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This question is somewhat related to a previous question of mine, but appeared in a different context.

Suppose given two morphisms of $S$-schemes $f,g:X \to Y$. Intuitively, I am interested on the locus of points of $X$ in which these two morphisms coincide. A schematic approach to that, would be to consider $Z$ as the pullback of the diagonal $\Delta: Y \to Y\times_S Y$ by the morphism $(f,g): X \to Y\times_S Y$.

Then, for example, if $Y\to S$ is separated ("Hausdorff"), then $Z \to X$ is a closed immersion, which makes sense.

Also, the composition of $Z \to X$ with $f$ and $g$ are the same (by the commutativity of the cartesian diagram defining $Z$). Hence, I would expect that the scheme $Z$ is what I'm searching.

But one could also take a naive, set-theoretic, approach (which we know is usually inappropriate when dealing with schemes) and define the set $A=\{x\in X \mid f(x)=g(x)\}$.

My question is: what is the relation between $Z$ and $A$?

The most strange thing is that $A$ can be empty, but $Z$ seems to be always well defined. In this case (when $A$ is empty), how this information is "captured" in the scheme $Z$?

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    What scheme structure do you propose to put on the _set_ $A$?2012-09-13
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    The limit you're computing is called an _equalizer._ You are trying to compare the equalizer in schemes over $S$ with the set-theoretic equalizer on points, and the problem with doing that is that the functor associating to a scheme its set of points is badly behaved (in particular it does not preserve limits). This is one reason not to use this functor at all...2012-09-13

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