Algebraic Geometry: Proceedings of the Japan-France by M. Raynaud (auth.), Michel Raynaud, Tetsuji Shioda (eds.)

By M. Raynaud (auth.), Michel Raynaud, Tetsuji Shioda (eds.)

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Additional info for Algebraic Geometry: Proceedings of the Japan-France Conference held at Tokyo and Kyoto, October 5–14, 1982

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D e n o t e b y As E k e d a h l showed, T2M the it turns o u t t h a t T2M 37 is a coherent is of finite finite sub-R-module length with length with Ekedahl, of define F F p-tors(M). nilpotent, bijective TIM Moreover, and p-tors(M)/T2M (and zero differential). to be the sub-R-module p-tors(M) such that potent on TIM/p-tors(M). 6. 1. see, is well adapted to the study of duality questions. operations. Internal In his tensor product. study of the K ~ n n e t h p r o b l e m m e n t i o n e d Ekedahl has introduced (graded) M WRN R-modules.

1. a) The and Fix an i n t e q e r following conditions are e q u i v a l e n t is f i n i t e l y qenerated over W for i+j = n ; - H J ( H i W ~ ") is f i n i t e l y generated over W for i+j = n ; - HJ(zw~) is f i n i t e l y qenerated over W for i+j = n o__rr n+l - HJ(Bw~) is f i n i t e l y generated over W for i+j = n o__rr n+l b) A s s u m e the c o n d i t i o n s P. on given Hn(x/w) compatible with ~ a) are s a t i s f i e d . 4 It was p r o p o s e d the c o n d i t i o n s b) of is H o d g e - W i t t in each d e g r e e varieties X consists is c a l l e d in [161 HJ(Bw~) = 0 as close as p o s s i b l e for all for example, group is of of K3 n height.

Spec(k) : D+(Spec(k)) D(X) are as in §i. W e a s s u m e m o r e o v e r is of p u r e d i m e n s i o n tr : Wn~[N] ~> ( W n f ) ! 2) , is the p r o j e c t i o n . Wn+I~[N] adjunction). Thus (resp. 4) (resp. 1). c o m p l e x on c o m p l e x is c o n c e n t r a t e d as is well k n o w n followinq one has a c Q m m u t a t i v e tr [ w h e r e the u p p e r These isomorphisms ~ i~Wn~[N] (Wn+if)'Wn+ 1 dualizing i s o m o r p h i s m of : Wn f : WnX i : Wn x r is a n a t u r a l WnX . The fact that such a in one d e g r e e is no s u r p r i s e since, is C o h e n - M a c a u l a y .

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