I have a quesion about the confusing using of the term "support of schemes" in Lemma 2.3.41 in Liu's "Algebraic Geometry" (page 53):
I know that the "support of a sheaf $F$ on a scheme $X$" is generally defined as the subset $S$ of $X$ so that for every $x \in S$ we have for the stalk $F_x \neq 0$. Why is this definition with the definition above the red line compatible (here for the closed subscheme of $\mathbb{P}_A ^n$ support $=$ underlying topol. space)?
Let $X$ be a scheme, and let $i:Z \rightarrow X$ be a closed subscheme. This entails several things: $Z$ is a scheme whose underlying topological space is a closed subset of $X$, and $i$ is a morphism of schemes such that $i^{\#}: \mathcal O_X \rightarrow i_{\ast} \mathcal O_Z$ is surjective as a morphism of sheaves on $X$.
In your notes, the support of this closed subscheme means the underlying space of $Z$ (I'll follow convention and identify $Z$ with its underlying space). And if $F$ is a sheaf of abelian groups on $X$, the support of $F$ is defined to be the set of $x \in X$ such that $F_x \neq 0$.
The connection between these two notions of support is this: $Z$ is exactly the support of the sheaf $i^{\ast} \mathcal O_Z$.
If $x \in Z$, then $(i_{\ast} \mathcal O_Z)_x$ is the direct limit of rings $\mathcal O_Z(V \cap Z)$, where $V$ runs through the set of open sets in $X$ containing $x$, ordered by reverse inclusion. But then $V \cap Z$ runs through the direct limit of open sets in $Z$ containing $x$, so this is just $\mathcal O_{Z,x} \neq 0$.
If $x \not\in Z$, then since $X - Z$ is open, we can shrink the direct system of open neighborhoods of $x$ to only include those contained in $X - Z$, and obtain the same direct limit. But for every such open neighborhood $V$, we have $\mathcal O_Z(V \cap Z) = \mathcal O_Z(\emptyset) = 0$.