By Donald Knutson
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Extra info for Algebraic Spaces
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. 5. 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).
L l 1 Y f:X + Y E D, then feD. 9: descent F be a sheaf. that for e a c h A stable if the Suppose i, f:X + Y of m a p s class following there the s h e a f E C, if and D of m a p s 6 C, and suppose the m a p W . , to be U. & D. 1 the say F = W" in the g l o b a l set of c l o s e d family i, f~i e D. 4a above, 6 D). i0: Let X e C. ~ cofinal set of c o v e r i n g s X is a set [[Xij ~ X } i e i . ] j ~ j c C o v T, such t h a t for 3 [U k ~ X ] k e K • C o v ~, t h e r e is a j E J and a m a p family n:I.
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. ) when it comes something, the t o p o l o g i c a l Luckily of u n i v e r s a l l y maps map and u n i v e r g a l l y to check note subcategories we d e f i n e is used g:Z + Y. should say in the case of open Similarly radiciel a given m a p for every p o s s i b l e it is s u f f i c i e n t this, bijective The above d e f i n i t i o n s it is a p p a r e n t l y spaces.
Algebraic Spaces by Donald Knutson