Tannakian Categories Research Articles

Constructing colimits by gluing vector bundles

As already observed by Gabriel, coherent sheaves on schemes obtained by gluing affine open subsets can be described by a simple gluing construction. An example due to Ferrand shows that this fails in general for pushouts along closed immersions, though the gluing construction still works for flat coherent sheaves.We show that by further restricting this gluing construction to vector bundles, we can construct pushouts along arbitrary morphisms (and more general colimits) of certain algebraic stacks called Adams stacks. The proof of this fact uses generalized Tannaka duality and a variant of Deligne's argument for the existence of fiber functors which works in arbitrary characteristic.We use this version of Deligne's existence theorem for fiber functors as a novel way of recognizing stacks which have atlases. It differs considerably from Artin's algebraicity results and their generalizations: rather than studying conditions on the functor of points which ensure the existence of an atlas, our theorem identifies conditions on the category of quasi-coherent sheaves of the stack which imply that an atlas exists.

A note on Tannakian categories and mixed motives

We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures. In other words, every non-zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated \'etale motives over F with integral coefficients, up to the tensor triangulated structure, in terms of the characteristic and the orderings of F.

Tannaka duality for enhanced triangulated categories I: reconstruction

We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal A$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $\mathcal A$-modules and of $C$-comodules. When $\mathcal A$ is Morita fibrant (i.e. an idempotent-complete pre-triangulated category), it is thus quasi-equivalent to the derived dg category of compact $C$-comodules. We give several applications for motivic Galois groups.

Depth-graded motivic Lie algebra

In this paper we suggest a way to understand the structure of depth-graded motivic Lie subalgebra generated by the depth one part for the neutral Tannakian category mixed Tate motives over Z. We will show that from an isomorphism conjecture proposed by K. Tasaka we can deduce the F. Brown's matrix conjecture and the nondegeneracy conjecture about depth-graded motivic Lie subalgebra generated by the depth one part.

The Drinfeld centre of a symmetric fusion category is 2-fold monoidal

We show that the Drinfeld centre of a symmetric fusion category over an algebraically closed field of characteristic zero is a bilax 2-fold monoidal category. That is, it carries two monoidal structures, the convolution and symmetric tensor products, that are bilax monoidal functors with respect to each other. We additionally show that the braiding and symmetry for the convolution and symmetric tensor products are compatible with this bilax structure.We establish these properties without referring to Tannaka duality for the symmetric fusion category. This has the advantage that all constructions are done purely in terms of the fusion category structure, making the result easy to use in other contexts.

The symmetric tensor product on the Drinfeld centre of a symmetric fusion category

We define a symmetric tensor product on the Drinfeld centre of a symmetric fusion category, in addition to its usual tensor product. We examine what this tensor product looks like under Tannaka duality, identifying the symmetric fusion category with the representation category of a finite (super)-group. Under this identification, the Drinfeld centre is the category of equivariant vector bundles over the finite group (underlying the super-group, in the super case). In the non-super case, we show that the symmetric tensor product corresponds to the fibrewise tensor product of these vector bundles. In the super case, we define for each super-group structure on the finite group a super-version of the fibrewise tensor product. We show that the symmetric tensor product on the Drinfeld centre of the representation category of the resulting finite super-groups corresponds to this super-version of the fibrewise tensor product on the category of equivariant vector bundles over the finite group.

On log motives

We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.

The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curves

In this paper, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.

Coherent Tannaka duality and algebraicity of Hom-stacks

We establish Tannaka duality for noetherian algebraic stacks with affine stabilizer groups. Our main application is the existence of Hom-stacks in great generality.

Higgs bundles and fundamental group schemes

Abstract Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

A Note on 1-Motives

Abstract We prove that for $1$-motives defined over an algebraically closed subfield of $\mathbf{C}$, viewed as Nori motives, the motivic Galois group coincides with the Mumford–Tate group. In particular, the Hodge realization of the Tannakian category of Nori motives generated by $1$-motives is fully faithful. This result extends an earlier result by the author, according to which Hodge cycles on abelian varieties are motivated (a weak form of the Hodge conjecture).

The formal theory of Tannaka duality

The formal theory of Tannaka duality

Tannakian Duality for Affine Homogeneous Spaces

AbstractAssociated with any closed quantum subgroup and any index set I ⊂ {1,…,N} is a certain homogeneous space , called an affine homogeneous space. Using Tannakian duality methods, we discuss the abstract axiomatization of the algebraic manifolds that can appear in this way.

Specialization map between stratified bundles and pro-étale fundamental group

Given a projective family of semi-stable curves over a complete discrete valuation ring of characteristic p>0 with algebraically closed residue field, we construct a specialization functor between the category of continuous representations of the pro-étale fundamental group of the closed fibre and the category of stratified bundles on the geometric generic fibre. By Tannakian duality, this functor induces a morphism between the corresponding affine group schemes. We show that this morphism is a lifting of the specialization map, constructed by Grothendieck, between the étale fundamental groups.

On the structure of affine flat group schemes over discrete valuation rings

We study affine group schemes over a discrete valuation ring $R$ using two techniques: Neron blowups and Tannakian categories. We employ the theory developed to define and study differential Galois groups of $\mathcal D$-modules on a scheme over a $R$. This throws light on how differential Galois groups of families degenerate.

UNIVERSAL MIXED ELLIPTIC MOTIVES

In this paper we construct a $\mathbb{Q}$-linear tannakian category $\mathsf{MEM}_{1}$ of universal mixed elliptic motives over the moduli space ${\mathcal{M}}_{1,1}$ of elliptic curves. It contains $\mathsf{MTM}$, the category of mixed Tate motives unramified over the integers. Each object of $\mathsf{MEM}_{1}$ is an object of $\mathsf{MTM}$ endowed with an action of $\text{SL}_{2}(\mathbb{Z})$ that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over ${\mathcal{M}}_{1,1}$ whose fiber over the tangential base point $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}q$ at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of $\mathsf{MEM}$ is determined and the lowest order terms of a set (conjecturally, a minimal generating set) of relations are deduced from computations of Brown. This set of relations includes the arithmetic relations, which describe the ‘infinitesimal Galois action’. We use the presentation to give a new and more conceptual proof of the Ihara–Takao congruences.

Tannaka duality and stable infinity-categories

We introduce a notion of fine Tannakian infinity-categories and prove Tannakian characterization results for symmetric monoidal stable infinity-categories over a field of characteristic zero. It connects derived quotient stacks with symmetric monoidal stable infinity-categories which satisfy a certain simple axiom. We also discuss several applications to examples.

Higgs varieties and fundamental groups

Abstract After reviewing some “fundamental group schemes” that can be attached to a variety by means of Tannaka duality, we consider the example of the Higgs fundamental group scheme, surveying its main properties and relations with the other fundamental groups, and giving some examples.

Differential Embedding Problems over Complex Function Fields

We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. We show that patching holds over function fields over the complex numbers. As the main application, we prove the solvability of all differential embedding problems over complex function fields, thereby providing new insight on the structure of the absolute differential Galois group, i.e., the fundamental group of the underlying Tannakian category.

Tannakian duality over Dedekind rings and applications

We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R is a complete DVR of equal characteristic we show that this category is Tannakian in the sense of Saavedra.