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In the past by the term "variety" people understood a subset of projective space locally closed for the Zariski topology. Now we have a more natural notion of abstract algebraic variety, i.e. a scheme that is so and so, and we can conceive non quasi-projective varieties.

Now we have the concept of constructible subset of a variety (or of a scheme), i.e. a finite union of locally closed subsets (subschemes). We know that the image of a morphism of varieties may fail to be a subvariety of the target, nevertheless it's always a constructible subset thereof.

Is there a reasonable notion of "abstract constructible space"? And would it be of any utility in algebraic geometry?

Edit: A side question. If we have a map $f:X\to Y$ of -say- varieties, we can put a closed subscheme structure on the (Zariski) closure $Z$ of $f(X)$, as described in Hartshorne's book. On the other hand we can consider the ringed space (that will not be, in genberal, a scheme) $W$ which is the quotient of $X$ by the equivalence relation induced by $f$. Will there be any relation between $Z$ and $W$?

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en.wikipedia.org/wiki/Constructible_sheaf Presumably then a constructible geometric space is a ringed space where the structure sheaf satisfies this property. –  Harry Gindi Mar 19 '10 at 18:21
    
Taking quotients by equivalence relations is not even really the right idea for locally ringed spaces. Rather, the way to extend taking quotients is to consider schemes as sheaves on the opposite category of commutative rings with the étale topology (functors of points). When we take quotients in this category by étale equivalence relations, we end up with algebraic spaces, which don't really have an interpretation as locally ringed spaces. If we want to take more quotients but still end up outside of our category, this is when we generalize further to stacks. –  Harry Gindi Mar 20 '10 at 6:55
    
@fpqc: My guess is that underlying your suggestion for a definition is the belief that constructible subsets of varieties have the property you suggest. But given a constructible subset of a variety how do you propose to put a ringed space structure on it so that the structure sheaf becomes a constructible sheaf? –  Kevin Buzzard Mar 20 '10 at 9:23
    
My first comment was really just a guess, because I can't find his definition of a constructible scheme (yes, I looked for about 15 minutes). Every definition I found had this requirement about a twisted locally constant sheaf. My second comment is true though. –  Harry Gindi Mar 20 '10 at 15:00
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@fpqc: as far as I know there is no such thing as a constructible scheme, so it doesn't surprise me that you couldn't find a definition. –  Kevin Buzzard Mar 20 '10 at 17:25
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