By Donald Knutson
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XXV, Grothendieck's (assumed The p r i m e DieuEGA. throughout spectrum to of R, object: Spec R is the set of all p r i m e a contravariant functor ideals of from the c a t e g o r y to that of sets. Spec R is a t o p o l o g i c a l g i v e n b y ideals of R. space with For an ideal I, the c l o s e d the c o r r e s p o n d i n g sets I. 2 39 closed set V(I) is the set of all p r i m e Spec is thus a functor to the c a t e g o r y of topological The topology can be d e f i n e d For any element modulo Bourbaki, equivalently I.
X n)) where of of schemes. is etale c V and the a s s o c i a t e d the J a c o b i a n R [ X I, in the of Schemes and an affine such the flat of finite p r e s e n t a t i o n ) . Topology any p o i n t x e X, morphism, of schemes take w a r n i n g defined corresponds Definition separated subscheme. should have b e e n (fppf = faithfully 4. the s c h e m e - t h e o r e t i c noetherian. 14: flat of a c l o s e d following topology: sheaf image of a q u a s i c o m p a c t open c o m p l e m e n t The constructions matrix (~fi/) is a 3 of I.
CX to For Ux' i s locally f(U~) is open any point v of V v so an open subset an open c o v e r i n g compact so there finite in V E Vx, and v • V v c Y w i t h one can c h o o s e V v c Vx. forms of of Vx. of V . x is a finite commutes x the c o n d i t i o n there . f(U a) c V . x x The r e s t r i c - and flat are affine open set V v l so by sets and U v nonempty. f ( U v) is an open Clearly subset The set of all such Vv, Since open Fix U' and V for the x x c Vv, each that are affine presentation there f(Uv) Again, diagram flat.