By Donald Knutson

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**Example text**

The fact that is an a f f i n e and w i t h map. of e x t e n d i n g over m u t a t i s explicitly (U) scheme U i an o p e n algebraic immersion, spaces, the one can do is find U ~ X etale. S o m e of the p r o b l e m s w h i c h For to a l g e b r a i c seem to c a r r y a p o i n t p in a s c h e m e X, and a m a p best theory all of the r e s u l t s exception around of s c h e m e h e r e on the p r o b l e m s instance, sheaf the G r o t h e n d i e c k - t o p o l o g i c a l is not r e l e v a n t be modified.

Of X and the m a p X r e d ~ X is u n i v e r s a l Applying following scheme Note ~ finite G X J ~ F + 0. definition that a l o c a l l y is n e c e s s a r i l y e X, scheme there is an o p e n sets I and J and an This clearly for a n o e t h e r i a n agrees affine free s h e a f on a n o e t h e r i a n coherent. : Let f:X ~ Y be a m a p f is of finite type of schemes. if it is locally of finite and q u a s i c o m p a c t . ii) f is of finite of finite p r e s e n t a t i o n , presentation if it is l o c a l l y quasicompact, and the induced A m a p X + X × X is q u a s i c o m p a c t .

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. (XXVIII) These include the notion of Nash manifold and Matsusaka's notion of Q-variety (xxI). 1) 28 in the case of varieties, algebraic spaces are a special have b e e n considered, case of Q - v a r i e t i e s .