Galois Cohomology

Galois cohomology is the cohomology of the absolute Galois group \(\Gamma = \operatorname{Gal}(\bar k/k)\) with values in \(\Gamma\)-modules — equivalently, the derived functor of "fixed points." For algebraic groups, it parametrises forms, twists, and classifies arithmetic invariants of \(G/k\).

Definition

Let \(k\) be a field with separable closure \(\bar k\) and absolute Galois group \(\Gamma = \operatorname{Gal}(\bar k/k)\) (a profinite group). For a continuous discrete \(\Gamma\)-module \(M\):

\[ H^i(k, M) \;:=\; H^i_{\mathrm{cont}}(\Gamma, M), \]

the continuous cohomology. Equivalently, étale cohomology of Spec \(k\) with values in \(M\): \[ H^i(k, M) \;\cong\; H^i_{\mathrm{\acute{e}t}}(\mathrm{Spec}\, k, M). \]

For an algebraic group \(G\) over \(k\) (possibly non-abelian), one defines:

\(H^0(k, G) := G(k)\), the rational points;
\(H^1(k, G)\) := equivalence classes of 1-cocycles \(c : \Gamma \to G(\bar k)\) satisfying \(c_{\sigma \tau} = c_\sigma \cdot \sigma(c_\tau)\), modulo \(c_\sigma \sim g^{-1} c_\sigma \sigma(g)\) for \(g \in G(\bar k)\).

For \(G\) abelian (commutative), \(H^i(k, G)\) is defined for all \(i \ge 0\) by standard cocycle/coboundary formulas. For \(G\) non-abelian, only \(H^0\) and \(H^1\) are defined, the latter as a pointed set.

Hilbert 90

Theorem (Hilbert 90). \[ H^1(k, \mathbb{G}_m) \;=\; H^1(k, \mathrm{GL}_n) \;=\; 0 \] for any field \(k\). For \(\mathbb{G}_m\) the proof goes back to Kummer/Hilbert; for \(\mathrm{GL}_n\) the result expresses the triviality of the Brauer-Severi variety in degree 0.

Consequence: every form of \(\mathrm{GL}_n\) over \(k\) twisted by a 1-cocycle is trivial, so all forms of \(\mathrm{GL}_n\) arise as inner forms (twists by \(\mathrm{PGL}_n = \mathrm{Aut}(\mathrm{GL}_n)\)).

Long exact sequence

For a short exact sequence of \(\Gamma\)-modules (or algebraic groups) \(1 \to A \to B \to C \to 1\):

\[ 1 \to H^0(k, A) \to H^0(k, B) \to H^0(k, C) \to H^1(k, A) \to H^1(k, B) \to H^1(k, C) \to \cdots \]

(further only if \(A\) is in the centre of \(B\); for non-central extensions the sequence stops after \(H^1\)).

The connecting map \(H^0(k, C) \to H^1(k, A)\) is the source of all obstructions in descent problems.

Cohomological dimension

A field \(k\) has cohomological dimension \(\mathrm{cd}(k) \le n\) if \(H^i(k, M) = 0\) for all torsion modules \(M\) and all \(i > n\). Key cases:

\(k\)	\(\mathrm{cd}(k)\)	\(H^1(k, G)\) for connected \(G\)
Algebraically closed	0	0
Finite	1	0 (Lang's theorem)
Function field \(k(C)\), \(C\)/\(\bar k\)	1	0
Non-archimedean local field	2	non-trivial for semisimple groups (e.g.\ \(\mathrm{PGL}_n\))
Number field	2 (away from real places)	non-trivial, but Hasse principle applies
\(\mathbb{R}\)	\(\infty\) (because \(\mathbb{Z}/2\))	real forms classified by \(H^1\)

The Brauer group

\(H^2(k, \mathbb{G}_m) = \mathrm{Br}(k)\), the Brauer group of \(k\). Classifies:

Central simple \(k\)-algebras of any rank (via Wedderburn) up to Morita equivalence;
Inner forms of \(\mathrm{PGL}_n\): \(\mathrm{Br}(k)[n]\) is the \(n\)-torsion;
Galois algebras of period \(n\).

For local fields: \(\mathrm{Br}(F) = \mathbb{Q}/\mathbb{Z}\) (canonical iso via Hasse invariant). For number fields: exact sequence \(0 \to \mathrm{Br}(F) \to \bigoplus_v \mathrm{Br}(F_v) \to \mathbb{Q}/\mathbb{Z} \to 0\) (Albert-Brauer-Hasse-Noether).

Galois cohomology of tori

For a torus \(T\) over \(k\) split by \(K\): \[ H^i(k, T) \;=\; H^i(\mathrm{Gal}(K/k), T(K)), \] computable via the character lattice \(X^*(T)\) as a Galois module.

For \(T\) a norm torus or Weil restriction: explicit computations via Shapiro's lemma.

Local-global principles

For a connected reductive group \(G\) over a global field \(F\), the Tate–Shafarevich set is \[ \Sha^1(F, G) \;:=\; \ker\Big(H^1(F, G) \to \prod_v H^1(F_v, G)\Big). \]

Hasse principle (when it holds): \(\Sha^1(F, G) = 0\), i.e.\ a \(G\)-torsor is trivial iff it is locally trivial everywhere. Holds for many \(G\) (Kneser, Harder, Chernousov: for simply connected semisimple \(G\) over number fields).

Failure of Hasse principle is measured by \(\Sha^1\) and is itself a deep arithmetic invariant.

Role in Langlands

Forms of \(G\): classified by \(H^1(k, \mathrm{Aut}(G))\); inner forms by \(H^1(k, G_{\mathrm{ad}})\); outer forms by the diagram automorphism component.
Endoscopy depends on \(H^1\) of \(G\) and its endoscopic groups.
Kottwitz invariants \(\kappa : B(G) \to \pi_1(G)_\Gamma\) and \(H^1\) are unified by Borovoi's abelianised Galois cohomology.
Global Langlands: \(\Sha^1\) is part of the input to the trace formula and stable trace formula.