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Remarks on Non-Abelian Cohomology of Proalgebraic Groups

Richard Hain

arXiv:1009.3662v2math.AG

TL;DR

Non-abelian cohomology must represent conjugacy classes of sections even when the defining extension has no splitting or preferred section. The paper constructs H1_nab(G, P) from a negatively weighted prounipotent coefficient group and an outer action, using weight filtrations, finiteness conditions, and compatible Gm-actions. Under the stated finiteness hypotheses, the paper obtains affine F-schemes representing the relevant section functors and identifies conditions making the unipotent GIT quotient a scheme.

  • Problem

    Non-abelian cohomology must represent conjugacy classes of sections even when the defining extension has no splitting or preferred section.

    Source: §1 · pp. 1, 3

  • Method

    The paper constructs H1_nab(G, P) from a negatively weighted prounipotent coefficient group and an outer action, using weight filtrations, finiteness conditions, and compatible Gm-actions.

    Source: §1 · pp. 1–2

  • Results

    Under the stated finiteness hypotheses, the paper obtains affine F-schemes representing the relevant section functors and identifies conditions making the unipotent GIT quotient a scheme.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework avoids choosing a preferred section, while compatible Gm-actions with negative weights yield unipotent GIT quotients that are schemes.

    Source: §1 · pp. 2–3

    The construction assumes negatively weighted cohomology with finite-dimensional weight-graded quotients, and the resulting cohomology scheme may be empty or unpointed.

    Source: §1 · pp. 1, 3

Abstract

from arXiv · show

These are detailed notes on a variant of the non-abelian cohomology developed by Minhyong Kim in arXiv:0409456 (published in Invent. Math.) to study rational points of varieties over number fields. The current variant is used in arXiv:1001.5008 to study rational points of the restriction of the universal curve to the generic point of M_{g,n}.

1. Introduction

The paper defines a non-abelian cohomology scheme representing P-conjugacy classes of sections for extensions determined by an outer action, without requiring a splitting. Under finiteness and negative-weight hypotheses, it obtains affine schemes, exact-sequence structures, and a GIT interpretation.

Hypotheses

  • The coefficient assumptions include negatively weighted H1(P) with finite-dimensional weight-graded quotients, inducing compatible filtrations on the Lie algebra and group.These hypotheses supply the weight-theoretic setting used throughout the construction.Source: §1 · p. 1

Contribution

  • The paper defines H1_nab(G, P) to represent P-conjugacy classes of sections of an extension of G by P.The coefficient group P is prounipotent with trivial center, and the extension is obtained by pulling back the automorphism extension along the outer action.Source: §1 · pp. 1–2

Main results

  • Under the stated finiteness condition, Theorem 1 produces an affine F-scheme of finite type H1_nab(G, P/W−NP) representing the relevant functor.The theorem applies when the finiteness condition holds for 1 ≤ m < N.Source: §1 · p. 2

GIT interpretation

  • Compatible Gm-actions and negative weights make the quotient by the unipotent group a scheme, providing a model for broader non-reductive GIT constructions.Each orbit contains a unique Gm-invariant section under a lift of the central cocharacter.Source: §1 · p. 2

Further structure

  • If the finiteness condition holds for all m ≥ 1, Corollary 2 gives the corresponding affine F-scheme without truncation.The introduction also states an exact sequence and Theorem 3 gives a principal action involving H1(g, GrW...).Source: §1 · pp. 2–3

Non-split extensions

  • The construction does not require the extension to split or select a preferred section, and H1_nab(G, P) may therefore be empty or unpointed.The paper presents this as essential because preferred sections can be unnatural in the relevant computations.Source: §1 · pp. 2–3

2. Weight Filtrations

Negatively weighted extensions provide lift-dependent splittings whose induced weight filtration is canonical. The centralizer of every lift of the central cocharacter is trivial, enabling invariant constructions.

2. Weight Filtrations

  • Levi’s theorem supplies lifts of the central cocharacter, and any two splittings are conjugate by an element of the prounipotent group.Thus the lift is not canonical, but its associated filtration is canonical.Source: §2 · p. 3
  • A central cocharacter lift splits each finite-dimensional G-module into isotypical weight spaces, while the induced weight filtration is independent of the lift.The filtration is by G-submodules and extends to ind- and pro-objects.Source: §2 · p. 3
  • The section defines the centralizer as the group representing the functor of elements commuting with the chosen cocharacter lift.This functorial formulation is used before proving the centralizer is trivial.Source: §2 · p. 4
  • For a negatively weighted extension, every lift of the central cocharacter has trivial centralizer.The proof reduces a commuting element to weight zero, which vanishes because all nontrivial extension weights are negative.Source: §2 · p. 4

3. Cohomology

The cohomology review relates algebraic-group cohomology to Lie algebra cohomology through de Rham cohomology and extension spectral sequences. It then extends these results to proalgebraic groups and negatively weighted extensions.

3.1. Cohomology of algebraic groups.

  • For connected affine algebraic groups, Hochschild’s theorem identifies group cohomology with Lie algebra cohomology up to the de Rham contribution.The natural map from group to Lie algebra cohomology is injective, while the discrepancy is controlled by H•_DR(G).Source: §3.1 · pp. 4–5
  • For unipotent groups, group and Lie algebra cohomology are naturally isomorphic for every rational module.This follows because the de Rham cohomology of a unipotent group vanishes.Source: §3.1 · p. 5
  • Hochschild’s connected-group result fails for disconnected groups, which requires separating the identity component and the finite component group.In particular, inclusion of the identity component induces an isomorphism on cohomology under the stated hypotheses.Source: §3.1 · p. 5
  • For a normal subgroup, the Grothendieck spectral sequence yields the spectral sequence governing cohomology of a group extension.The construction applies invariants under the normal subgroup before passing to the quotient.Source: §3.1 · p. 5
  • When the identity component acts without invariants, first cohomology of the group agrees with first Lie algebra cohomology.For negatively weighted extensions and nonzero weights, the same comparison applies; nonnegative weights force vanishing in the stated case.Source: §3.1 · pp. 5–6

3.2. Cohomology of proalgebraic groups.

  • For proalgebraic groups with finitely many components, the comparison map to Lie algebra cohomology is injective and becomes an isomorphism when the identity component has no invariants.Continuous cohomology is defined through finite-dimensional stages and compatible inverse limits.Source: §3.2 · p. 6
  • A lifted cocharacter preserves the Lie bracket, cochain differential, and resulting cohomological weight decomposition in negatively weighted extensions.The induced splitting decomposes cohomology into weight summands computed from the corresponding cochain complexes.Source: §3.2 · p. 7

4. Proofs of the Main Results

The proofs construct schemes of sections and quotient them by prounipotent conjugation using negative weights. This yields representable non-abelian cohomology schemes and an exact sequence description.

4.1. Setup.

  • The setup assumes a negatively weighted extension of a reductive group by a prounipotent group, giving the coefficient Lie algebra a natural negative weight filtration.Later the coefficient group is obtained by truncating the prounipotent group by a weight filtration step.Source: §4.1 · p. 7

4.2. Sections.

  • Every N-orbit of graded sections contains a unique section graded by a chosen lift of the central cocharacter.This canonical representative follows from conjugating the induced cocharacter lift and the trivial-centralizer property.Source: §4.2 · p. 8
  • R-invariant graded sections of the associated graded Lie algebra extension form an ind-affine scheme, finite type when the relevant H1 spaces are finite dimensional.The construction uses graded lifts from the abelianization and equations expressing compatibility with the Lie structure.Source: §4.2 · p. 9
  • The full section functor is represented by an ind-affine scheme independent of the chosen cocharacter lift, and it has finite type under the same cohomological condition.Conjugation identifies arbitrary sections with cocharacter-graded sections.Source: §4.2 · p. 10

4.3. Existence of non-abelian cohomology.

  • Theorem 4.6 constructs an ind-affine scheme H1_nab(G, N) representing sections modulo N(A)-conjugation, with finite type under finite-dimensional cohomology conditions.The construction combines graded section schemes with the conjugation parametrization.Source: §4.3 · p. 10

4.4. The exact sequence.

  • When W−N−1n = 0 and Z = W−Nn is non-zero, the connecting morphism δ obstructs lifting a graded section: δ(σ)=0 permits correction by an R-invariant 1-cochain.Source: §4.4 · p. 11
  • Under the stated finite-dimensionality assumptions, the exact-sequence terms are affine schemes of finite type and the action of H1(g,z) on the fibers is principal.The negative weight of z eliminates invariant 1-coboundaries, supporting the principal-action description.Source: §4.4 · p. 11

5. Relation to Kim’s work

The paper formulates Kim’s non-abelian cohomology in its own framework for S-integral points and shows agreement on Qℓ-rational points. Weighted completion and exactness properties provide the bridge between the constructions.

Relation to Kim’s work

  • The discussion places this equivalence in the context of Kim’s study of S-integral points of varieties, particularly curves, over number fields.The paper presents its variant as a natural expression of Kim’s ideas.Source: §5 · p. 12

Setup

  • The construction starts with a smooth projective scheme over a number field, a finite set of primes, and the associated Galois group Gk,S.The Galois group acts on the relevant cohomology, supplying the arithmetic input for weighted completion.Source: §5 · p. 12

Weighted completion

  • Weighted completion of Gk,S with respect to ρX and ω produces a proalgebraic group G extending the reductive group R by a prounipotent group.The cocharacter is defined from the weight −1 action on H1.Source: §5 · p. 12

Comparison

  • Exactness and injectivity identify the relevant pulled-back sequences used to compare the paper’s cohomology with Kim’s construction.The comparison uses exact sequences arising from proalgebraic completions and their pullbacks.Source: §5 · p. 12
  • The universal mapping property of weighted completion yields a bijection, so the two versions of non-abelian cohomology agree on Qℓ-rational points.The paper explicitly describes this extension as the kind considered by Kim in.Source: §5 · p. 12

Appendix A. Cohomology of Gm ⊗A

Appendix A proves vanishing of H1 for rank-one Gm-modules and uses it to establish conjugacy of sections for suitably weighted extensions. The proof proceeds from explicit cocycle calculations to induction along a central series.

Cohomology calculation

  • For every Z-algebra A, H1(Gm ⊗Z A, Vd) vanishes.This is Proposition A.1, the cohomological input for the section-conjugacy result.Source: Appendix A · p. 13
  • The cocycle calculation shows that when d ≠ 0 every cocycle has the form ad(td − 1), hence is a coboundary; when d = 0, only the zero cocycle occurs.The argument uses the cocycle condition and the resulting restrictions on Laurent-polynomial coefficients.Source: Appendix A · p. 13

Conjugacy of sections

  • If the unipotent kernel has a descending central series whose graded quotients are sums of Vd-modules, any two sections of G → Gm ⊗Z A are conjugate by an element of U(A).The result is proved inductively on the length of the lower central series.Source: Appendix A · p. 13
  • The induction passes to the quotient by the center, lifts a conjugating element, and then corrects the lift using a central element because the center is abelian.This combines the induction hypothesis with the exact sequence associated to the central subgroup.Source: Appendix A · pp. 13–14
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