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Unformatted text preview: INDEPENDENCE OF ` AND SURFACES MARTIN OLSSON 1. The main theorem 1.1. Let F q be a finite field of characteristic p , let k be an algebraic closure of F q , and let X/ F q be a separated, normal, 2-dimensional F q-scheme of finite type. For a prime ` 6 = p and i Z define P i ` := det(1- TF | H i ( X k , Q ` )) Q ` [ T ] and P i c,` := det(1- TF | H i c ( X k , Q ` )) Q ` [ T ] , where H i ( X k , Q ` ) (resp. H i c ( X k , Q ` )) denotes the etale cohomology of X k (resp. compactly supported etale cohomology of X k ), and F denotes the Frobenius endomorphism. Our aim in this note is to prove the following: Theorem 1.2. The polynomials P i ` and P i c,` are in Q [ T ] , and for any two primes `,` not equal to p we have P i ` = P i ` , P i c,` = P i c,` . 1.3. The same argument will show that if X/k is a separated normal surface over an alge- braically closed field k , then the dimensions of the cohomology groups H i ( X, Q ` ) , H i c ( X, Q ` ) are independent of ` . We apply this to study the Brauer group in this setting to obtain the following theorem: Theorem 1.4. Let k be a separably closed field of characteristic exponent p , and let X/k be a smooth 2-dimensional k-scheme. Let f Br( X ) denote the quotient of the Brauer group of X by its p-torsion, and let f Br( X ) div f Br( X ) be the subgroup of divisible elements. Then the quotient f Br( X ) / f Br( X ) div is a finite group, and there exists an integer r such that f Br( X ) div ' F r p , where F p denotes the quotient of Q / Z by its p-torsion subgroup. Remark 1.5. The results of this note seem well-known to experts (in particular the indepen- dence of ` for the Betti numbers of surfaces is tacitly indicated in [Ill, 1.4]), but no reference appears available. 1 2 MARTIN OLSSON 1.6. Notation. For an abelian group A and an integer N we write A [ N ] for Ker( N : A A ) . For a prime ` we write T ` A for the Z `-module lim - n A [ ` n ]) , where the projective limit is taken with respect to the maps ` : A [ ` n +1 ] A [ ` n ] . We write V ` ( A ) for the Q `-vector space V ` ( A ) := T ` ( A ) Q . Observe that V ` (- ) is a left exact functor in the sense that if A B C is a short exact sequence then the resulting sequence V ` A V ` B V ` C is also exact. Note also that if A is a finitely generated abelian group, then V ` A = 0. 1.7. Acknowledgements. The author was partially funded by NSF grant DMS-0714086, NSF CAREER grant DMS-0748718, and an Alfred P. Sloan Research Fellowship. 2. Kummer theory 2.1. Let k be an algebraically closed field, and let X/k be a proper smooth k-scheme of dimension 2. Let E,D X be divisors with simple normal crossings, and assume E D = ....
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