Vector bundles on elliptic curves

Question

Let $F$ be a stable vector bundle of degree $d$ and rank $r$, with $(r,d)$ coprime and $X$ an elliptic curve. I know that I can construct an extension $$0 \to H^0(F) \otimes O_X \to G \to F \to 0 $$ such that the boundary map of the associated long sequence in cohomology is given by $Id: H^0(F) \to H^0(F)$. Now know that G is a vector bundle of rank $r+d$ and degree $d$. What I'd like to do is proving that $G$ is stable (here I have problems), because it'll let me proceed by induction with the computation of $M(r,d)$.

Answer 1

edit: I'm guessing, since you say there is an induced LES, that you mean the sequence is exact. Note there is an isomorphism between U(r,d) and U(r+d,d) given by 4.9 which is giving you the extension (the proof there does the isomorphism starting from the other direction however). In any case, assuming this is how you get the extension, $G$ is indecomposable. Now an indecomposable vector bundle on an elliptic curve is stable iff $(r,d)=1$ by 4.19. Now since $(r,d) = 1$, then $k | (r + d) \implies (k, r) = (k, d) = 1$ so any divisors of $\ r + d$ cannot divide $\ d$ so $(r + d, d) = 1$