By Donald Knutson
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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.
For any subspace Y of X, the reduced closed subspace of X, whose set of points is the topological closure of Y. 3) For a given map f:Y + X, with separated, image. 10, (II. ~ 6 ~x-algebra. ) . If X immersion, this I. 2 49 4) For a g i v e n closed subscheme of X, its open complement. 5) For a given open subscheme of X, its r e d u c e d closed complement. 22: make the Definition i) that X r e d is a r e d u c e d of r e d u c e d schemes quasicompactness A quasicoherent if it is l o c a l l y s h e a f F on a n o e t h e r i a n if for any p o i n t x OXI our p r e v i o u s scheme w e can also and q u a s i c o m p a c t .
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.
Algebraic Spaces by Donald Knutson