By Donald Knutson

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**Example text**

XXVIII) These include the notion of Nash manifold and Matsusaka's notion of Q-variety (xxI). 1) 28 in the case of varieties, algebraic spaces are a special have b e e n considered, case of Q - v a r i e t i e s . , algebraic Artin's objects topology spaces (XXVII). seem theorems). ) such as M u m - these (for now) geometric structure CHAPTER THE ETALE l, Grothendieck 2. The Zariski 3. The Flat 4. The Etale 5. Etale i. Grothendieck (where Topologies Topology Topology Equivalence gory C consists families TOPOLOGY Topology Definition ONE and D e s c e n t A of S c h e m e s .

In the the following (C,Cov T) Definition (under ~) for all images of the U o. 5: A class if for a n y [U i + U} i, U 1 e S. 13) let C be a c a t e g o r y satisfying the a x i o m A 0. of o b j e c t s S c C is s t a b l e c Coy c S if and o n l y T, U and if I. 7: diagram A class if D is a c l o s e d [Yi + Y} E Cov in C, and D of m a p s subcategory Tp if each f e D, then f' E D. in C is s t a b l e and for any f. :X × Y. + Y. l l 1 Y f:X + Y E D, then feD. 9: descent F be a sheaf. that for e a c h A stable if the Suppose i, f:X + Y of m a p s class following there the s h e a f E C, if and D of m a p s 6 C, and suppose the m a p W .

Union We of any X of C X = ~ for each Xi-i,j say C has (finite) £ I, (finite) set of of C exists. 19: B of C m i g h t W e now list satisfy some in order axioms that to give a closed a nice subcategory topology. I. ~ ____t_> Y}iEI be a set of maps ~ X. exists, and let ~:X ~ Y be the induced l i£I Then ~ e B if and only if for all i c I, ~i E B. map. union X = (Thus if C has d i s j o i n t {U i + U] $2: and only unions, in Cov T B can be r e p l a c e d The r e s u l t i n g the of C for w h i c h lack of indices A map any c o v e r i n g by a c o v e r i n g often m a k e s f e B is a u n i v e r s a l family map arguments [_~ U.