By M. Raynaud (auth.), Michel Raynaud, Tetsuji Shioda (eds.)
Read or Download Algebraic Geometry: Proceedings of the Japan-France Conference held at Tokyo and Kyoto, October 5–14, 1982 PDF
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Additional info for Algebraic Geometry: Proceedings of the Japan-France Conference held at Tokyo and Kyoto, October 5–14, 1982
Module. -module, F and The lim M M. = together zero) FV = VF = p (graded) system Wn[d]-modules maps is an and c o h e r e n c e . ~ with : Mn+ 1 FdV = d is an n respectively and ) Mn . ) be called category. in an o b v i o u s way, If a M. 3) M • D(R) M is an the o p e r a t o r s denote L R. ) . The : D(R) in M^ L : = R ~ i m R. ) M in R-module, induced R. ® R M D(R), . set . 4) for all in b) n D-(R), If X/k complete on WnX . W e say M M ^ (which is a g a i n : this DEFINITION is c o m p l e t e is of finite follows .
6) t h e r e are i n c l u s i o n s • ' ~ (i) f o l l o w s from though parallel --co . . 13 Z3 c Z~3 (ii) and the first a s s e r t i o n of E2 R'-modules. E k e d a h l has p r o p o s e d gate s p e c t r a l sequence is not the c o h o m o l o g y from the o b j e c t (i) (ii). 3) y i e l d s a g a i n the last i s o m o r p h i s m of 3 . 2 . 4 g i v e n in , R'-module. 8 to a r b i t r a r y coherent complexes of R-modules. 3. 3). In the d o m i n o e s and h a v e the same d i m e n s i o n .
To The functor D(W). One defines in an obvious way the notion of sheaf of (graded) R-modules on the k-scheme D(X,R). 3) F X , and the corresponding derived category (applied componentwise) derives into a functor ) D+(R), and for M = RF(X,L) L E D+(X,R), the spectral sequence reads E~ 3 = HJ(x,L i') ----z2 H (X,~L) (provided that ~ commutes with RF , which will be the case in practice). 2. Examples. 1. The De Rham-Wit t complex. 2, the De Rham-Witt complex red as a sheaf of RF(X,W~') of R-modules D+(R) on d ) HJ(w~I) will be denoted RJF(w~" ) RF(X/W).
Algebraic Geometry: Proceedings of the Japan-France Conference held at Tokyo and Kyoto, October 5–14, 1982 by M. Raynaud (auth.), Michel Raynaud, Tetsuji Shioda (eds.)