Lemma 93.9.10. In Situation 93.9.9 there is an equivalence
\[ \mathcal{D}\! \mathit{ef}_ X = \mathcal{D}\! \mathit{ef}_{P_1} \times _{\mathcal{D}\! \mathit{ef}_{P_{12}}} \mathcal{D}\! \mathit{ef}_{P_2} \]
Proof. It suffices to show that the functors of Lemma 93.9.6 define an equivalence
\[ \mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{U_1} \times _{\mathcal{D}\! \mathit{ef}_{U_{12}}} \mathcal{D}\! \mathit{ef}_{U_2} \]
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
\[ g : F_1(A, V_1) \to F_2(A, V_2) \]
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. $\square$
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