# When is a graph isomorphic to the source?

Let $S$ be a scheme and let $f:X\to Y$ be a morphism of separated $S$-schemes. This MSE answer shows that, when $X$ is reduced, the obvious morphism $g_f:\Gamma_f \to X\times_S Y\to X$ is an isomorphism, where $\Gamma_f$ is the scheme-theoretic image of $\Delta_f:X\to X\times_s Y$.

Can we show that $g_f$ without assuming that $X$ is reduced?

I see how the reduced induced structure is being used there, but I do not see if $X$ being reduced affects the statament, $X$ cannot have double/triple/... points if $\Gamma_f$ does not! So, I wonder if there is a probably more elaborate proof that shows $g_f$ to be an isomorphism.

The only assumption I have is that $S$ is Noetherian.

A hint or a reference would be appreciated.

• I think one problem is that typically you give the graph the image scheme structure, which is the reduced induced structure... and this is an obstruction to the projection being an isomorphism. Note that if $Y$ is reduced, then $f$ induces a map from the reduction of $X$ to $Y$, and for this you can study the graph and get an isomorphism as before. How are you defining the graph? Feb 19, 2017 at 23:03
• @AreaMan: $\Gamma_f$ is not necessarily reduced, I take it to be the scheme-theoretic image of $\Delta_f$, that is the smallest subscheme in $X\times_S Y$ that factorises $\Delta_f$, as in Hartshorne's II.Ex.3.11(d). I think that $X\to \Gamma_f$ is a closed immersion, using a similar argument to stacks-project Tag 024T, and their underlying topological spaces are isomorphic as in the other MSE answer, referred to in the question. I just do not see how $\Gamma_f$ can be bigger. A counterexample would be appreciated! Feb 19, 2017 at 23:25
• Oh, you're right, I got my arrows backwards -- the reduction is a closed subscheme, which makes sense because it is smaller... I remembering thinking about this example: $Spec k[[x]] \to Spec k[x]$, or $\bigcup_{n \geq 0} k[x]/x^n \to Spec k[x]$. The scheme theoretic image in both cases is the entire line, which is certainly initially counter intuitive (though makes sense from the Taylor seris pov). Maybe you can use this example somehow? Feb 19, 2017 at 23:32

For any morphism of $S$-schemes $f:X\to Y$ the graph morphism $\Gamma_f:X\to X\times _SY$ is an immersion (but not necessarily a closed immersion).
Its associated image subscheme $G_f\subset X\times _SY$ is indeed isomorphic to $X$ under the restriction $g:G_f\to X$ of the first projection $p_X:X\times _SY \to X$.
This is explained in Corollaire (5.3.11) page 133 of $EGA_I$ and the seven following lines .