By Donald Knutson

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**Example text**

F is locally i • I, the following X. 1 ~ X Y. 1 >Y f. 8, the image moment. 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.

Are e a s i l y m o d i f i e d . A more theorems and p r o o f s b a s e d This problem sheaf cohomology The proofs arise can b e e a s i l y r e s o l v e d . l). I I - V can be d i a g r a m m e d : is that subsets. 16 have somewhat different proofs than in EGA. The arguments in the other steps are mostly from EGA and in fact most of Chapter V on formal algebraic spaces is practically a straight translation of EGA. 4) it should be noted that there have been other candidates considered in the search for more general algebraicgeometric objects.

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. , for 0 ~ M' ~ M ~ M '° ~ 0 of R - m o d u l e s , ~ ~ S: the 0 ~ M' ® S ~ M ~ S ~ M" ® S ~ 0 is R R R R exact. 2) type, For any exact the i n d u c e d sequence sequence ~of R-modules of finite ~ ® S is exact.