Cohomological obstruction theory for Brauer classes and the period-index problem

Cohomolo gical obstruction theory for Brauer classes and the period- index problem ∗ Benjamin Antieau October 24, 2018 Abstract Let U be a connected scheme of finite ´ etale cohomological dimension in which e very finite set of points is contained in an af fine open subscheme. Suppose that α is a class in H 2 ( U ´ et , G m ) tors . For each positive intege r m , the K -theory of α -twisted sheav es is used to identify obstructions to α being representable by an Azumaya algebra of rank m 2 . The ´ etale index of α , denoted eti ( α ) , is the least positi ve integer such th at all the obstructions vanish. Let per ( α ) be the order of α in H 2 ( U ´ et , G m ) tors . Methods from stable homotop y theory giv e an upper bound on the ´ et ale index that depends on the period of α and the ´ etale cohomological dimension of U ; this boun d is expresse d in terms of the exponen ts of the stable homotop y groups of spheres and the exponents of the stable homotopy groups of B ( Z / ( per ( α ))) . As a corollary , i f U is the spectrum of a fi eld of finite cohomo- logical dimension d , then eti ( α ) | pe r ( α ) ⌊ d 2 ⌋ , where ⌊ d 2 ⌋ is the integer part of d 2 , whene v er per ( α ) is div ided neither by the characteristic of k nor by an y primes that are small relativ e to d . Key W ords Brauer groups, twisted shea ves, higher algebraic K -theory , stable homotop y theory . Mathematics Subject Classification 2 000 Primary: 14F22 , 16K50 . Sec- ondary: 19D23 , 55Q10 , 55Q45 . 1 Introd uction Hypothesis 1.1. Througho ut this pape r , U den otes a con nected sch eme o f fin ite ´ etale cohomo logical d imension d having the prop erty that ev ery fin ite set of poin ts of U is contained in an affine o pen subscheme. For i nstance, any quasi-pro jectiv e scheme over a noetherian base satisfies this hypothesis. Definition 1 .2 (see Definition 6 .2 ) . For α ∈ H 2 ( U ´ et , G m ) , d efine eti ( α ) to be the positive generator of th e rank map K α, ´ et 0 ( U ) → Z , where K α, ´ et denotes α -twisted ´ etale K - theory defined in Definition 6.1 . ∗ This material is based upon work supported by the NSF under Grant No. DMS-0901373. 1 1 INTR ODUCTION 2 This paper is ded icated to proving the following theo rem, with the exception of the divisibility property , which is proven in [ 1 ]. Theorem 1.3. Let α ∈ H 2 ( U ´ e t , G m ) tors . Then, eti ( α ) has the following pr o perties: 1. computability : in the descent spectral sequ ence E s,t 2 = H s ( U ´ e t , K α t ) ⇒ K α, ´ et t − s ( U ) for α -twisted ´ etale K -theory , the inte ger eti ( α ) ∈ Z ∼ = H 0 ( U ´ et , K α 0 ) is th e smallest positive inte ger such that d α k ( eti ( α )) = 0 for all k ≥ 2 , wher e d α k is the k th differ ential in the s pectral sequence; 2. divisibility : p e r ( α ) | eti ( α ) , where per ( α ) is the or der of α in H 2 ( U ´ et , G m ) tors ; 3. obstruction : if A is an Azuma ya algebra in the class of α , then eti ( α ) | deg ( A ) , wher e deg ( A ) , the degr ee of A , is the positive square-r oot of the r ank of A ; 4. bound : if per ( α ) is prime to the characteristics of the r esidue fields of U , then eti ( α ) | Y j ∈{ 1 ,..., d − 1 } l α j . wher e l α j is the least common mu ltiple o f the exponents of π s j and π s j ( B Z / ( pe r ( α ))) . In particular , eti ( α ) is finite even if α is n ot r epr esen table by an Azumaya algebra. The first proper ty is shown in Lemma 6.4 . Th e obstruct ion pro perty is proven in Theorem 6.5 , and the bound property is established in Theorem 6.1 0 . An analysis of the in tegers l α j , tog ether with th e divisibility and bound p roperties ab ove and the fact that th e p eriod and index have the sam e p rime d i visors for Brauer classes on a field, giv es the following, Theor em 6.12 : Theorem 1.4 (Period- ´ Etale Index Theorem) . Let k be a field, a nd let α ∈ H 2 ( k , G m ) . Let S be th e set of prime div isors of per ( α ) , and suppose th at d = cd S k < 2 min q ∈ S ( q ) . Then, eti ( α ) | ( per ( α )) ⌊ d 2 ⌋ , wher e ⌊ d 2 ⌋ is the integer pa rt of d 2 . The theo rem should be viewed as a topological version of the period-ind ex conjec- ture, attributed to C olliot-Th ´ el ` ene . Conjecture 1.5 (Period-Index Conjectu re) . I f k is a field of dimension d , then ind ( α ) | ( per ( α )) d − 1 for a ll α ∈ Br( k ) , wher e i nd ( α ) is the squar e-r o ot o f th e rank o f th e uniq ue d ivision algebra r epr esenting α . 1 INTR ODUCTION 3 In the conje cture, the dimen sion might mea n eith er that k is C d , that k is the fu nc- tion field of a d -dimensional algebraic variety over an algebraically closed field, that k is the fu nction field of a ( d − 1) - dimensiona l v ariety over a fin ite field, or that k is th e function field of a ( d − 2) -dime nsional v ariety over a local field. It is no t known what the precise statement should be. Howe ver , the co njecture is known to be false if dimension is taken to be the cohom o- logical dime nsion of the field. For p rime powers l e and l f , with e ≤ f , a co nstruction of Merku rjev [ 19 ] can be used to constru ct a field k with cd l ( k ) = 2 , and a division algebra D over k with per ( D ) = l e and ind ( D ) = l f . For general backgroun d on the conjecture and its importance, see [ 1 6 ]. It is known to be true in the following cases, wh ere in fact the period and index coin cide: • p -adic fields, by class field theory; • number fields, by the Brauer-Hasse-Noether theo rem; • C 2 -fields, when per ( α ) = 2 a 3 b , by Artin and Harris [ 3 ]; • function fields k ( X ) o f alge braic surfaces X over an algebraically closed field k , by de Jong [ 10 ]; • quotient fields K of excellent henselian two-dimension al lo cal d omains with residue field k separably clo sed wh en α is a class of period prime to the char- acteristic of k , by Colliot-Th ´ el ` en e, Ojanguren, and P arimala [ 8 ]; • fields l (( t )) of transcend ence degree 1 over l , a ch aracteristic zero field o f coho - mological dimension 1 , by Colliot-Th ´ el ` en e , P . Gille, and Parimala [ 7 ]. The conjecture is also known in the following situation s. Saltma n [ 21 ] showed that ind ( D ) | per ( D ) 2 holds for d i vision algeb ras over the fu nction fields of curves over p -adic fields. Lieb lich, in [ 18 ] has shown that this is also true for the function fields o f sur faces over finite fields. Finally , Lieblich and Krashen have established in [ 16 ] the shar p relation ind ( D ) | per ( D ) d for the f unction fields of cu rves over d -local fields, such a s k (( t 1 )) · · · (( t d )) , w here k is algebraically closed. Moreover , in these examples, the exponent is th e best possible. Acknowledgments This paper is pa rt of my Ph.D. thesis, and I thank Henr i Gillet, my thesis advisor at UIC, as well as Da vid Gepner , Christian Haesemeyer, an d Brooke Shipley for discussions a nd suppo rt. Also, the referee of th e pape r has made a great number of suggestions resulting in the improvement of the exposition. 2 ST A CKS OF TWISTED SHEA VES 4 2 Stacks of twisted shea ves Proposition 2.1 (Artin [ 2 ]) . If U is a scheme such that every finite set of points is con- tained in some affine open subscheme, then the shea f cohom ology gr oup H 2 ( U ´ et , G m ) is co mputab le by covers ( instead o f hyper cover s); tha t is, ˇ H 2 ( U ´ et , G m ) ≃ → H 2 ( U ´ e t , G m ) . Remark 2.2 . The proposition e nsures that no in formatio n is lost by using o nly covers in the constructio ns an d theo rems below . Howe ver , at the expen se of anoth er level o f detail, all of the material in this paper can be m odified to app ly to any conn ected schem e of finite cohom ological dimension, p rovided that o ne uses 1 -h ypercovers instead of covers. Indeed , for any scheme U , the sma ll ´ e tale site U ´ et has fib er prod ucts and finite produc ts. Therefor e, by [ 4 , Theo rem V .7.4.1 ], H 2 ( U ´ e t , G m ) is compu table by 1 -hyp ercovers. T wo ´ etale stacks play a fun damental r ˆ ole in this pap er . For backgro und on stacks see [ 12 , Chapter 4]. The first stack is Pro j , the stack of locally fr ee finite rank coherent modules and isomorphisms. Thus an object in the category o f sections Pro j V on an ´ etale m ap V → U is a lo cally f ree fin ite ran k co herent O V -modu le. For brevity , such an object will be called a lffr sheaf. Fix a positi ve integer n . The second stack is the stack nSets of s heaves of finite and faithful µ n -sets, where µ n is th e sheaf of n th roots of unity in th e ´ etale top ology . The category of sections nSets V consist of sh eav es F with a faithfu l action of µ n | V such that F decomposes into finitely many orbits. Objects will b e called µ n -sheaves. The morph isms are isomorph isms. Every object of nSets V is a disjoint sum of µ n -torsors. Ther e is a m ap o f stacks, the u nit mo rphism, i : nSets → Pro j o btained b y sen ding a µ n -torsor to the associated G m -torsor, and then taking the sheaf of sections. Disjoint sums are taken to direct sums. Definition 2.3. Let α ∈ H 2 ( U ´ et , G m ) , and suppo se that U = ( U i ) i ∈ I is an ´ etale cover such that α comes from the ˇ Cech cocycle ( α ij k ) , where each α ij k ∈ Γ( U ij k , G m ) . An α -twisted coh erent O U -modu le consists of a c oheren t O U i -modu le F i for e ach i ∈ I , together with isomor phisms θ ij : F i | U ij ≃ → F j | U ij such that θ ki ◦ θ j k ◦ θ ij = α ij k ∈ G m ( U ij k ) . For properties of α -twisted sheav es, see [ 17 ] or [ 9 ]. The locally free and finite rank α -twisted coherent O U -modu les natur ally gi ve rise to a stack Pro j α , where the sections over V → U are the α | V -twisted lf fr sheaves . Lemma 2.4. Let α ∈ H 2 ( U ´ e t , G m ) . If V → U is ´ etale and V is co nnected, then ther e is a n A zumaya a lgebra of rank n 2 r epr e senting α | V if and only if ther e is a α - twisted lffr sheaf of r ank n in Pro j α V . Pr o of. See [ 17 , Proposition 3.1.2.1 ]. Lemma 2.5. Let α ∈ H 2 ( U ´ et , G m ) , an d let V → U be an ´ etale map. If α | V is trivial, ther e is an α -twisted lffr rank 1 sheaf in Pro j α V . Pr o of. This follows from [ 17 , Proposition 3.1.2.1.iv]. Similarly , if β ∈ H 2 ( U ´ et , µ n ) , then th ere is a twisted fo rm nSets β of nSets constructed in the same way as Pro j α is in Definition 2.3 . 3 K-THEOR Y 5 Lemma 2.6. If H 2 ( U ´ et , µ n ) → H 2 ( U ´ e t , G m ) send s β to α , th en the unit ma p i : nSets → Pro j twists to give a twisted unit map i β : nSets β → Pro j α . Pr o of. Suppose f or simplicity that β is defined on the cover U = ( U i ) i ∈ I by β ij k ∈ µ n ( U ij k ) . If F is a β -twisted µ n -set, then F i = F | U i is a µ n -set fo r all i ∈ I , and there are isomo rphisms θ ij : F i ≃ → F j . Thus, i ( F i ) is a lffr shea f, a nd i ( θ ij ) give isomorph isms i ( F i ) ≃ → i ( F j ) such th at i ( θ ki ) ◦ i ( θ j k ) ◦ i ( θ ij ) = β ij k , where now β ij k is v iewed a s a 2 -cocycle in G m , which is by hypo thesis cohomolo gous to α . Th us i ( F i ) and i ( θ ij ) g i ve the data of an α -twisted lffr sheaf. The details are left to the reader . Both stacks nSets β and Pro j α are stacks of symm etric mono idal c ategories in the following sen se. Each category of sections is a symm etric mono idal category , und er disjoint union and d irect sum respec ti vely , and the restriction is compa tible with this structure. 3 K-theory Definition 3.1. There is a functor K : S y mM on → Spt , from the category of symmetric monoidal categories an d lax functors to spectra. For details, see [ 23 , Section 1.6 ]. This K - theory is always co nnective. If T is a symmetr ic monoid al category , let K n ( T ) = π n ( K ( T )) for n ∈ Z . Example 3.2. If R is a commutative ring, and if Pro j R is the symmetric monoidal cat- egory o f finitely generated projective R -modu les and isomorp hisms, with direct sum, then K ( Pro j R ) agrees with Qu illen’ s hig her algeb raic K -theo ry of R [ 1 3 ]. I n p artic- ular , K 0 ( R ) is the usual Grothe ndieck group of R . Similarly , if X is a scheme, a nd Pro j X is the categor y o f locally free and finite ran k O X -modu les. Th en the Quillen Q -constru ction Q Pro j X of P ro j X has a natur al structu re of symmetric monoidal cat- egory under direct sum. Quillen’ s high er algeb raic K -th eory of X ag rees w ith the homoto py of Ω K ( Q Pro j X ) . Definition 3.3. F or β ∈ H 2 ( U ´ e t , µ n ) , let T β denote the presheaf of spectra V 7→ K ( nSets β V ) . Define T β k ( V ) = π k T β ( V ) , and let T β k be the sheafification of T β k . 3 K-THEOR Y 6 Definition 3.4. Similarly , for α ∈ H 2 ( U ´ e t , G m ) , let K α be the preshea f of spectra V 7→ K ( Pro j α V ) , with associated homotopy presheaves K α k ( V ) = π k K α ( V ) , and presheaves K α k . Remark 3.5 . Note that the pre sheaf of spectra K α is in some sen se th e wr ong cho ice of presh eaf. Th e correct version would be to take Tho mason-Trobaugh K -theor y [ 24 ]. Howe ver , all of the com putations in this paper h ave to d o with th e ´ etale sheafification of K α . Since the two versions agr ee on affine schemes, it f ollows that their ´ eta le sheafifications are isomorph ic in the h omotopy cate gory . If β 7→ α in H 2 ( U ´ e t , µ n ) → H 2 ( U ´ e t , G m ) , then the twisted unit morphism i β of Lemma 2.6 gives a morp hism of preshea ves of spectra K ( i β ) : T β → K α . This map is crucial to the proo f o f the bound property of the ´ etale index. Lemma 3.6. Let β ∈ H 2 ( U ´ e t , µ n ) . The n, the stalk of T β j at a geometric point x → U is naturally isomorphic to π s j B ( µ n ( k ( x ))) ⊕ π s j , wher e k ( x ) is the (separably closed ) r esidue field of x , π s j is the j th stable homoto py gr o up of S 0 , and B G den otes the topological classifying space of a gr oup G . Pr o of. It is enough to study the stalk ( T β j ) x , as this is isom orphic to ( T β j ) x . Sin ce the K - theory functor preserves filtered colimits, because the classifying space construction does, ( T β j ) x ∼ = colim x ∈ V → U T β j ( V ) = colim x ∈ V → U K j ( nSets β V ) ∼ = K j  colim x ∈ V → U nSets β V  . But, colim x ∈ V → U nSets β V is equi valent, by the arguments of [ 14 , EGA IV 8.5], to the category of finite and f aithful µ n ( O sh U, x ) ∼ = µ n ( k ( x )) -sets. Ther efore, ( T β j ) x ∼ = K j ( nSets x ) , where nSets x is the symmetric mo noidal category of finite and faithfu l µ n ( k ( x )) -sets and isomorphisms. This category is a groupoid equiv alent to a j ≥ 0 S j ≀ µ n ( k ( x )) , where S j is the sym metric g roup on j letters, and S j ≀ µ n is the wre ath pr oduct. T he notation means that the stalk is equiv alent to the g roupo id with conne cted components indexed by j ≥ 0 , where the automo rphism gr oup of an object in the j th com ponen t is S j ≀ µ n ( k ( x )) . 4 ST ABLE HOMOT OPY OF C LASSIFYING SP AC ES 7 Therefo re, by the Bar ratt-Priddy -Quillen-Kah n t heorem (see Thom ason [ 25 , Lemma 2 .5]), the K -theor y spectrum of this symm etric monoid al category is weak equ iv alent to the suspension spectru m Σ ∞ ( B µ n ( k ( x ))) + of the classifying space of B µ n ( k ( x )) with a disjoint b asepoint. This spectr um is weak ly equiv alent to Σ ∞  B µ n ( k ( x )) ∨ S 0  . This comp letes the p roof. If n is prime to the characteristic of k ( x ) , then µ n ( k ( x )) ∼ = Z / ( n ) . Oth erwise, let m b e the largest divisor of n that is prime to the ch aracteristic. Then , µ n ( k ( x )) ∼ = Z / ( m ) . 4 Stable homotopy of classifying spaces Proposition 4.1. Let 0 < k < 2 p − 3 . The n, the p - primary comp onent π s k ( p ) of π s k is zer o. An d, π s 2 p − 3 ( p ) = Z / ( p ) . Pr o of. This follows from th e c omputatio n of th e image of the J -mo rphism (see [ 20 , Theorem 1.1.1 3]) an d, for example, [ 20 , Theorem 1.1.14]. I thank Peter Bousfield for telling me about the next proposition. Proposition 4.2. F or 0 < k < 2 p − 2 , the stable ho motopy gr oup π s k ( B Z / ( p n )) is isomorphic to Z / ( p n ) for k odd and 0 for k even. Pr o of. Let p be a prime. Recall the stable splitting of Holzsager [ 15 ] Σ B Z / ( p n ) ≃ → X 1 ∨ · · · ∨ X p − 1 , where, if k > 0 , the redu ced homology of X m is ˜ H k ( X m , Z ) ≃ → ( Z / ( p n ) if k ∼ = 2 m mod 2 p − 2 , 0 otherwise. Define C m as the cofiber of M 1 → X m , where M 1 = M ( Z / ( p n ) , 2 m ) is the M oore space with ˜ H k ( M 1 , Z ) ≃ → ( Z / ( p n ) if k = 2 m , 0 otherwise, when k > 0 . The homolo gy of C m is ˜ H k ( C m , Z ) ≃ → ( Z / ( p n ) if k > 2 m an d k ∼ = 2 m mo d 2 p − 2 , 0 otherwise. 4 ST ABLE HOMOT OPY OF C LASSIFYING SP AC ES 8 Therefo re, the ma p M 2 = M ( Z / ( p n ) , 2 m + 2 p − 2) → C m is a (2 m + 4 p − 5) -equivalence. Thus, for k < 2 m + 4 p − 5 (r esp. k = 2 m + 4 p − 5 ), the map π s k ( M 2 ) → π s k ( C m ) is an isomorph ism (resp. su rjection). Therefor e, ther e is an exact sequence π s 2 m +4 p − 5 ( M 2 ) → π s 2 m +4 p − 6 ( M 1 ) → π s 2 m +4 p − 6 ( X m ) → π s 2 m +4 p − 6 ( M 2 ) → · · · → π s k ( M 1 ) → π s k ( X m ) → π s k ( M 2 ) → · · · (1) Let M ( Z / ( p n )) be the Mo ore spectrum. It is the cofiber of the multiplication by p n map on the sphere spectrum S . Thus, its stab le ho motopy gro ups fit into exact sequences 0 → π s k ⊗ Z Z / ( p n ) → π k ( M ( Z / ( p n ))) → T or Z 1 ( π s k − 1 , Z / ( p n )) → 0 . These sequen ces are in fact split wh en p is odd or when p = 2 and n > 1 . The Moor e spaces M 1 and M 2 are the le vel 2 m and (2 m + 2 p − 2) spaces o f M ( Z / ( p n )) . Thus, π s k ( M 1 ) = π k − 2 m ( M ( Z / ( p n ))) π s k ( M 2 ) = π k − 2 m − 2 p +2 ( M ( Z / ( p n ))) . By Prop osition 4.1 , the first p -torsion in π s k is a copy of Z / ( p ) in degree k = 2 p − 3 . Therefo re, the first two no n-zero stable homotopy gr oups of M 1 and M 2 are π s 2 m ( M 1 ) = Z / ( p n ) π s 2 m +2 p − 3 ( M 1 ) = Z / ( p ) π s 2 m +2 p − 2 ( M 2 ) = Z / ( p n ) π s 2 m +4 p − 5 ( M 2 ) = Z / ( p ) . Using th e exact sequen ce ( 1 ), it fo llows that the first non -zero stable homotopy grou p of X m is π s 2 m ( X m ) = Z / ( p n ) . The next potentially non-zero stable hom otopy gro up fits into the exact sequence ( 1 ) at degree 2 m + 2 p − 3 : Z / ( p n ) → Z / ( p ) → π s 2 m +2 p − 3 ( X m ) → 0 . It follows that π s k (Σ B Z / ( p n )) = ( Z / ( p n ) if 0 < k < 2 p − 1 and k is even, 0 if 0 < k < 2 p − 1 and k is odd . The theorem follows immediately . 5 HOMO TOPY S HEA VES ARE ISOMORPHIC 9 Corollary 4.3. If, Z / ( n ) = M q | n Z / ( q v q ( n ) ) , wher e q ranges over the prime divisors of n , then , fo r 0 < k < 2 min q | n ( q ) − 2 , π s k ( Z / ( n )) ∼ = Z / ( n ) whe n k is od d and π s k ( B Z / ( n )) = 0 when k is even. Pr o of. This follows from the prop osition, since B G ∼ → ∨ q | n B Z / ( q v q ( n ) ) . Corollary 4.4. Denote by m j the exponent of the finite abe lian gr o up π s j for j ≥ 1 . If β ∈ H 2 ( U ´ e t , µ n ) , then, for 0 < j < 2 min q | n ( q ) − 2 , the cohomology gr ou p H k ( U ´ e t , T β j ) is ann ihilated by n · m j when j is od d and by m j when j is even. Pr o of. The stalk of T β j at x → U is isomorp hic to π s j ( B µ n ( k ( x ))) ⊕ π s j . But, µ n ( k ( x )) ∼ = Z / ( m ) , wher e m is the largest divisor of n pr ime to the charac teristic of k ( x ) . The corollar y now follows from th e computation of Corollary 4.3 . 5 Homotopy shea ves ar e is omorphic Proposition 5.1. Fi x an element α ∈ H 2 ( U ´ et , G m ) . Then, for all n ≥ 0 , the homotopy sheaves K α and K are n aturally isomorphic. Similarly , if β ∈ H 2 ( U ´ et , µ n ) , then T β ∼ = T . Pr o of. Here is a proo f for the case of α ∈ H 2 ( U ´ e t , G m ) . The proof of the other case is identical. Let U = ( U i ) i ∈ I → U be a cover over which α is trivial (this is possible by the local triviality of sheaf co homolo gy). Then, by Lemm a 2.5 , there are α -twisted line bundles L i on each U i . These define equ i valences of stacks θ i : Pro j | U i → Pro j α | U i for all i given by θ i ( V )( P ) = L i ⊗ P , when V → U i . These equ iv alences ind uce point-wise we ak equivalences of K -th eory presheaves: θ i : K | U i → K α | U i . This means that for all ´ etale map s V → U i , ( θ i ) | V : K | V → K α | V 6 THE PERIOD-INDEX PR OBLEM 10 is a weak equiv alence. It fo llows th at on U i there ar e isomor phisms of homotopy presheaves: θ i : ( K n ) | U i ≃ → ( K α n ) | U i . In fact, the θ i glue at the le vel of homotopy sheaves. It suffices to check that, on U ij = U i × U U j , the auto -equivalence of Pro j | U ij giv en by tensoring by M ij = L − 1 i ⊗ L j is lo cally ho motop ic to th e iden tity . But, there is a trivialization of M ij , over a cover V of U ij . So, o n each elem ent V of V , the re is an iso morph ism σ V : O U V ≃ → M ij | V . This in duces a natural transformatio n f rom the identity to θ − 1 i ◦ θ j on V . But, th e K -functo r takes natural transformations to ho motop ies of maps of spectr a. So, on V , θ i | V = θ j | V : ( K n ) | V → ( K α n ) | V . It follows that the θ i glue to g iv e isomorphisms of sheaves θ : K n ≃ → K α n , as desired. 6 The period-ind ex probl em Definition 6.1. Let K α, ´ et (resp. T β , ´ et ) deno te the ´ etale sheafificatio n of K α (resp. T β ) with respect to the local mo del structure on preshe av es of spectra . Th is is the model stru cture in which co fibrations ar e given by cofibra tions of spectr a in the sense of Bousfield and Friedlan der [ 6 ], an d weak equiv alences a re morph isms that induce isomorph isms of all homotopy shea ves. Since U is of finite coh omolo gical d imension, specific models are gi ven by Tho mason [ 26 , Defin ition 1.33]. There are conv ergent spectral sequences, called Brown-Gersten or descen t spectral sequences, E s,t 2 = H s ( U ´ e t , K α t ) ⇒ K α, ´ et t − s ( U ) (2) E s,t 2 = H s ( U ´ e t , T β t ) ⇒ T β , ´ et t − s ( U ) (3) with differentials d α k of degree ( k, k − 1) ; see [ 26 , Proposition 1.36]. Definition 6.2. Let α ∈ H 2 ( U ´ e t , G m ) tors . Define th e ´ etale ind ex of α , eti ( α ) , to be the positiv e generator of the image of the edge map (o r rank map) K α, ´ et 0 ( U ) → H 0 ( U ´ e t , K α 0 ) ∼ = Z in the descent spectral sequence. Remark 6.3 . The map of presheaves K α, ´ et 0 → Z is ca lled the ran k map becau se the composite K α 0 → K α, ´ et 0 → Z is th e usu al ran k map on the preshea f of α - twisted Grothend ieck gro ups. Lemma 6 .4 ( Computability ) . Let α ∈ H 2 ( U ´ et , G m ) tors . Then , e ti ( α ) is the un ique smallest positive inte ger in H 0 ( U ´ e t , K α 0 ) ∼ = Z such that d α k ( eti ( α )) = 0 for all k ≥ 2 . Pr o of. This follows imme diately from the co n vergence o f th e descen t spectr al sequ ence ( 2 ). 6 THE PERIOD-INDEX PR OBLEM 11 Lemma 6.5 ( Obstruction ) . F o r α ∈ H 2 ( U ´ et , G m ) tors , eti ( α ) | deg ( A ) for any Azumaya algebra A in the class of α . Pr o of. Suppose that A is in the class of α and that m = deg ( A ) . Then , by Lemma 2.4 , there is an α -twisted lffr sheaf o f r ank m . Hence, m is in the image of rank : K α 0 ( U ) → Z . Since the rank hom omorp hism factors through K α, ´ et 0 ( U ) → Z , th e lemma follows fro m the definition of the ´ etale index. Theorem 6.6 ( Divisibili ty [ 1 ]) . F or α ∈ H 2 ( U ´ e t , G m ) tors , per ( α ) | eti ( α ) . Example 6.7 . I f D is a cyclic division algebra ( x, y ) ζ n over a field o f characteristic prime to n , so that per ( D ) = i nd ( D ) = n , then eti ( D ) = n . Example 6. 8. If D /k is a d ivision algebra, and if l/ k is a finite separab le field exten- sion of d egree p rime to p er ( D ) , the n a standard argu ment using norm maps says that eti ( D l ) = e ti ( D ) . Example 6.9. Le t Q be the non-separated quadric with α the no n-zero cohomolo gical Brauer class [ 11 ]. Then per ( α ) = eti ( α ) = 2 , wh ile ind ( α ) = + ∞ . Denote b y m j the exponent of π s j , the j th stab le h omotopy grou p of S 0 , and let n α j denote the exponen t of π s j ( B Z / ( pe r ( α ))) . Fina lly , let l α j denote the exponen t of π s j ⊕ π s j ( B Z / ( pe r ( α ))) . So, l α j is the least common multiple of m j and n α j . Theorem 6.10 ( Bound ) . Let U b e a conn ected scheme of cohomologica l d imension d . Let α ∈ H 2 ( U ´ et , G m ) tors be such th at per ( α ) is p rime to the characteristic of all r esidue fields of U . Then , eti ( α ) | Y j ∈{ 1 ,..., d − 1 } l α j . Pr o of. Because of the assumption o n pe r ( α ) and the residue character istics of U , the sequence of sheaves 1 → µ per ( α ) → G m per ( α ) − − − − → G m → 1 is exact. Th us, there is a lift β of α in H 2 ( U ´ et , µ per ( α ) ) . Th ere is a mor phism of descent spectral sequences [ 26 ] H s ( U ´ e t , T β t ) → H s ( U ´ et , K α t ) induced by K ( i β ) : T β → K α . Let d β k denote th e k th d ifferential in the de scent spec- tral sequence for T β . As the class 1 ∈ H 0 ( U ´ e t , T β 0 ) maps to the c lass 1 ∈ H 0 ( U ´ e t , K α 0 ) , if d β k ( m ) = 0 for 2 ≤ k ≤ k ′ , then d α k ( m ) = 0 for 2 ≤ k ≤ k ′ . The differential d β k lands in a su bquotien t of H k ( U, T β k − 1 )) . Therefore, d β k lands in a g roup of exponen t at most l α k − 1 , by Corollary 4.4 . Sin ce the she av es T β k are to rsion fo r k > 0 , the d ifferen- tials d β k vanish fo r k > d . 6 THE PERIOD-INDEX PR OBLEM 12 Definition 6.11. Let K be a field, and let S be a non-emp ty set of primes. Let cd S k be the supremum of all the cohomo logical dimen sions cd q k fo r all primes q ∈ S . Theorem 6.12. Let K b e a fi eld, and let α ∈ Br( K ) = H 2 ( K, G m ) be such that n = per ( α ) is prime to the characteristic of K . Let S be the set of prime divisors of n , and suppose that d = cd S k < 2 min q ∈ S ( q ) . Then , eti ( α ) | ( per ( α )) ⌊ d 2 ⌋ . Pr o of. Set c = ⌊ d 2 ⌋ . Combin ing Theo rem 6.10 a nd Cor ollary 4. 4 , it follows that, if d is ev en, then d β k ( an c ) = 0 for all k ≥ 2 , wher e a is prime to n . The same reasonin g sho ws that if d is odd, then d β k ( an c ) = 0 when 2 ≤ k ≤ d − 1 . By [ 22 ], the stalks of K α 2 j are torsion -free for j > 0 . Therefo re, the maps H m ( K, T 2 j ) → H m ( K, K 2 j ) are zero for j > 0 and all m . I t follows that if d β k ( m ) = 0 f or 2 ≤ k ≤ 2 j , the n d α k ( m ) = 0 for 2 ≤ k ≤ 2 j + 1 . The refore, when d is odd, d α k ( an c ) = 0 for 2 ≤ k ≤ d and he nce for all k ≥ 2 . Thus, eti ( α ) | an c , where a is relatively prime to n . On the other han d, as K is a field, the primes di visors of per ( α ) and eti ( α ) are the same since eti ( α ) | ind ( α ) . So, eti ( α ) | n f for some positi ve integer f . It follows that eti ( α ) | n min( c,f ) | n c . This comp letes the p roof. The cond ition d < 2 min S ( q ) excludes n o primes f or function fields of cu rves, surfaces, o r three-folds. It excludes the prime 2 for fu nction fields of four-folds an d fi ve-folds. The bound prope rty and the metho d o f the proo f of Theorem 6.12 can be used to giv e bou nds on eti ( α ) whenever the stable homotopy is known in a suf ficiently large range. But, the exponent ⌊ d 2 ⌋ will no lo nger suf fice ( with this method). For instance, if k is such that cd 2 k = 4 and k is not characteristic 2 , then for any α ∈ B r( k ) of per ( α ) = 2 , these a rguments gi ve eti ( α ) | per ( α ) 4 . T he extra factor of per ( α ) 2 comes from the fact that π s 3 = Z / (24) . REFERENCES 13 Let K α 0 ( X ) (0) = K α 0 / ker  K α 0 ( X ) r ank − − − → Z  K α, ´ et 0 ( X ) (0) = K α, ´ et 0 / ker  K α, ´ et 0 ( X ) r ank − − − → Z  . When α is trivial, the natural inclusion K α 0 ( X ) (0) → K α, ´ et 0 ( X ) (0) (4) is an isomorph ism. Corollary 6.13 . The ma p of Equatio n ( 4 ) is not su rjective in general when α is no t trivial. Pr o of. F or e xample, let k ( C ) be the function field of a cur ve over a p -adic field. Jacob and Tignol have sho wn in an appendix of [ 21 ] th at ther e are division algebras over k ( C ) for which ind ( α ) = per ( α ) 2 . Howe ver , since these fields are of co homolo gical dimension 3 , it follows that eti ( α ) = p er ( α ) . Thu s, th e map is n ot surjective for X = Sp ec k ( C ) . Conjecture 6.14. Let k = C (( t 1 )) · · · (( t d )) be an iterated Lau r ent series field over the complex numbers. Th en, for α ∈ Br ( k ) , eti ( α ) = ind ( α ) . One reason to believ e th is co njecture is that f or d -lo cal fields k , Becher a nd Ho ff- man ha ve established [ 5 ] that the index s atisfies ind ( α ) | per ( α ) ⌊ d 2 ⌋ , for all α ∈ B r( k ) . Refer ences [1] Benjamin Antieau, ˇ Cech appr oximation to the Br own-Gersten spectral sequence , submitted, http://arxiv .org/abs/09 09.3786 , 2010 . 1 , 6.6 [2] M. Artin, O n the joins of Hen sel rings , Advances in Math . 7 (1971) , 282–2 96 (1971 ). MR 02895 01 2.1 [3] , Brauer-Severi varieties , Brauer grou ps in ring th eory and algebraic ge - ometry (Wilrijk, 1 981) , Lecture Notes in Math ., vol. 917, Springer, Berlin, 1 982, pp. 194–21 0. MR65743 0 1 [4] M. Artin, A. Grothend ieck, an d J. L. V er dier (eds.), Th ´ eorie des topos et coho - mologie ´ etale des sch ´ emas. Tome 2 , Lecture Notes in Ma thematics, V ol. 27 0, Springer-V erlag, Berlin, 19 72, S ´ em inaire d e G ´ eo m ´ etrie Alg ´ ebriqu e du Bois- Marie 1963–19 64 (SGA 4 ), Dirig ´ e par M. Artin, A. Grothendieck et J. L. V erdier . A vec la collaboration de N. Bourb aki, P . Delign e et B. Saint-Do nat. MR03546 53 2.2 REFERENCES 14 [5] Karim Jo hannes Bech er an d Detlev W . Hoffmann, Symbol len gths in Milno r K - theory , Homolog y Homotopy Appl. 6 (20 04), n o. 1, 17–31 (electro nic). MR20615 65 6 [6] A. K. Bousfield and E . M. Friedland er , Homotop y th eory of Γ -spac es, spectra, and b isimplicial sets , Geom etric applications of h omotopy theo ry ( Proc. Conf., Evanston, Ill., 1977 ), II , Lecture Notes i n Math., v ol. 658, Springer , Berlin, 19 78, pp. 80–130 . MR51356 9 6.1 [7] J.-L. Colliot-Th ´ el ` ene, P . Gille, and R. Parimala, Arithmetic o f linear a lgebraic gr o ups over 2-dimension al geometric fi elds , Duke Math. J. 121 (2004 ), no . 2, 285–3 41. MR20 3464 4 1 [8] J.-L. Co lliot-Th ´ e l ` ene, M. Ojanguren , and R. Parimala, Quad ratic forms over fraction field s of two-dimensiona l Henselia n ring s a nd Brauer gr oups of r elated schemes , Alg ebra, arith metic an d ge ometry , Part I , II (M umbai, 20 00), T ata In st. Fund. Res. Stud. Math., vol. 16, T ata Inst. Fun d. Res., Bom bay , 2 002, pp . 1 85– 217. MR19406 69 1 [9] Andrei C ˘ ald ˘ araru , Derived cate gories of twisted sheav es on Calabi-Y a u manifolds , Ph.D. thesis, Cor nell University , May 20 00, http://www.m ath.wisc.ed u/˜andreic/ . 2.3 [10] A. J. de Jon g, Th e period- index pr oblem for the Brauer gr oup of an a lgebraic surface , Duke Math. J. 123 (2004), no. 1, 71–94. MR2060023 1 [11] Dan Edid in, Brendan Hassett, Andr ew Kresch, and An gelo V istoli, Brauer gr ou ps and quotien t stacks , Amer . J. Math. 123 (2001), no. 4, 7 61–7 77. MR184457 7 6.9 [12] Barbara Fantechi, Lothar G ¨ ottsche , Luc Illusie, Ste ven L. Kleiman, Ni tin Nitsure, and Ang elo V istoli, Fundam ental algebraic geo metry , Mathe matical Surveys and Monog raphs, vol. 12 3, American Mathematical So ciety , Providen ce, RI, 20 05, Grothend ieck’ s FGA e xplained. MR22226 46 2 [13] Daniel Grayson, High er algebraic K -theo ry . II ( after Daniel Quillen) , Alge- braic K -theory (Proc. Conf., North western Uni v ., Evanston, Ill., 1976), Springer, Berlin, 1976, pp. 217–2 40. Lectu re Notes in Math., V ol. 551. MR05740 96 3.2 [14] A. Groth endieck, ´ El ´ ements d e g ´ eom ´ etrie alg ´ ebrique. IV. ´ Etude locale des sch ´ emas et d es morp hismes de sch ´ emas. III , Inst. Ha utes ´ Etudes Sci. Publ. M ath. (1966 ), no. 28 , 255. MR02170 86 3.6 [15] Richard Holzsager, Stable splitting o f K ( G, 1) , Pr oc. Ame r . M ath. Soc. 31 (1972 ), 305–306. MR02 8754 0 4.2 [16] Max Lieblich, P eriod and inde x i n the br auer gr oup of an arith- metic surface (with an appen dix by Daniel Krashen) , 2007, http://arxiv .org/abs/ma th/0702240 . 1 REFERENCES 15 [17] , T wisted sheav es an d the perio d-index p r ob lem , Compos. Math. 1 44 (2008 ), no. 1, 1– 31. MR238855 4 2.3 , 2.4 , 2.5 [18] , The period- index p r o blem for fields of transcendence degr ee 2 , 200 9, http://arxiv .org/abs/09 09.4345 . 1 [19] A. S. Me rkurjev , Kap lansky’ s conjecture in the theory of qu adratic forms , Zap. Nauch n. Sem. Le ningrad . Otd el. Mat. Inst. Steklov . (LOMI ) 175 (1 989), no. K oltsa i Moduli. 3, 75–89, 163–16 4. MR1047 239 1 [20] Douglas C. Ravenel, Complex cobor dism and stable homotopy gr oups of sphe r es , Pure and App lied M athematics, vol. 121, Acad emic Pr ess I nc., Orlando , FL, 198 6. MR86004 2 4.1 [21] Da vid J. Saltman, Division algebr as over p -a dic curves , J. Ramanujan Math . Soc. (1997 ), no. 1, 25 –47. MR1462850 1 , 6.13 [22] Andrei A. Su slin, On th e K -theory of local fi elds , Proceedin gs of the L uminy confere nce on alg ebraic K -theo ry (Lum iny , 19 83), v ol. 34, 1984, pp. 3 01–3 18. MR77206 5 6.12 [23] R. W . T homason , Symmetric mon oidal cate gories model all connective spectra , Theory Appl. Categ. 1 (1995), No. 5, 78–118 (electronic). MR1337494 3. 1 [24] R. W . Thomason and Th omas Trobaugh, Highe r a lgebraic K - theory of schemes and of derived ca te gories , Th e Groth endieck Festschrift, Vol. III, Pro gr . Math., vol. 88, Birkh ¨ auser Boston, Boston, MA, 1990, pp. 247–4 35. MR11069 18 3.5 [25] Robert W . Tho mason, F irst qua drant spectral sequences in algebraic K - theory via h omotop y co limits , Comm. Alg ebra 10 (1 982) , n o. 15, 1589– 1668 . MR66858 0 3.6 [26] , Algebraic K - theory and ´ etale coho mology , Ann. Sci. ´ Ecole Norm. Su p. (4) 18 (1985) , no . 3, 437–5 52. MR82 6102 6.1 , 6.1 , 6.10 Benjamin Antieau [ antieau@mat h.ucla.edu ] UCLA Math Departmen t 520 Portola Plaza Los Angeles, CA 90095 -155 5