Forms of Algebraic Objects

A form of an object \(X\) over a field \(k\) (with respect to an extension \(K/k\)) is a \(k\)-object that becomes isomorphic to \(X\) after base change to \(K\). Forms are classified by Galois cohomology and underlie the entire theory of twisted objects in algebraic geometry.

General definition

Let \(K/k\) be a field extension and \(X_K\) an algebraic object over \(K\) (e.g.\ a scheme, group scheme, vector bundle, algebra, etc.). A \(K/k\)-form of \(X_K\) is a \(k\)-object \(X\) together with an isomorphism \(X \otimes_k K \cong X_K\) (over \(K\)).

Two forms are equivalent if there is a \(k\)-isomorphism between them inducing the trivial isomorphism on \(X_K\).

Galois forms

For \(K/k\) a Galois extension with group \(\Gamma = \mathrm{Gal}(K/k)\): \(K/k\)-forms of \(X_K\) are classified by \[ H^1(\Gamma, \mathrm{Aut}_K(X_K)) \;=\; \{\text{cocycles}\} / \{\text{coboundaries}\}. \]

Taking the limit over all finite Galois \(K/k\) gives the full absolute Galois form classification: \[ \{\text{forms of } X_{\bar k} \text{ over } k\} / \cong \;=\; H^1(k, \mathrm{Aut}(X_{\bar k})). \]

Examples

Vector spaces

For \(V_{\bar k} = \bar k^n\): forms over \(k\) are exactly \(n\)-dimensional \(k\)-vector spaces (by Hilbert-Speiser). All are isomorphic to \(k^n\), so there is just one form.

Algebras

For \(A_{\bar k} = M_n(\bar k)\) (matrix algebras): \(\mathrm{Aut}(M_n(\bar k)) = \mathrm{PGL}_n(\bar k)\), and \(H^1(k, \mathrm{PGL}_n) = \mathrm{Br}(k)[n]\). Forms of \(M_n(\bar k)\) are exactly central simple \(k\)-algebras of degree \(n\).

Quadratic spaces

\((V, q)\) with \(q\) a quadratic form: \(\mathrm{Aut}(V_{\bar k}, q) = \mathrm{O}_n(\bar k)\), and \(H^1(k, \mathrm{O}_n)\) classifies non-degenerate quadratic \(k\)-forms of dimension \(n\) up to isometry.

Algebraic groups

For \(G\) a split connected reductive group: \(\mathrm{Aut}(G)\) fits into \(1 \to G_{\mathrm{ad}} \to \mathrm{Aut}(G) \to \mathrm{Out}(G) \to 1\), and \(H^1(k, \mathrm{Aut}(G))\) classifies all \(k\)-forms of \(G\).

Galois algebras

For \(\bar k^n\) (with permutation action): forms are étale \(k\)-algebras of degree \(n\) (finite products of finite separable field extensions, total dimension \(n\)).

Twisting construction

Given a 1-cocycle \(c : \Gamma \to \mathrm{Aut}(X_{\bar k})\) and an object \(X_{\bar k}\), the twist \(X_{\bar k}^c\) has the same underlying set but with the Galois action \(\sigma \cdot_c x = c_\sigma \cdot \sigma(x)\).

If the original action admitted a \(k\)-form \(X\), the twisted action admits a different \(k\)-form \(X^c\). Two cocycles give equivalent twists iff cohomologous.

See cocycles and twisting.

Comparison of forms

Given two forms \(X, X'\) of \(X_{\bar k}\), the comparison cocycle is \(c : \Gamma \to \mathrm{Aut}(X_{\bar k})\), \(c_\sigma = i_\sigma \cdot \sigma_{X'}^{-1} \sigma_X\), where \(i_\sigma\) is the natural identification. \(c\) is a cocycle, and the cohomology class of \(c\) detects the equivalence class of forms.

Triviality criteria

\(X\) and \(X'\) are equivalent as forms iff \(c \in \mathrm{Aut}(X_{\bar k})\) is a coboundary, iff \(X \cong X'\) over \(k\).

Forms over an algebraically closed field: all forms are trivial.

Forms preserve invariants up to base change

For any "intrinsic" invariant of \(X_{\bar k}\) (e.g.\ dimension, genus, discrete cohomological invariants): all forms share it. Forms differ in "rational invariants" that depend on the field \(k\).

Role in Langlands

Forms of reductive groups: classified by Galois cohomology of the automorphism group; quasi-split + inner-form refinement organises the Langlands programme.
Forms of Shimura varieties: the rational structure of Shimura varieties is encoded by forms over various subfields.
Forms of automorphic representations: not quite a "form" question, but Langlands functoriality navigates between forms of reductive groups.