basic notion about schemes.. what is the difference between $s(x)$ and $s_x$ for $s \in \Gamma(U, O_X)$

Question

Let $X$ be a scheme and $U$ an open subset of $X$. Let $s \in \Gamma(U, O_X)$ and $x \in U$.
I am getting confused with what the difference is between $s_x$ and $s(x)$... Are they the same thing?

Answer 1

It is standard to write $s_x$ for the image of $s$ in the stalk $\mathscr{O}_{X,x}$ of $X$ at $x$, and I think it is more or less standard to write $s(x)$ for the image of $s$ in the residue field $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_x$ of $\mathscr{O}_{X,x}$.

Answer 2

$s_x$ is the image of $s$ in the stalk $\mathcal{O}_{X,x}$, so $s_x\in \mathcal{O}_{X,x}$, while $s(x)$ is the value of $s$ in the residue field at $x$. That is, the image of $x$ under the composition $\mathcal{O}_{X}\to \mathcal{O}_{X,x}\to \kappa(x)$, where $\kappa(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x$.