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

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2. Assume and let n ~i X is proper, . 3) is a perfect pairinq W -modules. 5) Hi(w~) ®HN-J(w~ -i) given by cup-product followed by noted in the introduction, are perfect, H*(W~*) Tr : ~ ( W ~ ) •W . 5) is not, in general, (hence finitely generated over even if W). 7]. 4) and H~(X/W) H ~ ( W ~ ~) of H ~ (W~), but rather mod F n . e. 4. 2. M u l t i p l i c a t i o n in ~" W n defines a map of D(Wn[d])~. 3) can in fact be viewed as a map of RF(Wnn') , D(Wn[d]) ) Wn(-N)[-N] : . e. defines an isomorphism of RF(Wn~') D(Wn[d]) ~ D(RF(Wnn'))(-N)[-N] D = RHom(-,Wn).

Under some mild restrictions, such a involving only Newton and Hodge polygons, has b e e n given by in answer to a question of Katz. His method uses in a curious way the formalism of t-structures of Deligne [2]. A concept that has g r a d u a l l y emerged as the main technical tool in the four questions above is the notion of graded module over a certain graded ring R O = W~[F,V]. 1), enlarging the usual Dieudonn~ ring §2 develops linear and homological algebra over this ring. For the convenience of the reader I have recalled in §i some 25 basic facts concerning the De Rham-Witt complex.

N . • Wn~ It uses ). 6) coincides n by Berthelot in [3, VII i]. 2. Autoduality ~ k . 7) , dual to it follows (for ~ grown j i+j = N , ~ / B n ~ for BnQJ+I from i+j =N), Fn ~ Z n n j ~ 0 . is dual to . 1. Z n ~J and of some compatigrnw~ is dual The reduction, 51 however, has to is perhaps not so trivial as it looks since in the process one use i~Wn~ the ~> ~ Using fact that, if i n : X e ~. 2. 2. Assume and let n ~i X is proper, . 3) is a perfect pairinq W -modules. 5) Hi(w~) ®HN-J(w~ -i) given by cup-product followed by noted in the introduction, are perfect, H*(W~*) Tr : ~ ( W ~ ) •W .

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