Cohomology Sets Research Articles

Lie bialgebras, fields of cohomological dimension at most 2 and Hilbert's Seventeenth Problem

We investigate Lie bialgebra structures on simple Lie algebras of non-split type A. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin–Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem.

Locally Equivalent Correspondences

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same covolume and pro-congruence completion. We also make effective a finiteness result of Prasad and Rapinchuk.

Cohomology with twisted coefficients of the classifying space of a fusion system

We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system F. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of F-stable elements to Fc-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group. We then study the problem of inducing an idempotent from an F-characteristic (S,S)-biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by Fc-stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a p-local finite group can be computed by these Fc-stable elements.

Rigid inner forms of real and p-adic groups

We dene a new cohomology set H 1 (u ! W;Z ! G) for an ane algebraic group G and a nite central subgroup, both dened over a local eld of characteristic zero, which is an enlargement of the usual rst Galois cohomology set of G. We show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered L-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the case of real groups, we show that this description is correct using Shelstad’s work.

TWISTED FORMS OF TORIC VARIETIES

We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to a construction of A. Merkurjev and I. Panin. This generalizes both a result of M.~Blunk for del Pezzo surfaces of degree 6, and the standard bijection between Severi-Brauer varieties and central simple algebras

Classification of quantum groups and Lie bialgebra structures on [formula omitted]. Relations with Brauer group

Given an arbitrary field F of characteristic 0, we study Lie bialgebra structures on sl(n,F), based on the description of the corresponding classical double. For any Lie bialgebra structure δ, the classical double D(sl(n,F),δ) is isomorphic to sl(n,F)⊗FA, where A is either F[ε], with ε2=0, or F⊕F or a quadratic field extension of F. In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n,F). In the second and third cases, a Belavin–Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F), up to gauge equivalence. The Belavin–Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed. For the Cremmer–Gervais r-matrix in sl(3), we also construct a natural map of sets between the total Belavin–Drinfeld twisted cohomology set and the Brauer group of the field F.

On quantum groups and Lie bialgebras related to sl(n)

Given an arbitrary field of characteristic 0, we study Lie bialgebra structures on sl(n, ), based on the description of the corresponding classical double. For any Lie bialgebra structure δ, the classical double D(sl(n, ),δ) is isomorphic to sl(n, ) ⊗A, where A is either [ε], with ε2 = 0, or ⊗ or a quadratic field extension of . In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n, ). In the second and third cases, a Belavin-Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n, ), up to gauge equivalence. The Belavin-Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed.

Weyl group actions on the Springer sheaf

We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence.

Non-Abelian Hopf Cohomology of Radford Products

We study the non-abelian Hopf cohomology theory of Radford products with coefficients in a comodule algebra. We show that these sets can be expressed in terms of the non-abelian Hopf cohomology theory of each factor of the Radford product. We write down an exact sequence relating these objects. This allows to compute explicitly the non-abelian Hopf cohomology sets in large classes of examples.

Indefinite extrinsic symmetric spaces II

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the classification difficult. We use the recently developed method of quadratic extensions for (𝔥, K )-invariant metric Lie algebras to tackle this problem. We obtain a one-to-one correspondence between isometry classes of extrinsic symmetric spaces and a certain cohomology set. This allows a systematic construction of extrinsic symmetric spaces and explicit classification results, e.g., if the metric of the embedded manifold or the ambient space has a small index. We will illustrate this by classifying all Lorentzian extrinsic symmetric spaces.

On the structure of pseudo-Riemannian symmetric spaces

Following our approach to metric Lie algebras developed in a previous paper we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semisimple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a triple consisting of: That leads to a classification scheme of indecomposable nonsimple pseudo-Riemannian symmetric spaces. In addition, we obtain a full classification of symmetric spaces of index 2 (thereby completing and correcting in part earlier classification results due to Cahen and Parker and to Neukirchner).

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green–Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers h q , 0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak–Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.

Chow-Künneth decomposition for some moduli spaces

In this paper we investigate Murre's conjecture on the Chow–Künneth decomposition for universal families of smooth curves over spaces which dominate the moduli space \mathcal M_g , in genus at most 8 and show existence of a Chow–Künneth decomposition. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow–Künneth decompositions.

Non-abelian Hopf cohomology II—The general case

We introduce and study non-abelian cohomology sets of Hopf algebras with coefficients in Hopf comodule algebras. We prove that these sets generalize as well Serre's non-abelian group cohomology theory as the cohomological theory constructed by the authors in a previous article. We establish their functoriality and compute explicit examples. Further we classify Hopf torsors.

Paramétrisation des courbes multiples primitives

Abstract The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Bănică and O. Forster in 1984. In 1995, D. Bayer and D. Eisenbud gave a complete description of the double curves. We give here a parametrization of primitive curves of arbitrary multiplicity. Let Z n = spec(ℂ[t]/(t n )). The curves of multiplicity n are obtained by taking an open cover (U i ) of a smooth curve C and by glueing schemes of type U i × Z n using automorphisms of U ij × Z n that leave U ij invariant. This leads to the study of the sheaf of nonabelian groups G n of automorphisms of C × Z n that leave the reduced curve invariant, and to the study of its first cohomology set. We prove also that in most cases it is the same to extend a primitive curve C n of multiplicity n to one of multiplicity n + 1, and to extend the quasi locally free sheaf D n of derivations of C n to a rank 2 vector bundle on C n .

Non-abelian Hopf cohomology

We introduce non-abelian cohomology sets of Hopf algebras with coefficients in Hopf modules. We prove that these sets generalize Serre's non-abelian group cohomology theory. Using descent techniques, we establish that our construction enables to classify as well twisted forms for modules over Hopf-Galois extensions as torsors over Hopf-modules.

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H2 Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.

Metric Lie algebras with maximal isotropic centre

A metric Lie algebra is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form. It is called indecomposable if it is not the direct sum of two metric Lie algebras. We are interested in describing the isomorphism classes of indecomposable metric Lie algebras. In the present paper we restrict ourselves to a certain class of solvable metric Lie algebras which includes all indecomposable metric Lie algebras with maximal isotropic centre. We will see that each metric Lie algebra belonging to this class is a twofold extension associated with an orthogonal representation of an abelian Lie algebra. We will describe equivalence classes of such extensions by a certain cohomology set. In particular we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre and the classification of metric Lie algebras of index 2.

Cohomology sets inside arithmetic groups

Cohomology sets inside arithmetic groups

Weak corestriction principle for non-abelian Galois cohomology

We introduce the notion of (Weak) Corestriction Principle and prove some relations between the validity of this principle for various connecting maps in non-abelian Galois cohomology over fields of characteristic 0. We also prove the validity of Weak Corestriction Principle for images of coboundary maps H(k, G) → H(k, T ), where T is a finite commutative k-group of multiplicative type, G is adjoint, semisimple and contains only almost simple factors of certain inner types. Introduction. Let G be a commutative algebraic group defined over a field k of characteristic 0. Let H(k, G) denote the usual Galois cohomology H(Gal(k/k), G(k)). It is well-known that there exists corestriction homomorphism Cores := Coresk′/k : Hi(k′, G) → H(k, G) for any i > 1 and any finite extension k′ of k, which gives rise to a map of functors (G 7→ Hi(k′, G)) → (G 7→ H(k, G)). In particular, if 1 → A j → B p → C → 1 is an exact sequence of commutative algebraic k-groups, {α1, α2, ...} (resp. {α′ 1, α′ 2, ...}) denotes the sequence of homomorphisms appearing in the long exact sequence of cohomology deduced from (∗) as cohomology of Gal(k/k)-modules (resp. as Gal(k/k′)-modules), then we have Cores ◦ α′ m = αm ◦ Cores for all m > 1. However, if in (∗) one of the groups is not commutative, then it turns out that there is no corestriction map between these two long exact sequences in general. (In [R1], C. Riehm has found some sufficient conditions for the existence of corestriction map.) It leads us to the following definition. Let A,B be algebraic groups defined over k. Assume that we are given a map of functors f : (L 7→ H(L,A)) → (L 7→ H(L,B)), where L denotes a field extension of k, i.e., a collection of maps of cohomology sets fL : H(L, A) → H(L,B), where fL is functorial in L. Received March 21, 2003, revised June 30, 2003; published on July 18, 2003. 2000 Mathematics Subject Classification: Primary 11G72; Secondary 18G50, 20G10.