Galois Groups and Descent Data

Galois descent is the technique of constructing \(k\)-objects from \(\bar k\)-objects with Galois-equivariant data. It is the elementary version of the more general descent theory (faithfully flat, étale) and the foundation of the theory of forms.

Setup

Let \(k\) be a field with separable closure \(\bar k\) and absolute Galois group \(\Gamma = \mathrm{Gal}(\bar k / k)\) (a profinite group).

For a category of "objects with field of definition" (e.g.\ vector spaces, algebras, varieties, group schemes):

\(\mathcal{C}(k)\) = \(k\)-objects;
\(\mathcal{C}(\bar k)\) = \(\bar k\)-objects, with \(\Gamma\)-action.

Galois descent for vector spaces

Theorem (Hilbert-Speiser). For \(V\) a finite-dimensional \(\bar k\)-vector space with a continuous semi-linear \(\Gamma\)-action, the \(\Gamma\)-invariants \[ V^\Gamma \;\subseteq\; V \] form a \(k\)-vector space \(V^\Gamma \otimes_k \bar k \;\stackrel{\sim}{\to}\; V\) (canonical isomorphism).

Semi-linear action: \(\sigma(\lambda v) = \sigma(\lambda) \cdot \sigma(v)\) for \(\lambda \in \bar k, v \in V, \sigma \in \Gamma\).

This gives an equivalence: \[ \{k\text{-vector spaces}\} \;\stackrel{\sim}{\to}\; \{\bar k\text{-vector spaces with continuous semi-linear } \Gamma\text{-action}\}. \]

Descent data

A descent datum on a \(\bar k\)-object \(X_{\bar k}\) is a collection of isomorphisms \[ \phi_\sigma : X_{\bar k} \;\xrightarrow{\sim}\; \sigma^* X_{\bar k}, \qquad \sigma \in \Gamma, \] satisfying the cocycle condition: \[ \phi_{\sigma\tau} \;=\; \sigma^*(\phi_\tau) \circ \phi_\sigma. \]

Theorem. Descent data on \(X_{\bar k}\) up to equivalence are in bijection with \(k\)-forms of \(X_{\bar k}\). The trivial descent datum (\(\phi_\sigma = \mathrm{id}\)) corresponds to \(X_{\bar k}\) itself if it is defined over \(k\).

Examples

Vector spaces

For \(V_{\bar k}\) with \(\Gamma\)-action, the \(k\)-form is \(V_{\bar k}^\Gamma\).

Algebraic varieties

For \(X_{\bar k}\) with a descent datum (= a "form" of a \(k\)-variety): the \(k\)-form is constructed as the closed subscheme of \(X\) cut out by Galois-equivariant equations.

Equivalently: \(X = \mathrm{Spec}(\mathcal{O}(X)^\Gamma)\) where \(\Gamma\) acts on the coordinate ring via the descent datum.

Algebraic groups

A descent datum on a \(\bar k\)-group \(G_{\bar k}\) is a \(\Gamma\)-equivariant collection of group isomorphisms; the resulting \(k\)-form \(G\) has \(G(\bar k) = G_{\bar k}\) with the new \(\Gamma\)-action.

Forms of \(\mathrm{GL}_n\)

For a 1-cocycle \(c : \Gamma \to \mathrm{PGL}_n(\bar k)\) corresponding to a central simple \(k\)-algebra \(D\) of degree \(n\):

Descent datum: \(\phi_\sigma = \mathrm{conj}(c_\sigma)\).
\(k\)-form: \(D^\times\) (the algebraic group of units of \(D\)).
Original \(\bar k\)-form: \(D^\times \otimes \bar k = \mathrm{GL}_n(\bar k)\).

Non-Galois descent (sketch)

Galois descent extends to:

fpqc descent: faithfully flat quasi-compact (most general).
fppf descent: faithfully flat finite presentation.
Étale descent: cover by étale morphisms.

These are the technical foundation for working with non-reduced schemes, stacks, and characteristic-\(p\) phenomena. SGA1 (Grothendieck) gives the general formalism.

Continuous vs.\ discrete

For Galois descent: the \(\Gamma\)-action on \(X_{\bar k}\) must be continuous with respect to the natural discrete topology on \(X_{\bar k}(\bar k')\) for finite extensions \(\bar k' / k\). Equivalently, \(X_{\bar k}\) descends through a finite Galois sub-extension.

For most objects (finite type, locally of finite type): this is automatic.

Connection to Galois cohomology

A descent datum on \(X_{\bar k}\) is equivalent to a 1-cocycle \(c : \Gamma \to \mathrm{Aut}(X_{\bar k})\); equivalence of descent data corresponds to cocycle cohomology. So: \[ \{k\text{-forms of } X_{\bar k}\}/\cong \;\stackrel{\sim}{\longleftrightarrow}\; H^1(k, \mathrm{Aut}(X_{\bar k})). \]

This is the bridge to Galois cohomology.

Role in Langlands

Forms of reductive groups: classified by descent data on the Chevalley split form.
Adelic descent: global objects (Shimura varieties, automorphic representations) are constructed via local-global descent.
Galois representations: \(\ell\)-adic Galois representations are descent data on certain cohomology groups; the global Langlands programme is largely about understanding these.

Mathlib

Not yet formalised. Galois descent for vector spaces / fields is implicit in Mathlib.FieldTheory.Galois.