By Donald Knutson

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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.

Exists, and let ~:X ~ Y be the induced l i£I Then ~ e B if and only if for all i c I, ~i E B. map. union X = (Thus if C has d i s j o i n t {U i + U] $2: and only unions, in Cov T B can be r e p l a c e d The r e s u l t i n g the of C for w h i c h lack of indices A map any c o v e r i n g by a c o v e r i n g often m a k e s f e B is a u n i v e r s a l family map arguments [_~ U. + U. ) effective epimorphism is then just given by if if it is an epimorphism. ) f X Let _~ Y Z be a c o m m u t a t i v e diagram [fl B-topology.

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 . 3 57 m a k i n g X. :X. ~ Y. such 1 1 1 Y. ~ Y; 1 and finally [Xi} is a c o v e r i n g of X in the Zariski topology ii) [Yi} is a c o v e r i n g of Y in the Zariski topology For each iv) Each Proof.