# Schemes, Spaces, and Points

Suppose you have two morphisms a, b : X —> Y and you want to know whether a = b. If

1. X, Y are schemes,
2. X is reduced,
3. a(x) = b(x) for all x in X, and
4. the induced maps on residue fields are the same too,

then a = b. If

1. X, Y are algebraic spaces,
2. Y is locally separated,
3. X is reduced, and
4. a(x) = b(x) in Y(K) for every x in X(K) and any field K,

then a = b. But the last statement does not hold if we replace condition 2 by the condition that Y is quasi-separated. Recall that quasi-separated algebraic spaces are the “usual” algebraic spaces, i.e., the ones in Knutson’s book, not some bizarre ultra general class of algebraic spaces.

This comes up when you consider quotient maps for groupoids in algebraic spaces, and it is just the first small sign that things get a little more confusing when dealing with algebraic spaces. Namely, the above means that if we have a groupoid in algebraic spaces (U, R, s, t, c) and a morphism U —> X then even if all of U,R,X are reduced to see whether U —> X is R-invariant (i.e. a quotient map), it is not enough to check that this holds on field valued points.

## 1 thought on “Schemes, Spaces, and Points”

1. The following holds though for arbitrary reduced algebraic spaces:

Let p: X’ -> X be a universal submersion. Then if fp=gp, then f=g. (i.e., p is a categorical epimorphism in the category of reduced algebraic spaces).

The result you quoted above can be stated as follows:

Let p: X’ -> X be surjective. Then p is a categorical epimorphism in the category of reduced locally separated spaces.

Finally, if we also require p to be qcqs and schematically dominant, we can remove the reduceness hypothesis. This can be found in my submersion paper (Prop. 7.2).