Stalks of the graph of a morphism

Question

I am interested in the graph $\Gamma_f$ of a morphism

$f: X\rightarrow Y$ between two sufficiently nice schemes $X,Y$.

One knows that it is a closed subset of $X\times Y$ (when the schemes are nice, say varieties over a field).

I would like to know the following: if you endow it with the reduced structure, what are the stalks of it's structure sheaf in a point $(x,f(x))$ ?

Thanks you very much!

Answer 1

Let $f:X\to Y$ be a morphism of $S$-schemes. The graph morphism $\gamma_f:X\to X\times_S Y$ is the pull-back of the diagonal morphism $\delta: Y\to Y\times_S Y$ along $f\times id_Y: X\times_S Y \to Y\times_S Y$. This implies that if $\delta$ is a closed embedding (i.e. $Y$ is separated over $S$) so is $\gamma_f$. So $\gamma_f$ induces an isomorphism between $X$ and its image $\Gamma_f \subset X\times Y$.