Vector-bundle maps on a regular punctured spectrum are recovered from parameter thickenings

Extend $f$ to a regular system of parameters $f,x_2,\dots ,x_d$. By [F1], $A$ has depth at least three, while $A\mathrm{/}{f}^{n}A$ has the regular sequence $x_2,x_3$: this follows by filtering it with the powers of $f$ and using regularity modulo $f$. Thus [F2] gives $\mathrm{\Gamma }(U,\mathcal{O})=A$ and $\mathrm{\Gamma }(U_n,\mathcal{O})=A\mathrm{/}{f}^{n}A$. Also $A$ is complete for the $f$-adic topology. An $f$-adic Cauchy sequence is $\mathfrak{m}$-adically Cauchy; its limit remains in every prescribed residue class modulo $f^n$, since $A\mathrm{/}{f}^{n}A$ is a separated complete finite module by [F2]. Injectivity follows from $\bigcap f^{n}A\subseteq \bigcap {\mathfrak{m}}^n=0$. Therefore .

Every finite locally free sheaf $H$ on the quasi-compact open $U$ admits an exact sequence $0\to H\to {\mathcal{O}}_U^r\to {\mathcal{O}}_U^s$ whose first cokernel is finite locally free. Here are the globalization details. For any quasi-coherent sheaf on $U$, its sections are a finite equalizer over principal affine opens $D(a_i)$ covering $U$ and their affine intersections. Localizing that equalizer shows $\mathrm{\Gamma }(U,H{)}_a=\mathrm{\Gamma }(D(a),H)$ for $D(a)\subseteq U$, since localization is flat. Take local bases of $H^{\ast }$ on a finite principal cover. Multiply each basis section by a sufficient power of its defining element to make it a global section; the resulting finitely many global sections still span $H^{\ast }$ on the corresponding opens, hence give a surjection ${\mathcal{O}}_U^r\to H^{\ast }$. Its kernel is locally free, since the quotient is locally free and the sequence splits locally. Dualizing gives an injection $H\to {\mathcal{O}}_U^r$ with locally free cokernel $K$. Apply the same construction to $K^{\ast }$ and dualize to embed $K\to {\mathcal{O}}_U^s$. This is the desired sequence, and its restriction to every $U_n$ remains exact because the local splitting makes it universally exact.

Taking sections expresses $\mathrm{\Gamma }(U,H)$ as the kernel of $A^r\to A^s$ using step 1.1. Similarly $\mathrm{\Gamma }(U_n,H{\mathrm{\mid }}_{U_n})$ is the kernel of $(A\mathrm{/}{f}^{n}A{)}^r\to (A\mathrm{/}{f}^{n}A{)}^s$. Inverse limits preserve kernels, and the compatible matrices on the right are reductions of the original matrix. Step 1.1 therefore gives . Apply this to the locally free sheaf $\mathcal{H}om(E,G)$ to obtain the claimed bijection. Apply it also to the Hom sheaf in the opposite direction: compatible inverse maps extend, and their compositions are identities because this bijection is injective. Multiplication and unit identities are maps between tensor products of locally free sheaves, so their equality is likewise detected on all $U_n$.