By Richard Sot (auth.)

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And With h y p o t h e s e s 10 ~ I 1 C ... C In_ 1 C I = A. n a n d notation as in Def. 1 w e ignore I r A, (*) (M), r > 0 t h e q u e s t i o n of w h e t h e r M are, of t h e p r e s e n t a t i o n (*) f o r of t h e p r e s e n t a t i o n to e s t a b l i s h T h i n . 6 . 6 a n d C o t . 1, although doing so would simplify the exposition, 6,2. Proposition. module a n d let presentation A-module. Let S b e a ring, M b e a finitely presented b e a multiplicative]y closed subset of (*) : A m Let A - An ; M (**) : (S-I(A)) m ---~ ) 0, m, n > 0 (S-I(A)) n ~ A.

Let e x t e n s i o n of k. (1) K k be a field and K be a finitely generated field T h e n t h e following two c o n d i t i o n s a r e e q u i v a l e n t : is a s e p a r a b l e f i e l d e x t e n s i o n of (2) T h e r e e x i s t s an i n t e g e r h o m o m o r p h i s m of n > 0 k-algebras k. a n d an ~tale ~ : kit 1. . . T n] ~ K. deg. whence K/k and k(A(T1) ..... ) 50 Proof. (I) ~ (2). Since K is a finitely generated field extension of view of (i) there exists a separating transcendence for K over k.

K. CHAPTER 4 Brief discussion of unramified and ~tale h o m o m o r p h i s m s In this chapter we present only the definitions and a few very basic properties of unramified and ~tale homomorphisms needed in the succeeding chapters in the treatment of simple homomorphisms. Those wishing to learn more of the basic properties of unramified and 4tale homomorphisms should consult one of the m a n y current texts in algebraic geometry. 1. Proposition. and put A = BQ. such that Bf Let k b e a field, B Suppose there exists is a f i n i t e l y g e n e r a t e d be a k-algebra, f ~ B such that Q m Spec(B) f ~ Q k-Mgebra and suppose and A = 0.