By David E. Dobbs (auth.)

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15 more fully, we shall need to develop the rudiments of the theory of Grothendieck topologies. The major portion of the next section is devoted to that task. We shall return to the functors 4. M in §5. GROTHENDIECK TOPOLOGIES AND ETALE ALGEBRAS We begin by recalling the basic definitions in ~4~ • Grothendieck ~ T with, for every object [U i ~ u J consists of a category U of Cat T , Cat T of morphisms in Cat T A together a set of families with codomain U This collection of sets of distinguished morphisms is called Coy T and is assumed to satisfy the following three conditions: (i) If a morphism f of Cat T is an isomorphism, then If) ¢ Coy T .

1' H (#') as G h(x) for x c an~ h c G . Since , i 2 (MF') G , the final assertion of Thm. e. is an equalizer diagram. is the inclusion map, then equal to the subset of (M 8)j (M*e)F In other words, if j : K ~ F is a monomorphism with image on which (M 8)~ I and (M 8)~ 2 agree. 15. Let f c ~(S,T) for fields S and T . e. -, (~*e)(~r mags ~sT) Since is a monomorphism. Me S to the face is a field, Prop. 14 implies that As the two maps implies t h a t the two maps S -* T @ST MeT Let on which the two maps into SI = f(S) .

Since the map g-setCk-alg( ]~ Kj,L),M) -~ g-set(k-algCKi,L),M) induced by Pi tested, sends M*( ~ K i) ~ is additive. v to w i, it is clear that the map to be M * (Ki) , sends m to itself. Hence M * - 27 - For the final assertion of the theorem, let be an inclusion map of fields in is identified wlth for g c g ; Vl, m Zl, m ¢ g-set(~/~',M) The map on to an element Z2,m(K2')= V2,m(i)' via R~marks. set(k-alg(K1,L),M ) g-set morphisms induced by z2, m c 9-set g/~',M) where i : ~ *L f sends Thus Then (M*f)(m) = is the inclusion map.