-Ginzburg model over a complete discrete valuation ring with potential induced by a uniformizer, the-adic realization of its singularity category is given by the inertia-invariant part of vanishing cohomology. We also prove a functorial and ∞-categorical lax symmetric monoidal version of Orlov's comparison theorem between the derived category of singularities and the derived category of matrix factorizations for a Landau-Ginzburg model over a noetherian regular local ring.

Higher traces, noncommutative motives, and the categorified Chern character

We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (∞,n)-categories, which is of independent interest.

Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^*$-algebras, and bivariant homology

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal \infty -categorical models for separable C^* -algebras \mathtt{SC_\infty^*} and noncommutative spectra \mathtt{NSp} using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of \mathtt{SC_\infty^*} and \mathtt{NSp} with respect to strongly self-absorbing C^* -algebras. We analyse the homotopy categories of the localizations of \mathtt{SC_\infty^*} and give universal characterizations thereof. We construct a stable \infty -categorical model for bivariant connective \mathtt E -theory and compute the connective \mathtt E -theory groups of \mathcal{O}_\infty -stable C^* -algebras. We also introduce and study the nonconnective version of Quillen's nonunital \mathtt K' -theory in the framework of stable \infty -categories. This is done in order to promote our earlier result relating topological \mathbb T -duality to noncommutative motives to the \infty -categorical setup. Finally, we carry out some computations in the case of stable and \mathcal{O}_\infty -stable C^* -algebras.

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 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 Hochschild 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.

Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) non-commutative motives

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 of small dimension satisfy Murre's conjecture (when ), Grothendieck's standard conjecture of Lefschetz type (when ), and Hodge's conjecture (when ).

Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) non-commutative motives

Гипотезы типа гипотезы Бейлинсона-Блоха предсказывают, что группы Чжоу с рациональными коэффициентами малой размерности у пересечения квадрик являются одномерными. Эта гипотеза была доказана А. Отвиновской (A. Otwinowska) в [1]. Используя гомологическую проективную двойственность и недавнюю теорию (якобианов) некоммутативных мотивов, дается другое доказательство этой гипотезы в случае полного пересечения двух квадрик или трех нечетномерных квадрик. Кроме того, доказывается, что в этих случаях единственным нетривиальным алгебраическим якобианом является средний из них. В качестве приложения, опираясь на работы Ч. Виала (C. Vial) [2], [3], описываются рациональные мотивы Чжоу этих полных пересечений и показывается, что для гладких расслоений над базой $S$ малой размерности, слои которых являются такими полными пересечениями, выполняются гипотеза Мюрре (Murre) ($\dim (S)\leq 1$), стандартная гипотеза Гротендика типа Лефшеца ($\dim (S)\leq 2$) и гипотеза Ходжа ($\dim (S)\leq 3$). Библиография: 29 наименований.

From Semi-Orthogonal Decompositions to Polarized Intermediate Jacobians via Jacobians of Noncommutative Motives

Let X and Y be complex smooth projective varieties, and D^b(X) and D^b(Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D^b(X) as in D^b(Y). Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism t of abelian varieties (up to isogeny) from the product of the intermediate algebraic Jacobians of X to the product of the intermediate algebraic Jacobians of Y. When the orthogonal complement of T in D^b(X) has a trivial Jacobian (e.g. when it is generated by exceptional objects), the morphism t is split injective. When this also holds for the orthogonal complement of T in D^b(Y), t becomes an isomorphism. Furthermore, in the case where X and Y have a single intermediate algebraic Jacobian carrying a principal polarization, we prove that the morphism t preserves this extra structure. As an application, we obtain categorical Torelli theorems, an incompatibility between two conjectures of Kuznetsov (one concerning Fourier-Mukai functors and another one concerning Fano threefolds), a new proof of a classical theorem of Clemens-Griffiths on blow-ups of threefolds, and several new results on quadric fibrations and intersection of quadrics.

$\mathbf A^1$-homotopy theory of noncommutative motives

In this article we continue the development of a theory of noncommutative motives, initiated in [30]. We construct categories of {\bf A}^1 -homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi–Villamayor's K -theory ( KV ) and Weibel's homotopy K -theory ( KH ). As an application, we obtain a complete classification of all the natural transformations defined on KV, KH . This leads to a streamlined construction of Weibel's homotopy Chern character from KV to periodic cyclic homology. Along the way we extend Dwyer–Friedlander's étale K -theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.

𝐾-theory of endomorphisms via noncommutative motives

We extend the K K -theory of endomorphisms functor from ordinary rings to (stable) ∞ \infty -categories. We show that K E n d ( − ) \mathrm {KEnd}(-) descends to the category of noncommutative motives, where it is corepresented by the noncommutative motive associated to the tensor algebra S [ t ] \mathbb {S}[t] of the sphere spectrum S \mathbb {S} . Using this corepresentability result, we classify all the natural transformations of K E n d ( − ) \mathrm {KEnd}(-) in terms of an integer plus a fraction between polynomials with constant term 1 1 ; this solves a problem raised by Almkvist in the seventies. Finally, making use of the multiplicative coalgebra structure of S [ t ] \mathbb {S}[t] , we explain how the (rational) Witt vectors can also be recovered from the symmetric monoidal category of noncommutative motives. Along the way we show that the K 0 K_0 -theory of endomorphisms of a connective ring spectrum R R equals the K 0 K_0 -theory of endomorphisms of the underlying ordinary ring π 0 R \pi _0R .

Bivariant cyclic cohomology and Connes’ bilinear pairings in noncommutative motives

In this article we further the study of noncommutative motives. We prove that the bivariant cohomology and the bivariant Chern character of any additive invariant E become representable in the category of noncommutative motives. This applies in particular to bivariant cyclic cohomology and its variants. When E is moreover symmetric monoidal we prove that the associated Chern character is multiplicative and characterize it by a precise universal property. In the particular case of bivariant cyclic cohomology the associated Chern character becomes the universal lift of the Dennis trace map. Then, we prove that under the above representability result, the composition operation in the category of noncommutative motives identifies with Connes’ bilinear pairings. As an application, we obtain a simple model, given by Karoubi’s infinite matrices, for the (de)suspension of these bivariant cohomology theories.

Homotopy-theoretically enriched categories of noncommutative motives

Waldhausen's $K$-theory of the sphere spectrum (closely related to the algebraic $K$-theory of the integers) is a naturally augmented $S^0$-algebra, and so has a Koszul dual. Classic work of Deligne and Goncharov implies an identification of the rationalization of this (covariant) dual with the Hopf algebra of functions on the motivic group for their category of mixed Tate motives over $\Z$. This paper argues that the rationalizations of categories of non-commutative motives defined recently by Blumberg, Gepner, and Tabuada consequently have natural enrichments, with morphism objects in the derived category of mixed Tate motives over $\Z$. We suggest that homotopic descent theory lifts this structure to define a category of motives defined not over $\Z$ but over the sphere ring-spectrum $S^0$.

Relations between the Chow motive and the noncommutative motive of a smooth projective variety

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.

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

We develop an algebraic formalism for topological $\mathbb{T}$-duality. More precisely, we show that topological $\mathbb{T}$-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an $E_\infty$-operad starting from any strongly self-absorbing $C^*$-algebra $\mathcal{D}$. Then we show that there is a functorial topological K-theory symmetric spectrum construction ${\bf K}_\Sigma^{top}(-)$ on the category of separable $C^*$-algebras, such that ${\bf K}_\Sigma^{top}(\mathcal{D})$ is an algebra over this operad; moreover, ${\bf K}_\Sigma^{top}(A\hat{\otimes}\mathcal{D})$ is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of $C^*$-algebras. We also show that $\mathcal{O}_\infty$-stable $C^*$-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of $ax+b$-semigroup $C^*$-algebras coming from number theory and that of $\mathcal{O}_\infty$-stabilized noncommutative tori.