Algebraic Geometry

Donald Knutson's Algebraic Spaces PDF

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By Donald Knutson

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6 ~x-algebra. ) . If X immersion, this I. 2 49 4) For a g i v e n closed subscheme of X, its open complement. 5) For a given open subscheme of X, its r e d u c e d closed complement. 22: make the Definition i) that X r e d is a r e d u c e d of r e d u c e d schemes quasicompactness A quasicoherent if it is l o c a l l y s h e a f F on a n o e t h e r i a n if for any p o i n t x OXI our p r e v i o u s scheme w e can also and q u a s i c o m p a c t . sequence scheme. : noetherian ii) of X, X r e d.

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. 3 58 union of the images of the c o r r e s p o n d i n g Let U x be the d i s j o i n t Uvl'''''Uvn cover Vx. union of Ux', U v l ' ' ' ' ' U v n Now let the index set I be the c o l l e c t i o n of p o i n t s and for each i e I, X.

2 41 s u b s c h e m e W of X is an o p e n of X. In these cases, called an o p e n subscheme of a c l o s e d the a s s o c i a t e d immersion, subscheme i n c l u s i o n Y ~ X is Z + X a closed immersion, and W ~ X an immersion. 6: A m o d u l e M over a s h e a f of m o d u l e s ~ taking F ( S p e c Rf,~) over a ring R g i v e s the s t r u c t u r e = M ® Rf. rise to s h e a f on Spec R, b y A general s h e a f of m o d u l e s F R on Spec R is c a l l e d some R - m o d u l e M. w e say ~ maps is coherent.

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