By Donald Knutson
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Extra resources for Algebraic Spaces
Is a r e g u l a r 2 % ,x X is of d i m e n s i o n of field). , is also is the s p e c t r u m for each p o i n t in its q u o t i e n t of X b y elements). , subspace U c Y, if for every f-l(u) = U × X Y I. 2 46 3) quasiseparated (f:X + Y is q u a s i s e p a r a t e d 4) m a p A : X + X x X is q u a s i c o m p a c t . 19: of m a p s are s t a b l e and local The following properties on the d o m a i n in the Z a r i s k i topology : i) locally of finite type (f:X -~ Y is l o c a l l y o f finite type if for any p o i n t x e X, subschemes x 6 U, there are affine open U = S p e c R c X and V = S p e c S c Y with f(x) e V, fJ :U -~ V and the a s s o c i a t e d m a p U S ~ R of rings m a k e s R f i n i t e l y g e n e r a t e d as an S-algebra.
Suppress for the p r o d u c t X × Y of S - s c h e m e s . S 3. The Flat Topology of S c h e m e s W e m a i n l y deal h e r e w i t h of a f f i n e schemes schemes. 12. on the c a t e g o r y to the c a t e g o r y The results quoted here of all consti- I. 2. descent, see (V~. Alg. schemes, f is flat if any, Ch. 1: Let all, (VIII), discussion f*:Spec or e q u i v a l e n t l y hence 1 SGA 60-61, of flat S ~ Spec R be a f:R ~ S a m a p of rings. of the f o l l o w i n g equivalent condi- tions hold: i) f,:(R-modules) any e x a c t induced sequence sequence ~ : ~ (S-modules) is exact.
5. subscheme immerBy (closed sub- of X. If X is a s u b s c h e m e X is a s u b s c h e m e respectively (respectively respectively subscheme) f:Y ~ X is an open immersion, sion, scheme, If n = i, F is c a l l e d sheaf. Definition immersion the rank of F. of Y and Y is a s u b s c h e m e of Z, then of Z. 11: and d i s j o i n t The c a t e g o r y sums. 6). we d e f i n e has fiber functor faithful, and p r e - of g:Z ~ Y (Schemes). A. the term necessary map for the n o t i o n s if f is injec- One open the induced open m a p s and closed open (in if for f':X X Z ~ Z is Y is then a c l o s e d universally closed.
Algebraic Spaces by Donald Knutson