I'm trying to understand the Lemma 4.6 of Bhatt-Morrow-Scholze's paper **Integral $p$-adic Hodge Theory**.
In the proof, for proving the restriction functor is fully faithful, it used a affine open cover ${U\_1,U\_2}$ of the punctured spectrum $U$. But why it is enough to prove $A\_{\inf} \cong \mathcal{O}(U)$ and $R=R\_1\cap R\_2\subset R\_{12}$? 


I have a sense that it is proving what we want on the affine subset, and then their intersection, but the detail is still a mess for me.


Also, why all vector bundle over $A\_{\inf}$ is free?



Equivalence of vector bundles over $Spec(A_{\inf})$ and the punctured spectrum
The ring $A\_{\text{inf}}$ is local, so every vector bundle (i.e. finite projective module, or finitely presented flat module) is free [Stacks, Tag [00NX](https://stacks.math.columbia.edu/tag/00NX)(4)].


Now let $(R,\mathfrak m)$ be any local ring, and let $X = \operatorname{Spec} R$, and $U = X \setminus\{\mathfrak m\}$. Assume that the natural map $R \to \Gamma(U,\mathcal O\_U)$ is an isomorphism.


To show that the restriction is fully faithful, let $\mathscr E$ and $\mathscr F$ be vector bundles on $X$. By the above, we have $\mathscr E \cong \mathcal O\_X^e$ and $\mathscr F \cong \mathcal O\_X^f$ for some $e, f$. Then $\operatorname{Hom}\_X(\mathscr E,\mathscr F) \cong \mathcal O\_X^{ef}$, so the natural map
$$\begin{array}{ccc}\operatorname{Hom}\_X(\mathscr E,\mathscr F) & \to & \operatorname{Hom}\_U(\mathscr E|\_U,\mathscr F|\_U) \\
|| & & ||\\
\Gamma(X,\mathcal O\_X^{ef}) & \to & \Gamma(U,\mathcal O\_U^{ef})\end{array}$$
is an isomorphism since $R \to \Gamma(U,\mathcal O\_U)$ is. $\square$




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In turn, assume $\mathfrak m = (x,y)$ for some elements $x, y \in R$. Set $R\_1 = R[1/x]$, $R\_2 = R[1/y]$, and $R\_{12} = R[1/xy]$, and let $U\_i$ (resp. $U\_{ij}$) be $\operatorname{Spec} R\_i$ (resp. $\operatorname{Spec} R\_{ij}$). Assume the maps $R \to R\_i$ and $R\_i \to R\_{12}$ are injective (e.g. $R$ is a domain).


Now if $R = R\_1 \cap R\_2 \subseteq R\_{12}$, then $R = \Gamma(U,\mathcal O\_U)$. Indeed, the sheaf condition on $\mathcal O\_X$ gives a short exact sequence
$$\begin{array}{ccccccc}0 & \to & \Gamma(U,\mathcal O\_U) & \to & \Gamma(U\_1,\mathcal O\_{U\_1}) \oplus \Gamma(U\_2,\mathcal O\_{U\_2}) & \to & \Gamma(U\_{12},\mathcal O\_{U\_{12}})\\
 & & || & & || & & ||\\
0 & \to & \Gamma(U,\mathcal O\_U) & \to & R\_1 \oplus R\_2 & \to & R\_{12}, \end{array}$$
realising $\Gamma(U,\mathcal O\_U)$ as the intersection $R\_1 \cap R\_2 \subseteq R\_{12}$ (as the second map is $(f,g) \mapsto f-g$). If this equals $R$, then the natural map $R \to \Gamma(U,\mathcal O\_U)$ is an isomorphism. $\square$




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**References.**


[Stacks] A.J. de Jong et al, [*The stacks project*](http://stacks.math.columbia.edu).


