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Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Denote by $U_i$ the fundamental affine schemes defined by the non-vanishing of the coordinates $X_i$ of $\mathbb{P}^n$ for $i=0,...,n$. The question is given a morphism $\phi \in \mathcal{H}om_{\mathbb{P}^3}(\mathcal{F}_1(U_i),\mathcal{F}_2(U_i))$ for some $i$ can we extend it to a morphism in $\mathcal{H}om_{\mathbb{P}^3}(\mathcal{F}_1,\mathcal{F}_2)$?

I was thinking in the following way: $\mathcal{F}_k(U_j)_{(X_i)}=\mathcal{F}_k(U_i \cap U_j)$ for any $j \not= i$ and $k=1, 2$ as $S$-modules where $S=\mathbb{C}[X_0,...,X_n]$. Denote by $T_j$ the multiplicative system generated by $X_j$. Then, $$\mathcal{F}_k(U_j)=\frac{T_i}{T_j}\mathcal{F}_k(U_i)$$ for $k=1, 2$. Then using $\phi$ and extending linearly, we can define morphisms $\phi_j$ from $\mathcal{F}_1(U_j)$ to $\mathcal{F}_2(U_j)$ for all $j$. As far as I understand this morphism glues over the open sets.

Is there some problem with these arguments? If so, is there any assumption which I am forgetting under which it would work?

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