mcolsson
commented
9 years ago Yes this line needs some expansion. The point is that the proof of 6.2.2 in fact shows the following (this is embedded in the discussion of 6.2.13 and not brought out enough): If in 6.2.2 the morphisms s and t are \'etale and X_1\rightarrow X_0\times X_0 is an imbedding then the projection X_0\rightarrow Y is \'etale surjective and X_0\times _YX_0=X_1. This implies that Y is the \'etale sheaf quotient of X_0 by X_1.
theorem 6.2.2 gives a sufficient condition for the existence of a universal (categorical) scheme quotient by an equivalence relation. Remark 6.2.6 states that this quotient is also universal in the category of locally ringed spaces. On the other hand, at the end of the proof of 6.4.1, this is used to state that the universal scheme quotient is also universal in the category of algebraic spaces (if I am reading this correctly), and I don't understand how this follows. Of course an algebraic space can be regarded as a locally ringed topos (for example, over big \'etale topos), but I don't see how it is a locally ringed topological space. Probably I am missing something here?