1976a Duality in the flat cohomology of surfaces

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1 1976a Duality in the flat cohomology of surfaces (Ann. Sci. Ecole Norm. Su. 9 (1976), ). By the early 1960s, the fundamental imortance of the cohomology grous H i.x;.z=`rz/.m// D def H i.x ; m et / `r for the study of the arithmetic of algebraic varieties was already clear. However, there was a ga in the theory: the grous are only defined when ` (the characteristic of the ground field k). What should the grous H i.x;.z= r Z/.m// be? Crystalline cohomology doesn t rovide the answer, because it is the analogue of de Rham cohomology, and is not (directly) useful, for examle, in studying the Brauer grou of a variety. For m D 0 there is no roblem: one can take H i.x;.z= r Z/.0// D def H i.x et ;Z= r Z/: For m D 1, the sheaf r is zero on X et, but its flat cohomology has the correct roerties: one can take H i.x;.z= r Z/.1// D def H i.x fl ; r /: This suggested trying H i.x;.z= r Z/.m// D H i.x fl ; m r / where m is the sheaf r r r on X fl ; but this is not romising because the sheaf r r is a big mess (in contrast, `r `r is just a twist of Z=`rZ). In my thesis, I studied the flat cohomology of r in the following way. Let f WX fl! X et be the continuous ma defined by the identity ma. The exact sequence of sheaves on X fl rovides an exact sequence of sheaves on X et 0! r! G m r! G m! 0 0! f G m r! f G m! R 1 f r! 0! 0! : Therefore R j f r D 0 for j 1, and so H i.x fl ; r / D H i 1.X et ; r / where r is the étale sheaf R 1 f r ' G m = r G m. In my thesis, I studied 1 using the exact sequence 1 C 0! 1! 1X;cl! 1 X! 0 (*) where C is the Cartier oerator. Eventually, this suggested the following to me: (a) Instead of looking for the mythical flat sheaves m, one should osit that R j f m r D r 0 for j m (since this is true m D 0;1), and instead look for the étale sheaf r.m/ D R def m f m. Thus, conjecturally, r H i.x;.z= r Z/.m// D def H i m.x et ; r.m///: 1

2 (b) To study 1.m/, one should look for a sequence like (*). In 1969, when Tate visited London (from Paris) to give a lecture, I told him that 2 should lay the role of the mythical cohomology grou H 2.X fl, /. A few days later, he sent me a letter (see below) saying that my idea seemed to be correct, because he had been able to define a new symbol (now called the Tate symbol) with values in 2 analogous to the Galois symbol which takes values in H 2.k; 2 /: ` In the early 1970s, Artin conjectured a flat duality theorem for the cohomology of r on a smooth rojective surface. I succeeded in roving the conjecture for r D 1 by using the sequence (*). In fact, for an arbitrary smooth rojective variety X, I defined the étale sheaves 1.m/ by the sequence 1 C 0! 1.m/! mx;cl! m X! 0: (**) and roved a duality theorem for the grous H i.x;.z=z/.m// D def H i m.x et ; 1.m///. Since 1.1/ ' R 1 f, this gave Artin s conjecture for a surface, but only for sheaves killed by. Sencer Bloch sent at the University of Michigan. In 1974, when I told him that I had a good theory for the grous H i.x;.z= r Z/.m// for r D 1 using the sheaves of differentials j, but that I didn t know how to extend it for r > 1 because the sheaves i are killed by, he was able to tell me that he had defined sheaves of differentials that are killed only by r. This was his work on what became known as the de Rham-Witt comlex. Using Bloch s work, I was able to comlete the roof the duality theorem for surfaces (Artin s conjecture). Moreover, for varieties of arbitrary dimension, I showed that the five-lemma would (trivially) extend the roof of the duality theorem from r D 1 to all r once one had exact sequences 0! 1! r! r 1! 0 (***) (cf. Remark 3.14 of the aer; also 1.7 and 1.11 of my 1986 AJM aer). Bloch s definition of the de Rham-Witt comlex was difficult to work with (tyical curves on K-grous). Deligne suggested a much simler construction, and in working out the details of Deligne s idea, Illusie was able to rove (***). 1 See also Milne1987, notes. Both Bloch and I soke on our work at the AMS Summer Institute on Algebraic Geometry, Arcata 1974, but neither of us was invited to contribute to the ublished roceedings 2 of the Institute. The sheaves r.m/ have roved to be very successful in laying the role of R m f m. r See, for examle, the notes for my aers 1986a and 1988b and my joint aers with Niranjan Ramachandran. Francohiles may refer the exosition of the roof of the duality theorem (case of surfaces only) in Berthelot Illusie, of course, is aware of all this, but nevertheless credits the duality theorem to one of his students. 2 Algebraic Geometry Arcata 1974 (Proc. Symos. Pure Math., Vol. 29), Amer. Math. Soc., Providence, R.I., Berthelot, P. Le théorème de dualité late our les surfaces (d arès J. S. Milne). Algebraic surfaces (Orsay, ), , Lecture Notes in Math., 868, Sringer, Berlin-New York,

3 Erratum Conditions on. I only needed the assumtions on (e.g., 186, > 2;m) because I had to rely on Bloch s aer for the de Rham-Witt comlex and he makes those assumtions (I should have made that clear). Once Illusie s aer became available they could be droed, as Berthelot made clear when he rewrote my aer in the case of surfaces (see footnote above). Remark 1.2, assumes that is if S is erfect, then i X=S D i X=F. This is incorrect, as illustrated by X D S D SecF.t/ al. The remark isn t used anywhere. Timo Keller oint out that there is no (1.16) (cited on 176). Further: In the statement of Lemma 1.7 it should read d m r 1. (add -1); X=S also on. 177 in the middle diagram, and in the diagram before Corollary 1.10: +1 should be -1. In the diagram in the roof of Lemma 1.7, it should read F. r / (move the ) u) X=S 3

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