Background:
(1) If C and D are categories and there is a forgetful functor U:C→D, then a C-ification functor F:D→C is an adjoint to U. For example, the (left) adjoint to the forgetful functor from groups to monoids is "groupification" of a monoid, given by formally adjoining inverses. The (left) adjoint to the forgetful functor from presheaves to sheaves is the usual "sheafification" functor.
Note that whenever you have a (left adjoint) C-ification functor F (whenever you have an adjunction, for that matter), you get a universal property. For any object X∈D, there is a canonical morphism (called the unit of adjunction) εX:X→U(F(X)) with the property that any morphism f:X→U(Y) factors as f=U(g)\circ εX for a unique morphism g:F(X)→Y in C.
(2) A scheme X is separated if the diagonal morphism X→XxX is a closed immersion. It is enough to check that the image of the diagonal is closed. Being separated is the algebro-geometric analogue of being hausdorff, which nothing in algebraic geometry ever is.
My question is whether there exists a "separification" functor adjoint to the forgetful functor U from the category of separated schemes to the category of schemes. Note that the forgetful functor U does not respect colimits (you can glue together separated schemes to get a non-separated scheme), so it has no hope of having a right adjoint. But U does respect limits (it's enough to show that an arbitrary product of separated schemes is separated and that fiber products of separated schemes are separated), so it might have a left adjoint.
To put it another way, given a scheme X, is there a canonically defined separated scheme Xs and a morphism X→Xs so that any morphism from X to a separated scheme factors uniquely through X→Xs?
Related questions I'd like to know the answer to:
- Is there a "relative separification" functor. That is, does an arbitrary morphism of schemes f:X→Y admit a canonical factorization through a separated morphism fs:X'→Y. This would be analogous to Stein factorization, which I regard as "relative affinification". An arbitrary (quasi-compact and quasi-separated) morphism f:X→Y canonically factors through the affine morphism SpecY(f*OX)→Y
- Is there a separification functor for algebraic spaces? Is it possible that the separification of a scheme is naturally an algebraic space?
- Is there a separification functor for algebraic stacks? (An algebraic stack is separated if the diagonal is proper.)
Best Answer
I think that it is highly unlikely that there exists a separification functor. What does exist is the following:
Theorem (Raynaud-Gruson): Let $S$ be a base scheme and work relative to $S$. Given a non-separated scheme $X$ of finite type, there is a blow-up (a proper birational morphism) $X'\to X$ such that $X'$ admits an étale morphism to a projective scheme $Z$ (in particular a separated scheme).
Note that there are non-separated schemes which do not even admit a quasi-finite morphism onto a separated scheme (e.g. take $\mathbf A^2$ with a double origin and blow-up one of the origins).
The Theorem is false as stated for non-locally separated algebraic spaces. There are 3 different solutions to this:
A) Take an alteration instead of a modification.
B) Replace étale with quasi-finite flat.
C) Allow $Z$ to be a proper stack with finite diagonal.