By Donald Knutson

ISBN-10: 3540054960

ISBN-13: 9783540054962

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**Extra resources for Algebraic Spaces **

**Example text**

Definition compact if it is of the form ~ If R is n o e t h e r i a n o f rings R ~ S) w e s p e a k sheaves such quasicoherent the F r e n c h meaning a finite quasicompact space. ) and s u f f i c i e n t is q u a s i c o m p a c t scheme covering covering is a u t o m a t i c a l l y to be q u a s i c o m p a c t a finite tradi- for the E n g l i s h that e v e r y open An affine follow it is n e c e s s a r y by a f f i n e schemes. I. 2 42 Definition ~X-mOdule. 9: (X, ~ X ) be a s c h e m e F is q u a s i c o h e r e n t is an open sequence Let ~X I ~ if for e v e r y p o i n t p of X s u b s e t U of X, w i t h p ~X J ~ F I and F an ~ 0 of ¢ U, and an e x a c t ~X-mOdules (where ~X I U and ~X J denote infinite index this c l e a r l y We the sums of the m o d u l e sets I and J) .

Of schemes determined is too fine to be able to g e n e r a l i z e following Let schemes, open. if Y is noetherian. on Y) ~ and The r e m e d y for this situation proposition: f:X ~ Y be of finite p r e s e n t a t i o n . 7. 10: Then schemes topology. on the c a t e g o r y of flat m a p s Proposition locally exact The t o p o l o g y of affine f:X ~ Y of schemes sheaves (respectively flat if and only f_flat) if the induced is exact on affine f:X + Y be a m a p of affine A map f*:(Quasicoherent of all m a p s in the X is n o e t h e r i a n Definition topology are • flat and of finite p r e s e n t a t i o n .

The Zariski 3. The Flat 4. The Etale 5. Etale i. Grothendieck (where Topologies Topology Topology Equivalence gory C consists families TOPOLOGY Topology Definition ONE and D e s c e n t A of S c h e m e s . . . . . . . . . . . . . 52 of S c h e m e s . . . . . . . . . . . . . 59 Relations ........................... 72 and D e s c e n t (Grothendieck) of a c a t e g o r y covering C = Cat the Theory ToDoloqv T and in C a t ~ on a c a t e - a set C o v ~ called r a n g e U of the m a p s ~ of coverings ~i is fixed) satisfying i) If ~ is an i s o m o r p h i s m 2) If {U i ~ U} each i then position 3) If 6 e Cov the then T and family {~] £ Cov ~.

### Algebraic Spaces by Donald Knutson

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