# Algebraic Spaces by Donald Knutson

By Donald Knutson

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X n)) where of of schemes. is etale c V and the a s s o c i a t e d the J a c o b i a n R [ X I, in the of Schemes and an affine such the flat of finite p r e s e n t a t i o n ) . Topology any p o i n t x e X, morphism, of schemes take w a r n i n g defined corresponds Definition separated subscheme. should have b e e n (fppf = faithfully 4. the s c h e m e - t h e o r e t i c noetherian. 14: flat of a c l o s e d following topology: sheaf image of a q u a s i c o m p a c t open c o m p l e m e n t The constructions matrix (~fi/) is a 3 of I.

12: Let X e C and ~ be a local c o n s t r u c t i o n L e t D be a stable class of m a p s of C satisfying tive d e s c e n t and suppose that in a family in the cofinal { (Xij) ~ Xij is in D. :X.. ~ X appearing 13 13 set associated to 6, the map T h e n ~ is e f f e c t i v e (and hence the map { (X) ~ X is in D). ). 4 and c o n s i d e r for a given c a t e g o r y C. fiber products. First definition. 15: effectively A family {U i ~ U}iEI epimorphic objects W of C, and m a p s W + U, family of maps of C is (UEEF) if for all and for all o b j e c t s V e C, the following d i a g r a m of sets is exact: HOmc(W,V) ~--~Hom (W X Ui,V) i C U ~ ~-~HOmc(W.

Topology. for w h i c h m M ~ M noetherian of f i n i t e open type of all m a x i m a l This construction scheme subscheme for w h i c h ~ and F Spec R = FI Spec R ideals m of R ( e q u i v a l e n t l y M ® R / m ~ 0). = Spec R/I. I local c o n s t r u c t i o n Put is e f f e c t i v e I. 3 52 and w e w r i t e ~ (X) = Supp F, c a l l e d is then the r e d u c e d closed only E X those p o i n t s x the s u p p o r t subspace for w h i c h of F. of X c o n t a i n i n g Supp F all and the stalk of F at x is n o n - zero.