What is $\mathcal{O}_{X,x}$?

Question

I read from Liu's "Algebraic Geometry and Arithmetic Curves the following definition:

A ringed topological space consists of a topological space $X$ endowed with a sheaf of rings $\mathcal{O}_X$ on $X$ such that $\mathcal{O}_{X,x}$ is a local ring for every $x\in X$. How do we define $\mathcal{O}_{X,x}$ as I was unable to find it from the book?

Answer 1

There are two ways to define the stalk ${\cal O}_{X,x}$:

• The space of germs lying over $x$. Such germs are equivalence classes of elements of $F(U)$, where $U$ ranges over open sets containing $x$, subject to the relation $a\sim b$ for $a\in F(U)$, $b\in F(V)$ and $a|_W=b|_W$ for some $W\subseteq U,V$ containing $x$.
• The direct limit $\varinjlim F(U)$ taken over open sets $U$ containing $x$.