Let $H$ be a hyperplane in $\mathbb P^n$ where $f:H\rightarrow \mathbb P^n$ is the closed immersion. Let $\mathcal F$ be a coherent subsheaf of $\mathcal E = \oplus\mathcal O_{\mathbb P^n}(l)$.
There is a short exact sequence: $$ 0\rightarrow \mathcal F(d-1)\rightarrow\mathcal F(d)\rightarrow\mathcal F_H(d)\rightarrow 0 $$ The author also claims that if we select $H$ which does not meet any associate point of $\mathcal E/\mathcal F$, then $i_H:\mathcal F_H\rightarrow \mathcal E_H$ is injective.
My questions are:
How to define the short exact sequence? Is this sequence always exact for any $H$?
Why $i_H: \mathcal F_H\rightarrow \mathcal E_H$ is injective if $H$ contains no associated point of $\mathcal E/\mathcal F$?
If we regard $H = \mathbb P^{n-1}$, why $\mathcal O_{\mathbb P^{n}}(l)|_H = \mathcal O_{H}(l)$? (i.e.Why the pullback is the twist sheaf of $H=\mathbb P^{n-1}$)
This question is from the proof of Mumford's theorem. Following are the statement of the theorem and related part of the proof:
Another parts:
