Let $k$ be a field and $\mathcal{F}$ coherent over $\mathbb{P}^{n}_{k}$, $H\subset\mathbb{P}^{n}_{k}$ a hyperplane.
In Nitsure's paper Construction of Hilbert and Quot schemes, page 9 right at the bottom, the author talks about a short exact sequence \begin{equation*} 0\longrightarrow\mathcal{F}(m-i-1)\longrightarrow^{\alpha}\mathcal{F}(m-i)\longrightarrow\mathcal{F}_{H}(m-i)\longrightarrow 0\text{,} \end{equation*} where
(...) the map $\alpha$ is locally given by multiplication with a defining equation of $H$, (...)
What exactly is meant by that: multiplication by a defining equation. I mean $H$ is locally given by some equation, sure, but what exactly do I have to multiply by that equation in order to get the map $\alpha$? I am confused.
Thank you in advance.
If you think of the sheaf $\mathcal{F}$ as a graded module $M$ then you can think of the above short exact sequence as a short exact sequence of graded components of modules
\begin{equation*} 0\longrightarrow M(m-i-1)\longrightarrow^{\alpha} M(m-i)\longrightarrow M(m-i)/\alpha M(m-i-1)\longrightarrow 0\text{,} \end{equation*}
where the map $\alpha$ is just the multiplication map by $\alpha$.