Let $C$ be a quasi-projective curve, $C_i$ for $i=1,...,r$ are the irreducible components of $C$. Assume that $C_i$ is non-singular and $F_i$ locally free sheaf on $C_i$ of the same rank for all $i$. Can we glue the sheaves $F_i$ i.e., does there exists a locally free sheaf $F$ on $C$ such that $F|_{C_i} \cong F_i$ for all $i$?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
By induction one can assume that $k = 2$. Let $Z = C_1 \cap C_2$ be the scheme-theoretic intersection. Choose an isomorphism of $\varphi:F_{1|Z} \to F_{2|Z}$ (it exists since both are free of the same rank) and define $F$ from the exact sequence $$ 0 \to F \to F_1 \oplus F_2 \to F_{1|Z} \to 0, $$ where the second arrow is $(\varphi,1)$. The sheaf $F$ is locally free since the sequence is locally isomorphic to $$ 0 \to O_C \to O_{C_1} \oplus O_{C_2} \to O_Z \to 0. $$
