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Let $f : X \to Y$ be a morphism of schemes. Is it possible to associate to every closed immersion $i : F \to X$ a closed immersion $f^* i : G \to Y$, such that in the affine case, $(Spec(A) \to Spec(B))^*$ is given taking ideals of $A$ to ideals of $B$ via preimages? Probably this is well-known to every algebraic geometer :-)

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It doesn't seem like the arrows line up to give you a diagram where you can push out. – Harry Gindi Apr 9 '10 at 17:09
up vote 6 down vote accepted

It sounds as though what you want is the closure of the image of $F$ under $f$. (That is, the minimal closed subscheme of $Y$ factoring $f$.)

If $X =$ Spec $A$, and $Y =$ Spec $B$, and $F =$ Spec $A/I$, and $f$ corresponds to the ring map $f':B\to A$, then we can consider the preimage $J$ of $I$ under $i'$. Consider the set of primes in $B$ containing $J$. Of course, any prime in the image of $F$ under $f$ must contain $J$, since its preimage under the ring map has to contain $I$. Thus, Spec $B/J$ contains the preimage. You can check that it's the biggest such ideal, noting that in order for us to have a map $B/J \to A/I$, $J$ should be contained in the preimage of $I$.

Being the closure of the image of $F$ under $f$ is a universal property of sorts (in particular, it's unique), which more or less allows us to argue that this construction generalizes to non-affine schemes. (Just apply this local construction, and uniqueness tells us that the local constructions glue together.)

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The name for this construction (the closure of the image) is the scheme-theoretic image. There is an extensive discussion in the notes for the stacks project, among other places. (Google "stacks project", and you should find them.) – Emerton Apr 9 '10 at 19:50
I didn't know the stacks project so far. Awesome! – Martin Brandenburg Apr 9 '10 at 20:11
Be careful: for things to localize well on the base you want that the ideal is the kernel of $O_Y \rightarrow f_{\ast} O_X$, so for this to be qcoh you want $f_{\ast} O_X$ to be qcoh; e.g., $f$ should be quasi-compact and quasi-separated. More generally it isn't clear that one gets a notion which localizes well. (Can make a construction globally with a minimality condition, but generally not local on the base, hence sort of useless.) – BCnrd Apr 10 '10 at 2:57
Brian-- thanks! I'd only ever thought through this in the case of varieties, and I hadn't thought to worry that things could go wrong more generally. – mvk Apr 10 '10 at 4:58

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