The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

A note on coinduction functors between categories of comodules for corings

In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case oflocally projective corings as a composition of suitable “Trace” and “Hom” functors and show how to derive it from a moregeneral coinduction functor between categories of type σ[M]. In special cases (e.g. the corings morphism is part of a morphism of measuringa-pairings or the corings have the same base ring), a version of our functor is shown to be isomorphic to the usual coinduction functor obtained by means of the cotensor product. Our results in this note generalize previous results of the author on coinduction functors between categories of comodules for coalgebras over commutative base rings.     

Cube structures and intersection bundles

We define the notion of a hypercube structure on a functor between two commutative Picard categories which generalizes the notion of a cube structure on a G_m-torsor over an abelian scheme. We prove that the determinant functor of a relative scheme X/S of relative dimension n is canonically endowed with a (n+2)-cube structure. We use this result to define the intersection bundle I_X/S(L₁,…,L_n+1) of n+1 line bundles on X/S and to construct an additive structure on the functor I_X/S:PIC(X/S)ⁿ⁺¹→PIC(S). Then, we construct the resultant of n+1 sections of n+1 line bundles on X, and the discriminant of a section of a line bundle on X. Finally we study the relationship between the cube structures on the determinant functor and on the discriminant functor, and we use it to prove a polarization formula for the discriminant functor.     

Interpolation categories for homology theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions, we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As an application, we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.     

Tannakian Categories, Linear Differential Algebraic Groups, and Parametrized Linear Differential Equations

We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group [18], [9]. This work was partially supported by NSF grant CCR-0096842 and by the Russian Foundation for Basic Research, project no. 05-01-00671.     

The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme

We define the algebraic fundamental group π ₁(G) of a reductive group scheme G over an arbitrary non-empty base scheme and show that the resulting functor G? π₁(G) is exact.     

Generalizations of the Sweedler Dual

As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.     

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C_c^∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c^∞(G)-C_c^∞(H)-bimodule C_c^∞(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c^∞gives an equivalence between the Morita category of étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.     

Gl2 of rings with many units

This paper describes the generiators and normal subgroups of GL₂(R) where R is a commutative ring with “many units”.     

Witt vectors and K-theory of automorphisms via noncommutative motives

We prove that the functor ring-of-rational-Witt-vectors W ₀(?) becomes co-representable in the category of noncommutative motives. As an application, we obtain an immediate extension of W 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.     

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.     

Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep2MatK(G) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π₀(G), π₁(G) and [α]∈H³(π₀(G),π₁(G)) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z]∈H²(π₀(G),K^∗)) of the first homotopy group π₀(G) as well as its category of representations on finite sets both live in Rep2MatK(G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep2MatK(G).     

When is the Diagonal Functor Frobenius?

Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or _R ?, the category of left modules over a ring R.     

Compact nilpotent groups

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π₂X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T _π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T _π-compact provided G/T _πG is π-complete, extending the abelian group result in a natural way.     

On 2-dimensional Nonaspherical Cell-like Peano Continua: A Simplified Approach

We construct a functor AC(?, ?) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)?dimensional; and (iii) For a path connected space X, π ₁(X, x) is trivial if and only if π ₂(AC(X, x)) is trivial. As a corollary, AC(S ¹, x) is a 2-dimensional nonaspherical cell-like Peano continuum.     

Locally (-1,1) Rings

If M is a finitely-generated module over a commutative noetherian ring, then it is well-known that the ideal-theoretic assassin and the ideal-theoretic support of M satisfy the following two conditions: (CD supp(M) and ass(M) have the same minimal elements ; (C2) supp(M) has only finitely-many minimal elements. The purpose of this note is to consider a sufficient condition on a noncommutative ring, generalizing the commutative case, for analogous results to hold for the torsion-theoretic assassin and the torsion-theore-tic support of suitable left R-modules M. Since the order in the lattice of torsion theories over the category R-mod of left R-modules is the reverse of the order in the lattice of left ideals of R, we would expect to substitute “maximal” for “minimal” in the above conditions. Also, noetherianness should be replaced by its torsion-theoretic counterpart, semi-noetherianness.     

A generalization of a theorem of A. Grothendieck

In this article we characterize the quasi‐barrelledness of the projective tensor product of a coechelon space of type one k ¹(A) with a Fréchet space, including homological conditions as exactness properties of the corresponding tensor product functor k ¹(A) ·: ? → ??, acting from the category of Fréchet spaces to the category of linear spaces, resp. the vanishing of its first right derivative Tor¹_π (k ¹(A),.). This generalizes and extends a classical theorem of A. Grothendieck ([13, Chap. II, §4, No. 3, Theorem 15]). Further we present an analogous theorem for complete coechelon spaces of type zero and the injective tensor product and results concerning the stronger property of barrelledness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

This article is the sequel to (Marcolli and Tabuada in Sel Math 20(1):315–358, 2014). We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub’s weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin–Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub’s weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky’s category of mixed motives.     

Differential Tannakian categories

We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces over the base field, then it is equivalent to the category of representations of a (pro-)linear differential algebraic group. Our treatment of the problem is via differential Hopf algebras and Deligne's fibre functor construction [P. Deligne, Catégories tannakiennes, in: The Grothendieck Festschrift, vol. II, in: Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195].