Bill Lawvere has asked whether the small étale site of a scheme is always equivalent (in the sense of giving the same topos) to a site where all idempotent endomorphisms are identity morphisms. It seems to me that what follows is a fairly simple proof that, as Bill expected, the answer is yes. But the proof uses EGA IV and is not so entirely simple that I can trust it without checking with other people. Lemma: The small etale site on a scheme S is equivalent to the site whose objects are connected affine schemes Spec(A)-->S etale over S, morphisms are all S-morphisms, and covering families are all surjective families. Proof, by the comparison lemma: This is a full subcategory of the small etale site, with the same covering families, and every scheme etale over S is covered in the small etale site by a family of connected affine schemes etale over S. Theorem: The only etale idempotent f:Spec(A)-->Spec(A) of a connected affine scheme is the identity. Proof: To say f:Spec(A)-->Spec(A) is idempotent is the same as saying f factors through the equalizer i:I>-->Spec(A) of f and the identity morphism on spec(A). That is, there is a retraction s:Spec(A)-->>I with section i:I>-->Spec(A). But then s is also etale because it is just base change of the etale f along i:I>-->Spec(A). Then I is connected since Spec(A) is. And s:Spec(A)-->>I is separated since the domain is affine. So EGA IV 17.9.4 applies to say the section i must be an isomorphism of I with a connected component of Spec(A). Since Spec(A) is assumed connected, i is iso, and f is the identity. And that finishes the proof, since an idempotent of g:Spec(A)-->S as etale S-scheme would be an etale idempotent of Spec(A). EGA IV 17.9.4 says: For f:X-->Y any separated etale morphism to a connected prescheme Y, every section g:Y-->X of f is an isomorphism of Y onto an open connected component of X. There is an issue of terminology. EGA IV says "prescheme" where today we say "scheme", and it says "scheme" where we say "separated scheme". But I don't think that causes any problem here, since every scheme we use is separated. (In fact, the image I is affine as the equalizer of morphisms between affines, but I don't think the proof needs that.) I think that is all right. But am I wrong? Did I miss a Noetherian condition somewhere in EGA IV? I do not think such conditions are taken as implicit in EGA IV but it is a long book. best, Colin 15-Feb-2005 16:23:56 -0400,6092;000000000001-00000000