(Torsor)
Let \(\mathcal{C}\) be a site with a terminal object \(X\), and \(\mathcal{G}\) a sheaf of groups on \(\mathcal{C}\). A \(\mathcal{G}\)-torsor is a sheaf of sets \(\mathcal{P}\) on \(\mathcal{C}\) with a right action \(\mathcal{P} \times \mathcal{G} \to \mathcal{P}\) such that
there is a covering \(\{U_i \to X\}\) with \(\mathcal{P}(U_i) \neq \emptyset\) for every \(i\), and
the morphism \(\mathcal{P} \times \mathcal{G} \to \mathcal{P} \times \mathcal{P}\), \((p, g) \mapsto (p, pg)\), is an isomorphism of sheaves.
A morphism of \(\mathcal{G}\)-torsors is a \(\mathcal{G}\)-equivariant morphism of sheaves. The trivial torsor is \(\mathcal{G}\) acting on itself by right multiplication.
A \(\mathcal{G}\)-torsor \(\mathcal{P}\) is trivial exactly when \(\mathcal{P}(X) \neq \emptyset\), and every morphism of \(\mathcal{G}\)-torsors is an isomorphism. Isomorphism classes of \(\mathcal{G}\)-torsors trivialized on a covering \(\mathfrak{U} = \{U_i \to X\}\) are in bijection with the Čech cohomology set \(\check{H}^1(\mathfrak{U}, \mathcal{G})\), and all torsors with \(\check{H}^1(X, \mathcal{G}) = \varinjlim_{\mathfrak{U}} \check{H}^1(\mathfrak{U}, \mathcal{G})\); for \(\mathcal{G}\) abelian this is \(H^1(X, \mathcal{G})\).
On a scheme \(X\) with the Zariski topology, a \(\GG_m = \OO_X^\times\)-torsor is the sheaf of nowhere-vanishing local sections of an invertible sheaf \(\mathcal{L}\), so \(\Pic X \cong H^1(X, \OO_X^\times)\) (PR-DIVLB). Likewise \(\operatorname{GL}_n\)-torsors correspond to locally free sheaves of rank \(n\), through their sheaves of local frames.
Let \(k\) be a field and \(n\) invertible in \(k\). On \((\Spec k)_{\mathrm{et}}\), the Kummer sequence of D-ETFPPF and \(H^1((\Spec k)_{\mathrm{et}}, \GG_m) = 0\) (Hilbert’s Theorem 90) give \(H^1((\Spec k)_{\mathrm{et}}, \mu_n) \cong k^\times / (k^\times)^n\). The torsor corresponding to \(a \in k^\times\) is \(\Spec k[z]/(z^n - a)\) with \(\mu_n\) acting by \(z \mapsto \zeta z\).
Dependencies
D-GROTHTOPSites, sheaves on a site, and topoiuses