Torsors
A torsor under a group \(G\) is a "principal homogeneous space" — a set with a free transitive \(G\)-action, but typically without a distinguished basepoint. Torsors are the geometric objects classified by \(H^1\), and the technical vehicle for descent and twisting.
Definition
Set-theoretic
For a group \(G\) acting on a non-empty set \(P\), \(P\) is a \(G\)-torsor if the action is free (\(g \cdot x = x \implies g = 1\)) and transitive (any two elements differ by a unique group element).
Equivalently, the map \(G \times P \to P \times P\), \((g, x) \mapsto (g \cdot x, x)\), is a bijection.
A torsor is "the same as" the group \(G\) but with the identity "forgotten."
Scheme-theoretic / sheaf-theoretic
For an algebraic group \(G/k\), a \(G\)-torsor (in the étale/fppf topology) is a scheme \(P/k\) with a \(G\)-action such that:
- the action map \(G \times P \to P \times P\) is an isomorphism;
- there exists an étale (or fppf) cover \(\{U_i \to \mathrm{Spec}\, k\}\) over which \(P\) becomes trivial: \(P \times_k U_i \cong G \times_k U_i\) as \(G\)-schemes.
The trivial torsor is \(G\) itself acting on itself by left translation; non-trivial torsors are forms of \(G\) twisted by a 1-cocycle.
Classification by \(H^1\)
Theorem. For a smooth algebraic group \(G\) over a field \(k\), \(G\)-torsors over \(k\) up to isomorphism are classified by \[ H^1_{\mathrm{\acute{e}t}}(k, G) \;=\; H^1(k, G). \] The trivial class corresponds to the trivial torsor.
For \(G\) commutative: \(H^1(k, G)\) is an abelian group. For \(G\) non-commutative: pointed set.
Examples
\(\mathbb{G}_m\)-torsors
A \(\mathbb{G}_m\)-torsor on \(\mathrm{Spec}\, k\) is a line bundle without a chosen trivialisation. By Hilbert 90, \(H^1(k, \mathbb{G}_m) = 0\), so all such torsors are trivial. (Over a non-trivial base scheme \(X\), \(\mathbb{G}_m\)-torsors are line bundles up to isomorphism, the Picard group.)
\(\mu_n\)-torsors
By Kummer theory, \(H^1(k, \mu_n) = k^\times / (k^\times)^n\). A \(\mu_n\)-torsor over \(k\) is a Galois algebra of exponent \(n\), i.e.\ \(k(\sqrt[n]{a})\) for \(a \in k^\times\) modulo \(n\)-th powers.
\(\mathrm{PGL}_n\)-torsors
\(H^1(k, \mathrm{PGL}_n) = \mathrm{Br}(k)[n]\). A torsor under \(\mathrm{PGL}_n\) is the Brauer-Severi variety of a central simple \(k\)-algebra \(D\) of degree \(n^2\); the variety becomes \(\mathbb{P}^{n-1}\) over an étale extension splitting \(D\).
\(\mathrm{O}(q)\)-torsors
Classify quadratic forms with the same dimension and discriminant (in characteristic \(\ne 2\)); concretely, the orthonormal frames of \(q\).
Twisting
Given a \(G\)-object \(X\) (any object on which \(G\) acts), the twist \(X^P := P \times^G X\) is the contraction \((P \times X) / G\) where \(G\) acts diagonally. \(X^P\) is naturally a \(k\)-form of \(X\). This gives a functor \[ P \;\mapsto\; X^P, \qquad \{G\text{-objects}\} \to \{k\text{-forms of } X\}. \]
In particular: for \(X = G\) itself, \(X^P = P\). For \(X = V\) a representation, \(X^P\) is the twisted bundle.
Geometric properties
For \(G\) smooth and a \(G\)-torsor \(P\):
- \(P\) is smooth (locally on étale cover, \(P \cong G\));
- \(\dim P = \dim G\);
- \(P\) has trivial fundamental group iff \(G\) does (and \(P\) is connected iff \(G\) is connected over \(\bar k\)).
If \(G\) is finite étale, \(P\) is also finite étale of the same order; the corresponding extension \(k(P)/k\) is a Galois algebra.
Bitorsors and biextensions
A bitorsor carries commuting left and right actions of (possibly different) groups; they classify bimodule structures and appear in extension theory of group schemes.
Biextensions generalise extensions of one group by another to "two-variable" extensions; they classify Weil pairings, dualities of abelian varieties, and the Heisenberg group in arithmetic geometry.
Role in Langlands
- Bun_G: the moduli stack of \(G\)-torsors on a curve, the central geometric object of geometric Langlands.
- L-packets and inner forms: the "rigid inner twists" of Kaletha are torsors under \(z\)-extensions.
- Endoscopy: endoscopic data include torsors under endoscopic groups (Kottwitz–Shelstad).
- Eisenstein series: Eisenstein cohomology is computed from parabolic torsor stacks on Bun_G.
References
- Milne, Étale Cohomology (1980), §III.
- Skorobogatov, Torsors and Rational Points (CUP 2001).
- Olsson, Algebraic Spaces and Stacks (2016), §12.