By Mark L. Green, Jacob P. Murre, Claire Voisin, Alberto Albano, Fabio Bardelli

ISBN-10: 354058692X

ISBN-13: 9783540586920

The most objective of the CIME summer time university on "Algebraic Cycles and Hodge conception" has been to assemble the main lively mathematicians during this zone to make the purpose at the current state-of-the-art. therefore the papers incorporated within the complaints are surveys and notes at the most crucial issues of this quarter of analysis. They comprise infinitesimal tools in Hodge thought; algebraic cycles and algebraic features of cohomology and k-theory, transcendental tools within the learn of algebraic cycles.

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Then C B C be and is unramified A. Proof. Immediate b y definition and by P r o p . 6. 7. P r o p o s i t i o n . 8 : A ' C Let A be a r i n g , B and be the structure homomorphism, C put be A - a l g e b r a s , let 52 ]_ _ 8' = 8 | B : B > B | C, let R e Spec(B | C) and put Q = 8'--(R). , ~tale) over A at Q. T h e n B | C A is u n r a m i f i e d ( r e s p . , ~ t a l e ) o v e r C a t R . Proof. In the unramified case, by Prop. 3 we have A whence by Prop. A 4 . 3 we c o n c l u d e On t h e o t h e r h a n d , a : A , B a' = a | C A : C A A B @ C is u n r a m i f i e d o v e r C a t A BQ is a f l a t A p - m o d u l e , w h e r e suppose is t h e s t r u c t u r e ~ B | C A homomorphism and and R 0 = a'-l(R).

4 a n d the fact that f is Thus B-linear l' o p' = 0. e ~ Ker(l'). a unique h o m o m o r p h i s m f o dB/A M) W e have in view of condition (I) in Prop. we conclude > HornC B y condition (3) in Prop. 4 there exists a B-modules Let l' M) Im(~') = Ker(l'). For this p u r p o s e let unique h o m o m o r p h i s m DerA(B, Since Hence where of B y condition (3) in Prop. B-modules e G Ker(l') e(b) = 0 w e conclude for all b E J, e'E DerA(C, Im(~') = Ker(l') f : pAl(B) M). 4 there exists ~ M such that f(dB/A(b)) whence e e = ~'(e').

A . 2 we c o n c l u d e t h a t A B b e an Q ~ Spec(B), A-algebra, P = X-I(Q) ~ : A and let > B Q' in B | ~(P) = B/~ApB. We s a y t h a t B is A u n r a m i f i e d o v e r A a t Q if a n d o n l y if B | ~ ( P ) i s @tale o v e r ~(P) A a t Q' a n d t h e r e e x i s t s f @ B s u c h t h a t f ~ Q a n d Bf is a f i n i t e l y presented A-algebra. , or B k We s a y t h a t is ~tale) i s ~tale o v e r B A is ~tale o v e r Q and BQ A at is a flat X is u n r a r n i f l e d ( r e s p . , at Q) B Q if a n d Ap-module.

### Algebraic cycles and Hodge theory: lectures given at the 2nd session of the Centro internazionale matematico estivo by Mark L. Green, Jacob P. Murre, Claire Voisin, Alberto Albano, Fabio Bardelli

by James

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