On the definition of support of a scheme

Question

I can't understand the definition of support of a scheme precisely (in Zariski site). For example, let $X=Spec(k[x,y])$ and $x\in O_X$ be a global section. Then what is the support of $x$?

Answer 1

If $A$ is a ring and $M$ an $A$-module, then the support of $m\in M$ (thought of as a section of the associated sheaf on $Spec(A)$ consists of all prime ideals $p$ such that there is no $u\in A-p$ with $um=0$. So for example if $A$ is an integral domain and $M$ torsion-free, then any nonzero $m\in M$ has support the whole of $Spec(A)$. (Your case is $A=k[x,y]$.)