Let $X$ be a quasiprojective scheme over a field $k$. We define that a $0$-dimensional subscheme $Z$ is of length $n$ if:
$\operatorname{dim}_k \operatorname{H}^0(Z,\mathcal{O}_Z)=\sum_{p\in Supp(Z)}\operatorname{dim}_k(\mathcal{O}_{Z,p})=n$
I don’t understand how can we get the first equality? I know there is an isomorphism between sections $\Gamma(Z,\mathcal{O}_Z)$ and $\operatorname{H}^0$, but I can’t go further. Hope someone could help. Thanks!
Presumably you've left off the condition that $Z$ should be a closed subscheme. As closed subschemes of noetherian schemes are noetherian, $Z$ is a zero-dimensional noetherian scheme which must be finite discrete (see here, for example - or it is not so hard to prove yourself). This means it's actually affine, as it's a finite coproduct of affine schemes. In particular, each point is open, so the stalk at a point is just the value of the sheaf on the open which consists of just that point, and picking a global section is equivalent to specifying the value in the stalk at each point.
