By Donald Knutson

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6) etale 7) universally 8) locally quasifinite locally of (see def. 20: descent In this case, in the Z a r i s k i U of Y, sheaf uniquely determined To p r o v e affine m a p s Spec (g'A) of A= ~y-modules by A. classes of topology: if for every affine = U X X is affine. Y f*(~'X' A is a q u a s i and X and We write are effective, for all schemes f are X = S~ A. ) Y immersions. 2) Open 3) Closed 4) Immersions 5) Quasiaffine exists such f-l(u) if w e w r i t e coherent that following (f:X ~ Y is affine subscheme show The immersions maps (f:X ~ Y is ~ u a s i a f f i n e a scheme W, that f = hg, and h is affine.

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 . . . . . . . . . . . . . 52 of S c h e m e s . . . . . . . . . . . . . 59 Relations ........................... 72 and D e s c e n t (Grothendieck) of a c a t e g o r y covering C = Cat the Theory ToDoloqv T and in C a t ~ on a c a t e - a set C o v ~ called r a n g e U of the m a p s ~ of coverings ~i is fixed) satisfying i) If ~ is an i s o m o r p h i s m 2) If {U i ~ U} each i then position 3) If 6 e Cov the then T and family {~] £ Cov ~.

Rise to s h e a f on Spec R, b y A general s h e a f of m o d u l e s F R on Spec R is c a l l e d some R - m o d u l e M. w e say ~ maps is coherent. ~vl ) is an a f f i n e scheme. ) Definition has is of finite of q u a s i c o h e r e n t A scheme for e v e r y p o i n t p map U ~ X with U tion Similarly and M for of algebras. Definition compact if it is of the form ~ If R is n o e t h e r i a n o f rings R ~ S) w e s p e a k sheaves such quasicoherent the F r e n c h meaning a finite quasicompact space.