NNON-TORSION BRAUER GROUPS IN POSITIVE CHARACTERISTIC
LOUIS ESSER
1. I
NTRODUCTION
One way of extending the notion of the classical Brauer group of a field to any scheme X is by defining the Brauer-Grothendieck group Br ( X ) = H ( X ´et , G m ) . Just as for fields,this group is torsion for any regular integral noetherian scheme [5, Corollaire 1.8]. How-ever, this no longer holds for singular schemes. In fact, there exist affine integral normalcomplex surfaces X with one singular point such that Br ( X ) contains the additive groupof C [3, Example 8.2.2]. Specifically, the affine cone over a curve of degree d ≥ in P C hasthis property. However, these examples leave open the following question, suggested byColliot-Th´el`ene and Skorobogatov: Question 1.1. If X is an integral normal quasi-projective variety over a field k of positivecharacteristic, is Br ( X ) a torsion group?To analyze this question, we use a result concerning the Brauer group of a normal vari-ety X with only isolated singularities p , ..., p n . Suppose X is defined over an algebraicallyclosed field k of arbitrary characteristic. Let K be the function field of X , R i = O X,p i bethe local ring at each singularity, and R sh i its strict henselization. Then, Br ( X ) is given bythe exact sequence (see [3, Section 8.2], elaborating on [5, §1, Remarque 11 (b)])(1) → Pic ( X ) → Cl ( X ) → n (cid:77) i =1 Cl ( R sh i ) → Br ( X ) → Br ( K ) . This sequence indicates that one way for Br ( X ) to be large is when a singularity has alarge henselian local class group with divisors that do not extend globally to X . This ideais illustrated by a counterexample given by Burt Totaro, which shows that the answerto Question 1.1 is “no” for threefolds: suppose that X ⊂ P is a hypersurface of degree d ≥ with a single node p . Then Y = Bl p ( X ) is a smooth, ample divisor in Bl p P . By theGrothendieck-Lefschetz theorem for Picard groups, the restriction Pic ( Bl p P ) → Pic ( Y ) isan isomorphism.If E is the exceptional divisor, then the sequence → Z · [ E ] → Pic ( Y ) → Cl ( X ) → yields Cl ( X ) ∼ = Z · O X (1) , so that the restriction map Cl ( X ) → Cl ( R sh ) is zero. However,since a threefold node is ´etale-locally the cone over a smooth quadric surface, one canshow that the henselian local class group contains a copy of Z . Thus, Br ( X ) is not torsion.We will show that counterexamples also exist in dimension 2, provided that one worksover a large algebraically closed field k . a r X i v : . [ m a t h . AG ] F e b LOUIS ESSER
Acknowledgements.
I thank Burt Totaro for suggesting this problem to me and for hisadvice. 2. A S
URFACE C OUNTEREXAMPLE
The following construction is taken from [6], Example 1.27. Take a smooth cubic curve D in the projective plane and a quartic curve Q that meets it tranversally. Let Y = Bl q ,...,q P , where q , ..., q are the points of intersection. On Y , the strict transform C of D satisfies C = − . Unlike rational curves, not all negative self-intersection highergenus curves may be contracted to yield a projective surface. Rather, the contractionmight only exist as an algebraic space. However, in this case, C is contractible. Proposition 2.1.
There exists a normal projective surface X and a contraction morphism Y → X whose exceptional locus is C .Proof. The Picard group of Y is the free abelian group on H , the pullback of a generalline in P , and the exceptional divisors E , ..., E . Then, we claim that the line bundle O Y (4 H − (cid:80) i E i ) = O Y ( H + C ) defines a basepoint-free linear system on Y . Indeed,no point outside of the E i can be a base point, and the transforms of both D + a lineand Q belong to the linear system. These don’t intersect on the exceptional locus bythe transversality assumption. Therefore, the system defines a morphism Y → X (cid:48) to aprojective surface X (cid:48) .This contracts only C : if C (cid:48) is another (irreducible) curve with C (cid:48) · ( H + C ) = 0 , clearly C (cid:48) is not supported on the E i , or the intersection would be positive. Therefore, H · C (cid:48) > ,meaning C · C (cid:48) < . Therefore, C (cid:48) = C , as desired. Finally, passing to the normalization X of X (cid:48) , we may assume X is a normal projective surface. (cid:3) The resulting singularity p of X has minimal resolution with exceptional set exactly C , asmooth elliptic curve. Singularities satisfying this condition are simple elliptic singularities .Over C , such singularities are completely classified. A simple elliptic singularity with C = − is known as an ˜ E singularity, and is complex analytically isomorphic to a coneover C [7]. Here, we present a computation of the henselian local class group Cl ( R sh ) ofthe singularity that works in any characteristic.Consider the pullback of this desingularization to a ”henselian neighborhood”: Y h Y Spec ( R sh ) X. The scheme Y h is regular and Y h \ C ∼ = Spec ( R sh ) \ { m } , so we have an exact sequence → Z · [ C ] → Pic ( Y h ) → Cl ( R sh ) → . It suffices, therefore, to compute Pic ( Y h ) . To do so,we’ll first consider infinitesimal neighborhoods of C in Y .The sequence of infinitesimal neighborhoods C = C ⊂ C ⊂ · · · is defined by powersof the ideal sheaf I C . Notably, these C n are the same regardless of whether we consider ON-TORSION BRAUER GROUPS IN POSITIVE CHARACTERISTIC 3 them inside Y or inside the henselian neighborhood Y h . The normal bundle to C in Y gives obstructions to extending line bundles to successive neighborhoods, but we’ll showthat all line bundles extend uniquely. The group lim ←− n Pic ( C n ) in the proposition below isalso the Picard group of the formal neighborhood of C in Y . Proposition 2.2.
The restriction map lim ←− n Pic ( C n ) → Pic ( C ) is an isomorphism.Proof. It’s enough to show that the maps Pic ( C n +1 ) → Pic ( C n ) are all isomorphisms for n ≥ . Each extension C n ⊂ C n +1 is a first-order thickening, since C n is defined in C n +1 by the square-zero ideal sheaf I nC / I n +1 C . Associated to such a thickening is a long exactsequence in cohomology [8, 0C6Q] · · · → H ( C, I nC / I n +1 C ) → Pic ( C n +1 ) → Pic ( C n ) → H ( C, I nC / I n +1 C ) → · · · . We may take the outer cohomology groups to be over C since the underlying topologi-cal spaces are the same. As sheaves of abelian groups on C , we have I nC / I n +1 C ∼ = O C ( − nC ) ,a multiple of the conormal bundle. But C = − in Y so this last bundle has degree n > .Therefore, its higher cohomology vanishes and Pic ( C n +1 ) → Pic ( C n ) is an isomorphismfor all n . (cid:3) Now, we need only compare lim ←− n Pic ( C n ) to Pic ( Y h ) . Using the Artin approximationtheorem [1, Theorem 3.5], the map Pic ( Y h ) → lim ←− n Pic ( C n ) is injective with dense image.However, the topology of the latter group is discrete in this setting because each group ofthe inverse limit is Pic ( C ) . Therefore, the map is surjective also and Pic ( Y h ) ∼ = Pic ( C ) . Theorem 2.3.
Let k be an algebraically closed field that is not the algebraic closure of a finite field.Then, Br ( X ) is non-torsion.Proof. From the above, we have the identification Cl ( R sh ) ∼ = Pic ( Y h ) / Z · O Y h ( C ) ∼ = Pic ( C ) / Z · O C ( C ) . Since deg C O C ( C ) = 3 , the class group is then an extension of Z / byPic ( C ) ∼ = C ( k ) . If k = F p , then C ( k ) is torsion, because every point is defined over F p m for some m . In contrast, if k satisfies the hypothesis of the theorem, C ( k ) has infinite rank[4, Theorem 10.1].However, the global class group of X is quite small: Cl ( Y ) = Pic ( Y ) ∼ = Z since Y is the blow up of P in 12 points and Cl ( X ) ∼ = Cl ( Y ) / Z · [ C ] . Therefore, the cokernel ofthe restriction map Cl ( X ) → Cl ( R sh ) in (1) contains non-torsion elements, so Br ( X ) doestoo. (cid:3) To complement the above result, we also prove
Theorem 2.4.
Suppose that X is an integral normal projective surface over the algebraic closure k = F p of a finite field. Then Br ( X ) is torsion.Proof. The strategy is similar to the previous theorem. Here, the crucial fact is that allpossible “building blocks” of the henselian local class group - abelian varieties over k , theadditive group of k , and the multiplicative group of k - are all torsion. LOUIS ESSER
In the exact sequence (1), Br ( K ) is always torsion, so if we can prove ⊕ ni =1 Cl ( R sh i ) is aswell, the result follows. Therefore, we focus on the desingularization π : Y h → Spec ( R sh ) of the henselian local ring at just one point p . We may choose Y h such that the exceptionalset E = π − ( p ) is a union of irreducible curves E j which are smooth and meet pairwisetransversely, with no three containing a common point.The following argument is due to Artin [2, p. 491]. Suppose G is the free abeliangroup of divisors supported on E , and consider the map Pic ( Y h ) → Hom ( G, Z ) given by L (cid:55)→ ( D (cid:55)→ D · L ) . This restricts to a map G → Hom ( G, Z ) that is injective because theintersection matrix of the curves E j is negative definite. In particular, the first map in theexcision sequence of class groups → G → Pic ( Y h ) → Cl ( R sh ) → is injective. Since G and Hom ( G, Z ) are free abelian groups of equal rank, G → Hom ( G, Z ) also has finitecokernel. This allows us to find a cycle Z = (cid:80) j r j E j with all r j > and O Z ( − Z ) ample.We’ll examine infinitesimal neighborhoods of the closed subscheme Z .As before, for every n ≥ , there is an exact sequence · · · → H ( Z, O Z ( − nZ )) → Pic ( Z n +1 ) → Pic ( Z n ) → H ( Z, O Z ( − nZ )) → · · · . Because dim( Z ) = 1 , the last group is always zero. Since O Z ( − Z ) is ample, the firstgroup is zero for n (cid:29) . Therefore, the inverse limit lim ←− n Pic ( Z n ) is constructed as afinite series of extensions of Pic ( Z ) by finite-dimensional k -vector spaces. While Z is gen-erally non-reduced and not equal to E , we may still apply Artin approximation. Thisis because for any large n , the scheme E n is nested between two infinitesimal neigh-borhoods of Z , where all restrictions of Picard groups are surjective (use a similar ex-act sequence to the above, e.g. [8, 09NY]). It follows that lim ←− n Pic ( Z n ) ∼ = lim ←− Pic ( E n ) , soPic ( Y h ) ∼ = lim ←− n Pic ( Z n ) .Next, let ¯ Z be the disjoint union of the schemes r j E j . Then f : ¯ Z → Z is a finitemap that is an isomorphism away from the finite set of intersection points and such that O Z ⊂ f ∗ O ¯ Z . It follows (see [8, 0C1H]) that Pic ( Z ) is a finite sequence of extensions ofPic ( ¯ Z ) by quotients of ( k, +) or ( k, ∗ ) . Lastly, Pic ( ¯ Z ) = ⊕ j Pic ( r j E j ) , each of which is builtfrom finite-dimensional k -vector spaces, and Pic ( E j ) ∼ = Z ⊕ Pic ( E j ) . Since the Pic ( E j ) are groups of k -points of abelian varieties over k , they are all torsion.Taken together, all of this implies that G and Pic ( Y h ) are equal rank. Because G → Pic ( Y h ) is injective, the quotient Cl ( R sh ) is a torsion group, as desired. (cid:3) R EFERENCES [1] M. Artin,
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