surfaces - INDEPENDENCE OF ` AND SURFACES MARTIN OLSSON 1....

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = ....
View Full Document

This note was uploaded on 02/05/2011 for the course MATH 224b taught by Professor Grunbaum,f during the Spring '08 term at University of California, Berkeley.

Page1 / 18

surfaces - INDEPENDENCE OF ` AND SURFACES MARTIN OLSSON 1....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online