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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?

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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(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. $\square$


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. $\square$


References.

[Stacks] A.J. de Jong et al, The stacks project.

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