Morphisms from spectra to schemes

Question

Let $X$ be a scheme. Show that for any $x \in X$ there exists a canonical morphism $\textrm{Spec}\, \mathcal{O}_{X,x} \rightarrow X$. If $k(x)=\mathcal{O}_{X,x}/\frak{m}_{x}$ is the residue field at $x$, conclude that there is a canonical morphism $\textrm{Spec}\, k(x) \rightarrow X$.

I came across this exercise while reading through one of my AG texts. I'm a little stuck on how to begin here. Any help would be appreciated!

Answer 1

First off, the canonical homomorphism is almost never an isomorphism. Just see what Matt said; if $X$ is not a point, then they're not even isomorphic as topological spaces.

Hints:

• Since $x$ is a point, it is contained in some affine piece of $X$, say $Spec (A)$.
• $x$ corresponds to some prime ideal $\mathfrak p_x$ of $A$, and the local ring $\mathcal O_{X,x}$ corresponds to the localization $A_{\mathfrak{p}_x}$ (invert everything not in $\mathfrak p_x$). We have a canonical localization homomorphism $A \to A_{\mathfrak p_x}$.

Can you see the rest?