Theorem Tag 04I7 now has a complete proof. It is the case of schemes for the result I mentioned in this post. It says that given two schemes X, Y any morphism of locally ringed topoi

(Sh(X_{etale}), O_X) —> (Sh(Y_{etale}), O_Y)

comes from a morphism of schemes X —> Y. To prove it you use that an affine scheme V etale over Y can be embedded into A^n_Y for some n (and that it is cut out by polynomial equations in there).

Of course, it would perhaps be quicker to try and directly prove the corresponding result for algebraic spaces or Deligne-Mumford stacks (haven’t worked out the details yet), but I want mostly to stick with the philosophy that each result is proved in various levels of generality: commutative algebra, schemes, algebraic spaces, algebraic stacks, higher topos theory, etc, etc.

In a related discussion Brian Conrad pointed me to Theorem A.4.1 of the preprint by Conrad-Lieblich-Olsson entitled “Nagata compactification for algebraic spaces”. This theorem states that the category of all first order thickenings of algebraic spaces is equivalent to the category of pairs (X, A —> O_X) where X is an algebraic space and A –> O_X is a surjection of sheaves of rings on X_{etale} with quasi-coherent square zero kernel.

It seems to me that it is useful to think about the locally ringed small etale topos of an algebraic space in order to formulate and prove such results, even though it will not necessarily simplify, or shorten the proofs. Namely, in that language Theorem A.4.1 can be reformulated as follows:

- if X —> X’ is a first order thickening of algebraic spaces, then X_{etale} = X’_{etale}, i.e., the topos doesn’t change,
- define a locally ringed topos (Sh(C), A) to be an algebraic space if it is equivalent to (Sh(X_{etale}), O_X) for some algebraic space X, and
- if (Sh(C), A) is an algebraic space and A’ –> A is a surjection of rings with square zero quasi-coherent kernel then (Sh(C), A’) is an algebraic space.

The functoriality takes care of itself by the result discussed higher up.

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