Algebraic vector bundles on the punctured spectrum: an exact reference for a result

Question

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=Spec(R)$ denote the affine- scheme with structure sheaf $\mathcal O_X$ and $U=Spec(R)\setminus \{\mathfrak m\}$ be the punctured spectrum and write $\mathcal O_U=\mathcal O_X|_U$. Then it is known that $\Gamma_U(\mathcal O_U)\cong R$. Let $\mathfrak Vect(U)$ be the category of Algebraic vector bundles on $U$ and $\mathcal C$ denote the category of finitely generated reflexive $R$-modules that are locally free on the punctured spectrum. Let $F: \mathcal C \to \mathfrak Vect(U)$ be the functor which sends a module $M$ to $\tilde M |_U$ (here $\tilde M$ is the sheaf defined by $M$ on $X$) .

From Horrock's famous paper https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-14.4.689 , I think it does follow that $\Gamma_U :\mathfrak Vect(U) \to \mathcal C$ is an equivalence of categories and in fact we have an isomorphism of functors $\Gamma_U \circ F \cong Id_{\mathcal C}$ and $F \circ \Gamma_U \cong Id_{\mathfrak Vect(U)}$. Is this indeed true ? And if it is, what is a precise explicit reference for this (because the paper of Horrocks does not explicitly state this result) ?

Answer 1

One modern account can be found in this thesis of Majidi-Zolbani, especially Chapter 1 and Appendix A. The point is that if $E$ is a vector bundle on $U$, then the global sections $\Gamma_U(E)$ is a finite $R$ module that is locally free on $U$. Sheafifying gets you back $E$. To prove that this induced a true equivalence between categories may require a bit more details, but for many practical purposes this plus some facts about local cohomology is enough. For example you can deduce Horrocks splitting criterion.