The Stacks project

Proof. It suffices to show that the functors of Lemma 93.9.6 define an equivalence

because then we can apply Lemma 93.9.7 to translate into rings. To do this we construct a quasi-inverse. Denote $F_ i : \mathcal{D}\! \mathit{ef}_{U_ i} \to \mathcal{D}\! \mathit{ef}_{U_{12}}$ the functor of Lemma 93.9.6. An object of the RHS is given by an $A$ in $\mathcal{C}_\Lambda$, objects $(A, V_1) \to (k, U_1)$ and $(A, V_2) \to (k, U_2)$, and a morphism

Now $F_ i(A, V_ i) = (A, V_{i, 3 - i})$ where $V_{i, 3 - i} \subset V_ i$ is the open subscheme whose base change to $k$ is $U_{12} \subset U_ i$. The morphism $g$ defines an isomorphism $V_{1, 2} \to V_{2, 1}$ of schemes over $A$ compatible with $\text{id} : U_{12} \to U_{12}$ over $k$. Thus $(\{ 1, 2\} , V_ i, V_{i, 3 - i}, g, g^{-1})$ is a glueing data as in Schemes, Section 26.14. Let $Y$ be the glueing, see Schemes, Lemma 26.14.1. Then $Y$ is a scheme over $A$ and the compatibilities mentioned above show that there is a canonical isomorphism $Y \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) = X$. Thus $(A, Y) \to (k, X)$ is an object of $\mathcal{D}\! \mathit{ef}_ X$. We omit the verification that this construction is a functor and is quasi-inverse to the given one. $\square$