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This may be pretty trivial, but I can't figure it out. Suppose that $S$ is any scheme, and $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ are two morphisms of $S$-schemes, such that the closed image of each one exists. That is, there is a smallest closed subscheme $Z_i$ of $Y_i$ over which $f_i$ factorizes (for $i=1,2$).

Is it true then that the closed image of $f_1\times_S f_2:X_1\times_S X_2\to Y_1\times_S Y_2$ exists? In this case, is it equal to $Z_1\times_S Z_2$ (with its natural closed immersion into $Y_1\times_S Y_2$)?

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    $\begingroup$ the closed image always exists. $\endgroup$ – Martin Brandenburg Apr 22 '10 at 23:24
  • $\begingroup$ To quote chapter and verse, this is Hartshorne II, Ex. 3.11(d) $\endgroup$ – Mariano Suárez-Álvarez Apr 23 '10 at 1:11
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    $\begingroup$ Ok, this is true, sorry. In EGA I (9.5) everything is stated as if the closed image does not always exist. One sufficient condition for this is that the direct image of the structure sheaf is quasi-coherent, which is not always true. But in any case, one can take the smallest ideal containing it which is quasi-coherent and that works. $\endgroup$ – unknown Apr 23 '10 at 8:04
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This is already false in the affine case.

The closed image of $Spec(B) \to Spec(A)$ is the spectrum of $A/K$, where $K$ is the kernel of $A \to B$. Let $A',B',K$ be analogously defined. The closed image of $Spec(B \otimes_R B') \to Spec(A \otimes_R A')$ is the spectrum of $(A \otimes_R A')/L$, where $L$ is the kernel of $A \otimes_R A' \to B \otimes_R B'$. We have to compare this ring with $A/K \otimes_R A'/K = (A \otimes_R A')/\langle K,K' \rangle$. If $K=K'=0$, this asks if the tensor product of two injective rings maps is injective, which is false in general.

This discussion shows that everything is fine if, in your notation, $X_1,Y_2$ or $X_2,Y_1$ are flat over $S$.

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  • $\begingroup$ The reason for the q-c hypotheses in EGA is that the "closed image" as a purely categorical thing without a link to pushforward sheaf has no reason to have good behavior under "localization" operations or good relation with geometry of the map (and so is of limited interest). For example, it can fail to commute with Zariski localization on the target. Why should it behave well with respect to products under these indicated flatness hypotheses? (How to extrapolate from affine case in absence of q-c hypotheses?) $\endgroup$ – BCnrd Apr 25 '10 at 15:54

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