Cocycles and Twisting

Twisting is the operation that constructs a new \(k\)-form of an object from a 1-cocycle valued in its automorphism group. It is the hands-on counterpart of \(H^1\) classifying forms.

Setup

Let \(k\) be a field with separable closure \(\bar k\) and Galois group \(\Gamma\). Let \(X\) be a \(k\)-object (an algebraic group, a variety, a vector space with structure, etc.), and let \(A = \mathrm{Aut}(X_{\bar k})\) be the \(\bar k\)-automorphism group of \(X_{\bar k}\) (a \(\Gamma\)-group).

Cocycles

A 1-cocycle of \(\Gamma\) with values in \(A\) is a continuous function \[ c : \Gamma \;\longrightarrow\; A \] satisfying the cocycle condition \[ c_{\sigma \tau} \;=\; c_\sigma \cdot \sigma(c_\tau) \qquad \text{for all } \sigma, \tau \in \Gamma. \]

Two cocycles \(c, c'\) are cohomologous if there exists \(a \in A\) with \(c'_\sigma = a^{-1} c_\sigma \sigma(a)\) for all \(\sigma\).

Cohomology classes form the pointed set \(H^1(\Gamma, A) = H^1(k, A)\).

Twisting construction

Given a 1-cocycle \(c \in Z^1(\Gamma, A)\), the twisted form \(X^c\) is the \(k\)-object whose \(\bar k\)-points are \(X(\bar k)\) but with the new Galois action: \[ \sigma \cdot_c x \;:=\; c_\sigma \cdot \sigma(x). \]

That this defines an action of \(\Gamma\) is precisely the cocycle condition; descent from \(\bar k\)-points to a \(k\)-form requires \(X\) to be acted upon by \(A\) in a "good" way (which is automatic if \(X\) is functorial in its structure).

Examples

Inner twists of \(\mathrm{GL}_n\)

\(A = \mathrm{PGL}_n\). A cocycle \(c : \Gamma \to \mathrm{PGL}_n(\bar k)\) lifts to a 2-cocycle in \(\bar k^\times\), giving a central simple algebra \(D_c\) of dim \(n^2\) over \(k\). Then \(\mathrm{GL}_n^c = D_c^\times\) (the unit group of \(D_c\) as an algebraic group).

Forms of vector spaces

For \(X = k^n\) with quadratic form \(q\), \(A = \mathrm{O}(q)\). Cocycles in \(H^1(k, \mathrm{O}(q))\) classify quadratic forms of the same rank and (over \(\mathbb{R}\)) the same discriminant.

Torsors and twisted forms

If \(X = G\) is an algebraic group acting on itself by left multiplication, then \(A = G\) (right translations centralise left). A 1-cocycle in \(G\) gives a torsor \(P\), and the twisted form of \(X = G\) is \(P\) itself.

Forms of root data

For \(X = (G_0, T_0, B_0)\) a pinned reductive group, the cocycle takes values in \(\mathrm{Out}(G_0) = \mathrm{Aut}(\text{based root datum})\). Outer twists produce non-split forms (e.g.\ \({}^2 A_n\), \({}^3 D_4\)).

Triviality of twisting on \(H^0\)

For a \(\Gamma\)-set \(Y\) on which \(A\) acts, twisting by \(c\) gives a new \(\Gamma\)-action on \(Y(\bar k)\), and there is a bijection \[ H^0(\Gamma, Y^c) \;=\; \{y \in Y(\bar k) : c_\sigma \cdot \sigma(y) = y \text{ for all } \sigma\}. \]

This identifies "\(k\)-rational points of the twist" with "\(c\)-twisted \(\Gamma\)-invariants of \(Y\)."

Twisting and long exact sequences

For a short exact sequence \(1 \to A \to B \to C \to 1\) and an element \(c \in H^1(k, C)\), twisting gives a new short exact sequence \[ 1 \to A^c \to B^c \to C \to 1 \] where \(C\) acts on \(A\) (and \(B\)) by inner conjugation in \(B\). This is the mechanism behind the comparison of \(H^1\) under twists, a basic tool for explicit calculations.

The "form" interpretation

The bijection between cocycles modulo coboundaries and \(k\)-forms can be summarised as:

\[ \{k\text{-forms of } X\} \;\stackrel{\sim}{\longleftrightarrow}\; H^1(k, \mathrm{Aut}(X)), \]

with the trivial cocycle going to \(X\) itself. This is the most concrete face of [[node:forms.galois_cohomology|Galois cohomology]].

Role in Langlands

Inner forms of reductive groups (Aut = adjoint): twisting by cocycles in \(G_{\mathrm{ad}}\).
Outer forms (e.g.\ \({}^2 E_6\)): twisting by cocycles in \(\mathrm{Out}(G)\), the diagram automorphism component.
Rigid inner twists (Kaletha): cocycles with values in \(z\)-extensions or in the "Galois gerbe."
Pure inner forms: cocycles in \(G\) itself rather than \(G_{\mathrm{ad}}\); coincide with inner forms when \(G\) is adjoint but differ otherwise (Vogan's distinction).

References

Serre, Cohomologie Galoisienne (1964), §I.5, II.
Borel–Serre, Sur les sous-groupes d'algèbres de Lie (Toulouse 1958).
Kaletha, Rigid inner forms of real and p-adic groups (Annals 2016).