Lemma 81.12.2. Let $(R \to R', f)$ be a glueing pair, see above. The functor (81.12.0.1) restricts to an equivalence between the category of affine $Y/X$ which are glueable for $(R \to R', f)$ and the full subcategory of objects $(V, V', Y')$ of $\textit{Spaces}(U \leftarrow U' \to X')$ with $V$, $V'$, $Y'$ affine.
Proof. Let $(V, V', Y')$ be an object of $\textit{Spaces}(U \leftarrow U' \to X')$ with $V$, $V'$, $Y'$ affine. Write $V = \mathop{\mathrm{Spec}}(A_1)$ and $Y' = \mathop{\mathrm{Spec}}(A')$. By our definition of the category $\textit{Spaces}(U \leftarrow U' \to X')$ we find that $V'$ is the spectrum of $A_1 \otimes _{R_ f} R'_ f = A_1 \otimes _ R R'$ and the spectrum of $A'_ f$. Hence we get an isomorphism $\varphi : A'_ f \to A_1 \otimes _ R R'$ of $R'_ f$-algebras. By More on Algebra, Theorem 15.90.16 there exists a unique glueable $R$-module $A$ and isomorphisms $A_ f \to A_1$ and $A \otimes _ R R' \to A'$ of modules compatible with $\varphi $. Since the sequence
\[ 0 \to A \to A_1 \oplus A' \to A'_ f \to 0 \]
is short exact, the multiplications on $A_1$ and $A'$ define a unique $R$-algebra structure on $A$ such that the maps $A \to A_1$ and $A \to A'$ are ring homomorphisms. We omit the verification that this construction defines a quasi-inverse to the functor (81.12.0.1) restricted to the subcategories mentioned in the statement of the lemma. $\square$
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