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By Donald Knutson

ISBN-10: 3540054960

ISBN-13: 9783540054962

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Example text

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. ~vl ) is an a f f i n e scheme.

A consequence of this d e f i n i t i o n is that a map of rings R + S is etale if and only if S is an R - a l g e b r a of the form S = R[XI, . ,fm ) m, w h e r e the ideal in S g e n e r a t e d bY the n × n m i n ° r s °f the d e t e r m i n a n t I ~ f i / ~ j ~ i S x the unit ideal. for separable is a field, In particular, algebras by the usual J a c o b i a n over a field, if R ~ S is etale and R then S is a finite p r o d u c t field e x t e n s i o n s Alternative criterion of finite separable of R.

This is not too s e r i o u s following we are a restriction lemma: L e t S be a s e p a r a t e d U an S - s c h e m e w i t h U + S l o c a l l y noetherian of finite scheme type. and T h e n U is quasiseparated. Proof. U × U must be is an immersion. 27: The s h e a f on X. following is a local L e t X be a n o e t h e r i a n For any a f f i n e let M be the R - m o d u l e L e t I be the i n t e r s e c t i o n ~ ( S p e c R) and U + U × U H e n c e U ~ U × U is q u a s i c o m p a c t . topology.

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Algebraic Spaces by Donald Knutson


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