Cech approximation to the Brown-Gersten spectral sequence

ˇ Cech approxim ation to the Bro wn-Gerst en spectral sequence Benjamin Antieau ∗ April 7, 2022 Abstract In this pa per , we show that the ´ etale inde x of a torsion cohomologica l Brauer class is di visible by the period of the class. The tool used to make this comp utation is the ˇ Cech ap proximation of the title. T o create the approximation, we use the folklore theorem that the homotop y li mit and Postnikov spectral sequen ces for a cosimplicial space ag ree be ginning with the E 2 -page. As far we kno w , this folklore theorem has no proof in the literature, so we include a proof. Key W ords Homotop y spectral sequences, cosimplicial spaces, Brauer groups, and twisted sheav es. Mathematics Subject C lassification 2010 Primary: 14F22 , 18G40 . Sec- ondary: 16K50 , 19D23 . Contents 1 Introduction 2 2 Spectral Sequences Associated to T owers of Fibrations 4 3 Cosimplicial Spaces 7 4 Spectral Sequences for Cosimpli cial Spa ces 9 5 The ˇ Cech Appr oximat ion 19 6 Differentials in the Holim Spectral Sequence 22 7 Description of d 2 for the ˇ Cech Spectral Sequence 25 8 Divisibility Theorem 25 ∗ This material is based upon work supported by the NSF under Grant No. DMS-0901373. 1 1 INTRODUCTION 2 9 Index and Spectral Index 30 1 Introd uction Let X be a g eometrically conn ected scheme, and let B r ′ ( X ) = H 2 ( X ´ et , G m ) tors be the cohom ological Brauer group of X . There are two in teger invariants of a Brauer class α ∈ Br ′ ( X ) . The first is the perio d o f α ; it is the order of α in the group Br ′ ( X ) , and it is generally written as per ( α ) . The second is the index of α . If A is an Azumaya algebra on X , then it has ran k n 2 as a lo cally free O X sheaf. The integer n is called the ind ex of A . F or a class α , its index ind ( α ) is defined as th e least integer n such that there is an Azumaya algebra A in the class of α of ind ex n . If no such Azum aya algebra exists, then the index is defin ed to be + ∞ . In general, per ( α ) | ind ( α ) . (1) If X = Sp ec k for a field k , then the ind ex is finite, a nd the p eriod and th e index have the same prime divisors. Therefore, there is a least integer e ( α ) such that ind ( α ) | ( per ( α )) e ( α ) . As α r anges over all Brauer classes, it is a n interesting o pen question to determine the values that e ( α ) can take on. In [ 6 ], Colliot-Th ´ el ` ene ask s whether the follo wing is true. Conjecture 1.1 (Period-In dex Conjecture) . Let k be a field of dimension d > 0 . The n, ind ( α ) | ( per ( α )) d − 1 , (2) for all α ∈ Br( k ) . The no tion of dimen sion in the con jecture is not entire ly settled. The co njecture is made specifically [ 6 ] f or func tion fields of d -d imensional varieties over algebr aically closed fields, or for C d fields. But it is k nown to be false for cohom ological dimen sion. Sev eral cases in low dim ensions are kn own, but the question is open in gener al. For example, it is no t known for any fun ction field C ( X ) of an algebraic 3 -fold X . The impo rtance o f the conjectur e for k ( X ) is that it gives infor mation both about the geometry o f X and about the arith metic o f k ( X ) when X is an algebraic variety over an algebraically closed field k . In [ 1 ], we intro duced a new in variant eti ( α ) for α ∈ B r ′ ( X ) , which we call the ´ etale i nd ex. By the definition, eti ( α ) | ind ( α ) . The main theor em of that pap er is an analogu e of Equ ation ( 2 ) wh en X = Spec k . Specifically , if k is a field of finite cohom ological dimension d = 2 c or d = 2 c + 1 , then eti ( α ) | ( per ( α )) c (3) when the prime di visors of per ( α ) are “large” with respect to d . Specifically , the statement is true when d < 2 q for all prim es q that d i vide per ( α ) . 1 INTRODUCTION 3 The m ain result of this paper, Theorem 8.7 , is that the analogu e of Equa tion ( 1 ) holds for the ´ etale index as well: per ( α ) | eti ( α ) . In particular, it is not equal to 1 if α is non-trivial. Therefore, the ´ etale i nd ex possesses similar for mal pr operties to the in dex. Howe ver , the ´ etale ind ex is always fin ite f or schemes X of finite ´ etale cohom ological dime nsion, while certain Brauer classes on non-q uasi-projective scheme s ha ve infinite index. Here is an applicatio n o f Theorem 8.7 . Define K α 0 ( k ) (0) = K α 0 ( k ) / ker (rank : K α 0 ( k ) → Z ) K α, ´ et 0 ( k ) (0) = K α, ´ et 0 ( k ) / ker  rank : K α, ´ et 0 ( k ) → Z  , where K α is twisted co nnective K -the ory , a nd K α, ´ et is the ´ etale sheafification. Th e image of the rank on K α 0 ( k ) is gener ated by ind ( α ) , while th e image of th e r ank on K α, ´ et 0 is generated by eti ( α ) . There is an inclusion K α 0 ( k ) (0) ⊆ K α, ´ et 0 ( k ) (0) (4) with cokernel F = Z /  ind ( α ) eti ( α )  . In the untwisted c ase ( α = 0 ), t he cokernel is always zero. On the other hand , Eq uation ( 3 ), together with the sharpn ess of some k nown cases of the period- index conje cture, imply that F is in general not zero when α 6 = 0 . Suppose that the period- index conjecture of Colliot- Th ´ el ` ene is true. Then , the fact that per ( α ) | eti ( α ) implies th at the cokern el F is o f order at most pe r ( α ) d − 2 . Co n- versely , if one could prove that the co kernel was of order a t m ost per ( α ) c when d is odd or per ( α ) c − 1 when d is even, then Eq uation ( 3 ) would imply the period- index con- jecture, at least away from some small primes. Thus, o ur result allows a tra nslation of the period-index conjectur e into a question ab out the arithmetic failure of ´ etale descent for twisted K -th eory . The p roof of Theorem 8.7 is based o n the computatio n of a differential in the de- scent spectral sequence associated to K α . Recall that the descent s pectr al s equ ence, or Brown-Gersten spe ctral sequence, is E s,t 2 = H s ( X ´ et , K α t ) ⇒ K α, ´ et t − s ( X ) , with d ifferentials d α r of d egree ( r , r − 1) . In [ 1 ], we proved that the twisted K -theory sheaves a re isomorphic, in a natur al way , to the untwisted K -theo ry sheav es. In par- ticular , K α 0 ∼ = Z , a nd K α 1 ∼ = G m . The ´ etale index is de fined to be th e least integer n such that d α r ( n ) = 0 , r ≥ 2 , where we view n as an elem ent in H 0 ( X, K α 0 ) . T he proof of the main theo rem, that per ( α ) | eti ( α ) , is g i ven by proving that d α 2 (1) = α , where α ∈ H 2 ( X ´ et , G m ) ∼ = H 2 ( X ´ et , K α 1 ) . Indeed, on ce this is don e, the least in teger such that d α 2 ( n ) = 0 is n = per ( α ) . This compu tation is an alogous to one gi ven by Kahn and Levine in [ 13 , Pr oposition 6.9.1 ] for the ´ etale moti vic spectral sequen ce conv erging to ´ etale twisted K -theory and another given b y Atiyah and Segal in [ 3 , Prop osition 4.6 ] in the topological twisted case. 2 SPECTRAL SEQUENCES ASSOCIA TED TO TO WERS OF FIBRA TIONS 4 T o make the computatio n of the differential, we introduce a ˇ Cech appro ximation to the descen t sp ectral sequence. This in turn relies on the com parison theorem, The- orem 4.7 , which says that the h omotopy limit a nd Postnikov spe ctral seq uences for cosimplicial spaces agree beginning with th e E 2 -page. This is a folklore theorem, and we inclu de a proo f since we k now of no ref erence fo r it. Both types of spec- tral sequences arise frequen tly in practice in hom otopy theory d ue to the mechanism of cosimp licial replacement of presheaves of spaces, and th e theorem easily a pplies in this situation to say that f or a p resheaf of space s on a small category , the homo topy limit and Postnikov spectral sequences agree beginning with the E 2 -page. Once this approx imation is e stablished, it remains to describe the d 2 differential for the homotopy limit spectral sequen ce of a cosimplicial space, and then to translate th is description to the ˇ Cech approxim ation spec tral sequence. Some backgrou nd is given in Sections 2 and 3 . Sev eral spectral seq uences for cosimplicial spaces are introduced, and a comparison the orem p roven, in Section 4 . This is applied to presheaves of spectra on a Grothend ieck site in Section 5 . The dif fer- entials of the homo topy limit spectral sequ ence are described in Section 6 , an d tho se of the ˇ Cech ap proxima tion in Section 7 . The main theorem is proven in Section 8 . Fi- nally , in Section 9 , the cokernel F of Equatio n ( 4 ) is related in d etail to the perio d-index conjecture . Acknowledgments This paper is par t of my Ph.D. dissertation, which was written under th e directio n of Henri Gillet at UIC. I owe him a gr eat deal of tha nks fo r his support. I have also spoken with Peter Bousfield, David Gepner, Christian Haesemeyer, Jumpei Nogami, and Bro oke Shipley about this work. They have a ll been very helpful and enco uraging . Ad ditionally , the referee did a very careful job which certainly has resulted in an improvement in the exposition . 2 Spectral Sequences Associated t o T ow ers of Fibrations The first q uadran t spectral sequen ces of this section are generalized in the sense that not ev ery term is an abelian grou p. They hav e the proper ty E s,t 1 is a group when t − s = 1 and it is a pointed set when t − s = 0 . Noneth eless, there is a g ood notion of such spectral seq uences. For details on these gen eralized spectral sequences as well as all the other material and notation in this section, see [ 4 , Section IX.4, esp. 4. 1]. Construction 2.1. Let · · · → X n → X n − 1 → · · · → X 1 → X 0 → ∗ be a tower of fibrations of pointed spaces. Let F s be the fiber of X s → X s − 1 , and denote by X the limit of the tower . There are long exact sequences associated to the fibrations F s → X s → X s − 1 : · · · → π t − s F s k − → π t − s X s i − → π t − s X s − 1 j − → π t − s − 1 F s → · · · . (5) 2 SPECTRAL SEQUENCES ASSOCIA TED TO TO WERS OF FIBRA TIONS 5 These sequences continue all the way do wn to π 0 X s − 1 : π 2 X s − 1 j − → π 1 F s k − → π 1 X s i − → π 1 X s − 1 j − → π 0 F s k − → π 0 X s i − → π 0 X s − 1 , and π 1 X s − 1 j − → π 0 F s extends to an actio n of π 1 X s − 1 on π 0 F s so tha t j is the map onto th e orbit o f the basepoint of F s under the action of π 1 X s − 1 . Besides the usual condition s of ker = im in the range that this makes sense, the exactness of Equation ( 5 ) means • that the qu otient of π 0 F s under this action injects into π 0 X s , • that the cokern el (quotient set) of π 0 F s k − → π 0 X s injects into π 0 X s − 1 , • that the stabilizer of the action of π 1 X s − 1 on π 0 F s at the base-point of F s is the quotient of π 1 X s by the image of π 1 F s → π 1 X s , and • that π 2 X s − 1 maps to the center of π 1 F s . The tower of fibrations and the e xact sequenc es above define an exact coup le D 1 D 1 E 1 i j k where D an d E are big raded groups: D s,t 1 = π t − s X s E s,t 1 = π t − s F s . The maps i , j , and k are of bi-degrees ( − 1 , 1) , (1 , − 2) , and (0 , 0) : i : π t − s X s → π t − s X s − 1 j : π t − s X s − 1 → π t − s − 1 F s k : π t − s − 1 F s → π t − s − 1 X s . As usual, the exact coup le g iv es rise to a dif ferential d = j ◦ k on E . It is o f bi-degree (1 , − 2) . The first derived exact coup le is π t − s X (1) s := D st 2 = im( i ) = im( π t − s X s +1 i − → π t − s X s ) ⊆ π t − s X s π t − s F (1) s := E st 2 = H( d ) = ker( π t − s F s k − → π t − s X s / im( i )) / ker( π t − s +1 X s − 1 i − → π t − s +1 X s − 2 ) . 2 SPECTRAL SEQUENCES ASSOCIA TED TO TO WERS OF FIBRA TIONS 6 When s = t , then the de finition of E st 2 should be inter preted as the qu otient o f the pointed set ker( π 0 F s k − → π 0 X s / im( i )) by the action of k er( π 1 X s − 1 i − → π 1 X s − 2 ) ⊆ π 1 X s − 1 . Then, the sequences π 2 X (1) s − 1 j − → π 1 F (1) s k − → π 1 X (1) s i − → π 1 X (1) s − 1 j − → π 0 F (1) s k − → π 0 X (1) s i − → π 0 X (1) s − 1 , are also exact in the generalized sense above. Repeating this process, one o btains a generalized s pectr al sequ ence E s,t 1 { X ∗ } asso- ciated to the to wer, with E s,t 1 = π t − s F s ⇀ π t − s X , where F s is th e fib er o f X n → X n − 1 . The differential d r is of d egree ( r , r − 1) . See [ 4 , Chapter I X]. The harpoo n ⇀ means that the spectral sequence may n ot con verge in the usual sense. In stead, there is a filtration Q s π i X = k er( π i X → π i X s ) with successi ve quotients e s,t ∞ = ker( Q s π t − s X → Q s − 1 π t − s X ) , and inclusions e s,t ∞ ⊆ E s,t ∞ . (6) Write E s,t 1 ⇒ π t − s X when the spectral sequen ce does hold in the usual sense, in which case equality h olds in Eq uation ( 7 ). In this c ase, say that th e spectral seq uence con verges completely . There is a second spectral sequence ˜ E s,t 2 { X ∗ } , which is simply a re-in dexed version of the first: ˜ E s,t 2 = E t, 2 t − s 1 = π t − s F t ⇀ π t − s X . This is derived fro m the e xact coup le ˜ D 2 ˜ D 2 ˜ E 2 , i j k W ith ˜ E 2 as above, an d ˜ D s,t 2 = π t − s X t . The filtration is the same: ˜ Q s π i X = Q s π i X , but the s uccessive quotients are ˜ e s,t ∞ = e t, 2 t − s ∞ . 3 COSIMPLICIAL SP A CES 7 Here, the differentials are als o of degree ( r , r − 1) . The sp ectral sequ ence and th e filtration Q s π i are fu nctorial for towers of pointed spaces. W e view the spe ctral sequence as inc luding the in formation of the abutment and the filtration on the abutment. Th us, a morp hism of sp ectral sequ ences in cludes a filtration-respe cting mo rphism of the abutment. Remark 2.2 . Under certain co nditions, these spectral seq uence do converge in some range to some of the homo topy g roups of X . For instance, suppose that i ≥ 1 and that for each s ≥ 0 there is an integer N ( s ) ≥ 1 such that E s,s + j M = E s,s + j ∞ (7) for all M ≥ N ( s ) when j = i and j = i + 1 . Th en, E s,t ∗ { X ∗ } conver ges to π i X . This is also tr ue for i = 0 wh en all of the h omotopy sets of the spaces in th e tower are abelian groups. Ag ain, for details see [ 4 , Section IX.5]. Remark 2.3 . Note th at in the application to K -theo ry , the spaces in th e to wer will have homoto py sets π t which are abelian group s f or all t ≥ 0 . Moreover , the c on vergence condition s of Equ ation ( 7 ) will always hold un der the fin ite coh omologica l dimension condition s used in th is paper . 3 Cosimplicial Spaces Definition 3.1. Let ∆ be the category of finite simplices. Objects of ∆ are no n-empty finite ordered sets, and morphisms are set m orphisms that preserve order . The category sSets o f simplicial sets is the functor category F un (∆ op , Sets ) . In general, if C is a category , then s C is the category F un (∆ op , C ) , the categor y of simp licial objects in C . Objects of sSe ts w ill b e c alled space s. The category sSe ts ∗ is the categor y of pointed spaces. Definition 3.2. If C is a cate gory , then denote by c C the f unctor category F un (∆ , C ) , the categor y of cosimplicial ob jects in C . The category of cosimplicia l spaces is the category csSets . Write csSets ∗ for the category of cosimplicial pointed spaces. Example 3.3. For a space X , let the same symbo l X denote the constant cosimplicial space n 7→ X . Example 3.4. T he cosimp licial space ∆ is the f unctor n 7→ ∆ n , where ∆ n is the simplicial space ∆ n . Example 3 .5. Let U • be a hypercover in a site C , and let X be a p resheaf o f simplicial sets on C ( a preshe af of spaces). Then, X U • denotes the c osimplicial space given by ev aluating X a t each lev el of U • in the usual way . Definition 3.6. If F is an en dofunc tor of sSets , then one extends F to an endofu nc- tor o n csSets by level-wise application. That is, for a cosimplical space X , define F ( X ) n = F ( X n ) . The typical examples are the s - skeleton functor s X 7→ X [ s ] a nd the Ex -functor . 3 COSIMPLICIAL SP A CES 8 Definition 3.7. Let P be a property of spaces. The n, a cosimplicial space X is le vel P if each space X n is P for n ≥ 0 . Similarly , if Q is a prop erty of morphisms of spaces, then a morp hism f : X → Y of cosimplicial spaces is level Q if f n : X n → Y n is Q for all n ≥ 0 . There is a go od model structure, the Reed y structure, on cosimplicial sp aces. Let X be a cosimplicial space so that X n is a simp licial set for n ≥ 0 . A morphism f : X → Y is a weak equivalence if each f n : X n → Y n is a weak equ i valence; that is, if f is a level weak equiv alence. The m aximal a ugmentatio n of a cosimplicial space is the simplicial set that equalizes d 0 , d 1 : X 0 → X 1 . A map f o f co simplicial spaces is called a cofibration if it is a lev el cofibration (level monomorp hism) and if it induces an isomorphism on the maximal augmentation s. The fibration s ar e all those morph isms with the righ t lifting pr operty with respect to acyclic cofibratio ns. A pr oof that this is a model category may be found in [ 4 , section X.5]. Example 3.8. As examples of cofib rant objects, consider ∆ an d ∆[ s ] . Ind eed, ∆ 0 is a single po int, and ∆ 1 is the 1 -simplex. The coface maps d 0 and d 1 send the unique p oint of ∆ 0 to the vertices 1 and 0 resp ecti vely o f ∆ 1 . The refore, the maximal augmentation is the empty simplicial complex. This also shows that ∆ [ s ] → ∆ is a cofibration. Let X be a cosimplicial space. Then, define the n th matching object of X to be M n X = lim ← − φ : n → k X k , where φ runs over all surjectio ns n → k in ∆ . There is a natu ral map X n +1 → M n X . Proposition 3.9 ( [ 4 , Section X.4.5]) . A morph ism f : X → Y is a fibration if a nd on ly if the induced map X n +1 → Y n +1 × M n Y M n X is a fibration of simplicial s ets for all n ≥ − 1 . The c losed mo del structur e on cosimplicial spac es is simplicial. Th at is, there is a functor Map : csSets op × csSets → sSe ts defined by Map ( X , Y ) : n 7→ Hom( X × ∆ n , Y ) . The sp ace Map ( X , Y ) is called the function comp lex fr om X to Y . Similarly , if X and Y are cosimplicial pointed spaces, then there is a pointed fun ction complex Map ∗ ( X, Y ) ∈ sSets ∗ defined by Map ∗ ( X, Y ) n = Hom( X ∧ ∆ n + , Y ) . Proposition 3.10 ([ 4 , Section X.5] ) . The simplicial mo del category axiom SM7 is sat- isfied: if A → B is a cofib ration o f cosimplicial spaces and if X → Y is a fibration of cosimplicial spaces, then Map ( B , X ) → Map ( A, X ) × Map ( A,Y ) Map ( B , Y ) is a fibration. 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 9 There is a functor csSets → csSe ts ∗ defined by X = ( n 7→ X n ) 7→ X + = ( n 7→ X n + ) , where X n + is the space X n with a disjoint basepoint attached. Definition 3.11. For each integer n ≥ 2 , there is a functo r π n : csSe ts ∗ → cAb , where cAb is the category of cosimplicial abelian groups, defined by π n ( X ) m = π n ( X m ) . There are also functor s, defined by the same equation, π 1 : csSe ts ∗ → cGroups π 0 : csSe ts ∗ → cSets ∗ , where cSets ∗ is the category of cosimplicial pointed sets and cGroups is the categor y of cosimplicial groups. Definition 3 .12. Let A be a co simplicial abe lian gr oup, co simplicial g roup, o r co sim- plicial pointed set. A pointed cosimp licial spa ce X is called a K ( A, n ) -cosimplicia l space if π n X ∼ = A , while π m X ∼ = ∗ for m 6 = n . 4 Spectral Sequence s f or Cosimplicial Spaces Let X be a pointed cosimplicial space. Define po inted simplicial complexes T ot ∞ X = Map ∗ (∆ + , X ) , T ot s X = Map ∗ (∆[ s ] + , X ) . By axiom SM7 an d E xample 3.8 , if X is fibrant, then T ot s X → T ot s − 1 X gi ves a tower o f pointed fibratio ns. The inverse limit o f this tower is T ot ∞ X when X is fibrant. Definition 4.1 . For an arbitrary cosimp licial po inted spa ce X let X → H c X be a pointed fibrant resolution . Then, the tota l spac e spectral sequence of X , T E 1 X , is defined to be the spectral sequence of the tower T ot ∗ H c X : T E s,t 1 X = E s,t 1 { T ot ∗ H c X } ⇀ π t − s T ot ∞ H c X . The fiber F s of T ot s H c X → T ot s − 1 H c X and the homotopy gro ups o f the fiber F s are identified in [ 4 , Propo sition X.6.3] ( see also Section 6 ): F s ≃ Map ∗ ( S s , N X s ) , 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 10 where N X s is the fiber of the fibration H c X s → M s − 1 H c X ; see Proposition 3.9 . Moreover , π i N X s ∼ = π i H c X s ∩ ker s 0 ∩ · · · ∩ ker s s − 1 , where the maps s i are the cosimplicial degeneracies: s i : H c X s → H c X s − 1 . Therefo re, there is a natural identification T E s,t 1 X ∼ = π t H c X s ∩ ker s 0 ∩ · · · ∩ ker s s − 1 t ≥ s ≥ 0 . Note that for t ≥ 2 , T E s,t 1 X is the s -degree of the normalized cochain complex N ∗ π t H c X associated to π t H c X . It is tedious but not hard to check tha t under this identification, the d ifferential d 1 of the spectral sequen ce is chain h omotop ic to th e differential of the normalized cochain comp lex. Ther efore, there are natur al identifica- tions T E s,t 2 X ∼ = H s ( N ∗ π t H c ) ∼ = H s ( C ∗ π t H c X ) ∼ = H s ( C ∗ π t X ) t ≥ 2 , t ≥ s ≥ 0 , (8) where C ∗ π t X d enotes the un normalized cochain complex associated to the cosimpli- cial abelian group π t X . It is no t hard to extend these identifications for t = 0 and s = 0 , and for t = 1 and s = 0 , 1 . For a detailed discussion, see [ 4 , Section X.7]. Define, for a cosimplicial abelian group A , the s th cohom otopy gr oup for s ≥ 0 as π s A = H s ( C ∗ A ) . If G is a co simplicial pointed set or cosimplicial group, then define π 0 G as the equalizer of ∂ 0 , ∂ 1 : G 0 ⇒ G 1 . This is a gro up if G is. Similarly , define a pointed cohomotopy set π 1 G for G a co simplicial group as follows. Let Z 1 G = { g ∈ G 1 : ( ∂ 0 g )( ∂ 1 g ) − 1 ( ∂ 2 g ) = 1 } . There is an action of G 0 × Z 1 G → Z 1 G given by ( g 0 , g 1 ) 7→ ( ∂ 1 g 0 ) g 1 ( ∂ 0 g 0 ) − 1 . The set Z 1 G is pointed by th e element 1 ∈ G 1 . Let π 1 G be the quotient s et of Z 1 G by this action , po inted by the orb it of 1 ∈ Z 1 G . Th en, by [ 4 , Paragrap h X.7.2], th ere ar e natural identifications T E s,t 2 X ∼ = ( π s π t X if t ≥ s ≥ 0 , 0 otherwise. (9) There is another spectral sequence for c osimplicial spaces, which is useful when X is level Kan. T his is the homotopy limit sp ectral sequ ence, which , in fact, exists in much greater generality . See [ 4 , Chap ter XI]. Th e main tool is a func tor fr om cosim- plicial spaces to cosimplicial spaces called Π . Let N ∆ be the nerve of the category 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 11 ∆ . Th en an element o f N ∆ n is a simp lex i ∗ : i 0 → · · · → i n , where each i k is non-n egati ve integer, and the the arrows are order-preservin g m aps of th e ord ered sets associated to i k : { 0 , · · · , k } . For X an arbitrary cosimplicial sp ace, let Π X d enote the space whose n th le vel is Π n X = Y i ∗ ∈ N ∆ n X i n . So, the n th level of Π X is the pro duct over all com positions i 0 → i n of X i n . The face map ∂ j for j < n com posed with pro jection onto i ∗ is pro jection onto ∂ j ( i ∗ ) fol- lowed by the identity . The face ma p ∂ n composed with projection on to i ∗ is projection onto ∂ n ( i ∗ ) followed by X ( i n − 1 → i n ) . Similarly , th e degener acy s j followed by projection onto i ∗ is projection onto s j ( i ∗ ) followed by the i den tity . The important thing a bout the cosimplicial replacement fu nctor Π is tha t it takes lev el fibrations into cosimplicial fibration s an d preserves weak equ i valences. See [ 4 , Proposition X.5.3]. Definition 4 .2. Let X b e a co simplicial p ointed sp ace. Let Ex ∞ X den ote the cosim pli- cial po inted space o btained from X b y applyin g the Ex ∞ -functo r to each level. Th en, Π Ex ∞ X is fibr ant. Define the homo topy limi t spectral sequence of X , HL E 1 X , to be HL E s,t 1 X = E s,t 1 { T ot ∗ Π Ex ∞ X } ⇀ π t − s T ot ∞ Π Ex ∞ X . The space T ot ∞ Π Ex ∞ X is called the homoto py limit of X , and will be wr itten as holim ∆ X . Lemma 4 .3. If X is a cosimplicial pointed spa ce tha t is level Kan, then th e n atural morphism X → E x ∞ X induces an isomorphism of spectral sequences E s,t 1 { T ot ∗ Π X } ≃ → E s,t 1 { T ot ∗ Π Ex ∞ X } . This morphism is natural in morphisms of cosimplicial pointed le vel Kan spac es. Let X be an arb itrary pointed cosimplicial space. T he functor Π can be defined on cosimplicial objects in any category with finite products. In particular, on po inted sets, group s, and a belian g roups. There are n atural iso morphisms, o f cosim plicial po inted sets fo r n = 0 , cosimplicial g roups for n = 1 , and co simplicial abelian gro ups for n > 1 , π n Π X ≃ Π π n X , where π n X is the cosimplicial object obtained b y evaluating π n at each cosimplicial lev el. For X a co simplicial ob ject in a categor y with finite produ cts, there is an natural morph ism X → Π X . Th e maps X n → Y i ∗ ∈ N ∆ n X i n are described as fo llows . The sim plex i ∗ determines a morphism (0 → 1 → · · · → n ) → i n , by taking th e images of 0 fro m each i i , 0 ≤ i < n . This induces the map X n to the produ ct, an d it extends to a cosimplicial map X → Π X . 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 12 Proposition 4.4 ([ 4 , Paragraph XI. 7.3]) . The ca nonica l map C ∗ π n X → C ∗ Π π n X is a quasi-isomorph ism. Proposition 4.5 ([ 4 , Paragr aph X I.7.5]) . If X is a fibrant cosimplicial poin ted space, then the natural morphism T E 1 X → HL E 1 X of spectral sequences is an is omo rphism. There is also the Postnikov tower of a cosimplicial s pace . Definition 4.6. Let X be a level-fibrant pointed c osimplicial space. Denote by X ( n ) the level-wise application of the coskeleton fu nctor . Then each X → X ( n ) and X ( m ) → X ( n ) , m ≥ n is a level fib ration. Ther efore, Π X ( m ) → Π X ( n ) is a fibration for m ≥ n . The spe ctral sequence of this tower is called the Postnik ov spec- tral sequence for X : P E s,t 2 X = ˜ E s,t 2 { T ot ∞ Π X ( ∗ ) } ∼ = π t − s T ot ∞ G ( t ) ⇀ π t − s holim ∆ X , where G ( t ) is the fiber of Π X ( t ) → Π X ( t − 1 ) . T his fiber is a fibran t resolution of a cosimplicial K ( π t X , t ) -space. By [ 4 , Paragraphs XI.7.2-3], t her e are natural isomorphism s π t − s T ot ∞ G ( t ) ≃ π s π t X for t ≥ s ≥ 0 . Th us, HL E s,t 2 X ∼ = P E s,t 2 X . In fact, this isomorphism comes from an isomorphism of spectral sequences. Theorem 4.7. Let X be level Kan. Then, ther e is a na tural isomorphism φ of spectral sequences φ : HL E 2 X → P E 2 X fr om the ho motopy limit s pectral s equ ence beginning with th e E 2 -page to the P ostn ikov tower spectral sequence. Pr o of. Recall that to create an iso morphism of spectra l sequences that come from exact couples I D 2 I D 2 I E 2 , i j k I I D 2 I I D 2 I I E 2 i j k 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 13 it suffices to create a morp hism of exact couples that is an isom orphism just on the E -terms: φ : I E 2 ≃ → I I E 2 . Indeed , this is enough to gu arantee that the morphism in duces an isomor phism on H(E) and a morphism of the deriv ed couples. So, it fo llows ind uctively that this is sufficient. Since X is le vel Ka n, there is a double tower o f fibration s, of which a typical square is T ot s +1 Π X ( t − 1) ← − − − − T ot s +1 Π X ( t )   y   y T ot s Π X ( t − 1) ← − − − − T ot s Π X ( t ) . These fit into a bigger diagra m T ot ∞ Π X ( t − 1) ← − − − − T ot ∞ Π X ( t ) ← − − − − T ot ∞ Π X   y   y   y T ot s +1 Π X ( t − 1) ← − − − − T ot s +1 Π X ( t ) ← − − − − T ot s +1 Π X   y   y   y T ot s Π X ( t − 1) ← − − − − T ot s Π X ( t ) ← − − − − T ot s Π X. (10) The horizontal inverse limits ar e T ot s Π X a nd the vertical in verse limits are T ot ∞ Π X ( t ) . Thus the homotopy limit spectral s equ ence comes the tower of fibrations at the horizo n- tal limit, while the Postnikov spectr al sequence comes fr om the tower o f fibrations at the vertical limit. Define F ( s + 1) to be the fiber of T ot s +1 Π X → T ot s Π X , and define G ( t ) be the fiber of T ot ∞ Π X ( t ) → T ot ∞ Π X ( t − 1) . First, construct a morp hism HL D s,t 2 = im( π t − s T ot s +1 Π X → π t − s T ot s Π X ) → P D s,t 2 = π t − s T ot ∞ Π X ( t ) . Let [ x ] ∈ HL D s,t 2 be represented by x : S t − s → T ot s +1 Π X → T o t s Π X . By adjunction , v iew th is as x : ∆[ s ] + ∧ S t − s → ∆[ s + 1] + ∧ S t − s → Π X. T o create the morphism, one must “lift” this to a mo rphism φ ( x ) : ∆ + ∧ S t − s → Π X ( t ) . The mor phism x con sists of a compatible collection of morphisms x n,i ∗ : ∆ n [ s ] + ∧ S t − s → ∆ n [ s + 1] + ∧ S t − s → X i n , one for each i ∗ ∈ N ∆ n . The key point underlying the details belo w is that X i n ( t ) is a Kan comp lex and also h as trivial hom otopy groups π k X i n ( t ) wh en k > t . At v ario us points one needs to make ch oices to extend maps. These need to be compatible with the cosimplicial stru cture. At any given poin t, this will in volve finitely many choices differing in simplicial d egrees greater th an t . Thus, it w ill always be p ossible to make the choices compatibly . 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 14 Define φ ( x ) 0 ,i ∗ : ∆ 0 [ r ] + ∧ S t − s → X i 0 ( t ) for all r and i ∗ ∈ N ∆ 0 as the comp osition ∆ 0 + ∧ S t − s = ∆ 0 [ s + 1 ] + ∧ S t − s x 0 ,i ∗ − − − → X i 0 → X i 0 ( t ) . Since the coskeleton map is a functor, this definition is functo rial. Now , suppo se that φ ( x ) h,i ∗ : ∆ h + ∧ S t − s → X i h ( t ) , is defined f or 0 ≤ h ≤ k and all i ∗ ∈ N ∆ h , comp atible with all coface and code- generacy maps in this range . In o ther words, suppose that we have defined compatible maps φ ( x ) h, ∗ : ∆ h + ∧ S t − s → Π h X ( t ) for 0 ≤ h ≤ k . If i ∗ ∈ N ∆ k +1 is degenerate, then define φ ( x ) k +1 ,i ∗ : ∆ k +1 + ∧ S t − s → X i k +1 ( t ) by forcing the diagram ∆ k +1 + ∧ S t − s φ ( x ) k +1 ,i ∗ − − − − − − − → X i k +1 ( t ) s j   y    ∆ k + ∧ S t − s φ ( x ) k,i ′ ∗ − − − − − → X i k +1 ( t ) to be commutati ve for every j such that s j ( i ′ ∗ ) = i ∗ for som e i ′ ∗ ∈ N ∆ k . Making th ese simultaneou sly c ommutative fo r all choices of j is possible b ecause of the simplicial relations s i ◦ s j = s j +1 ◦ s i for i ≤ j . If i ∗ ∈ N ∆ k +1 is not d egenerate, then the c osimplicial stru cture r equires that the diagrams ∆ k + ∧ S t − s φ ( x ) k,∂ j ( i ∗ ) − − − − − − − → X i k +1 ( t ) ∂ j   y    ∆ k +1 + ∧ S t − s φ ( x ) k +1 ,i ∗ − − − − − − − → X i k +1 ( t ) for 0 ≤ j < k and ∆ k + ∧ S t − s φ ( x ) k,∂ k ( i ∗ ) − − − − − − − − → X i k ( t ) ∂ k   y X ( i k +1 → i k )   y ∆ k +1 + ∧ S t − s φ ( x ) k +1 ,i ∗ − − − − − − − → X i k +1 ( t ) be commu tati ve. In o ther words, the map φ ( x ) k +1 ,i ∗ is already determ ined on ∂ ∆ k +1 + ∧ S t − s . Thus, to make the induc ti ve step, on e must fill in the dashed line so that the diag ram 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 15 ∆ k +1 [ s + 1] + ∧ S t − s X i k +1 ∆ k +1 + ∧ S t − s X i k +1 ( t ) ∂ ∆ k +1 + ∧ S t − s x k +1 ,i ∗ φ ( x ) k +1 ,i ∗ is comm utativ e. If s ≥ k , then ∆ k +1 [ s + 1 ] + = ∆ k +1 + , so there is nothing to do. If s < k , then ∆ k +1 [ s + 1] + ∧ S t − s ⊆ ∂ ∆ k +1 + ∧ S t − s , and the outer squar e comm utes by induction . Th erefore, it suffices for the dashed arr ow to com mute in the bottom triang le. As s < k , the arrow ∂ ∆ k +1 + ∧ S t − s → X i k +1 correspo nds to a map S k + t − s +1 → X i k +1 ( t ) . But, k + t − s + 1 > t . Hence, π k + t − s +1 X i k +1 ( t ) = 0 . As X ( t ) is a Kan complex, such a fi ll exists (see [ 11 , page 35]). Choose a fill for all choices of i ∗ . This comp letes the induction, gi ving φ ( x ) ∗ : ∆ ∗ + ∧ S t − s → Π ∗ X ( t ) , for 0 ≤ ∗ ≤ k + 1 . Th e p rocess o utlined earlier in the proo f gives a base case. So, induction provides the desired map φ ( x ) : ∆ + ∧ S t − s → Π ∗ X ( t ) . If y is ano ther morphism S t − s → T ot s +1 Π X repr esenting the class [ x ] , then it is straightfor ward, using a similar inductive a rgument, to lif t a homotopy between x and y to a h omotopy between φ ( x ) and φ ( y ) so that the map is well-de fined on the level of hom otopy group s. T hus, th e con struction above determines a well-defined th e map φ : HL D s,t 2 → P D s,t 2 . It is easy to see that φ commutes with i . Indeed, on HL D 2 , i is given by restriction from ∆[ s ] to ∆[ s − 1] . On P D 2 , i is g iv en by mapping X ( t ) → X ( t − 1) . Now , composin g φ ( x ) with Π X ( t ) gives a map that solves the lifting problem to d efine 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 16 φ ( i ( x )) : ∆[ s − 1] + ∧ S t − s i ( x ) − − − − → Π X   y    ∆[ s ] + ∧ S t − s x − − − − → Π X   y    ∆[ s + 1] + ∧ S t − s − − − − → Π X   y   y ∆ + ∧ S t − s φ ( x ) − − − − → Π X ( t )      y ∆ + ∧ S t − s φ ( i ( x )) − − − − → Π X ( t − 1) . Therefo re, φ commutes with i . Extend φ to HL E 2 by lifting x : S t − s → F ( s ) to x : S t − s → T ot s +1 Π X and applyin g φ (recall the description in Section 2 of E 2 ). In other words, apply k and then φ . This defines an element o f P D s,t 2 , and, by constru ction, φ ( x ) actually factors throug h th e fiber G ( t ) . Ind eed, since x : ∆[ s + 1 ] ∧ S t − s → Π X comes from the fiber F s , it restricts to the tri vial map on ∆[ s − 1 ] ∧ S t − s . Therefo re, x is tri vial on the t − 1 - skeleton. Extending this to a map x : ∆ ∧ S t − s → Π X ( t ) as above does not change this, so th at w hen one co mposes with Π X ( t ) → Π X ( t − 1) , the map is homotopic to the co nstant map on th e basepoin t. T o ch eck that this determines a well-defined map on E 2 -terms, one must check that if x = y + j ( z ) , where z ∈ ker( π t − s +1 X s − 1 → π t − s +1 X s − 2 ) and x, y ∈ ker( π t − s F ( s ) → π t − s X s /i ( π t − s X s +1 )) , 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 17 then φ ( x ) = φ ( z ) . But, since φ ( j ( z )) = φ ( k ( j ( z ))) = φ (0) , it follows th at this definition of φ on HL E 2 is indeed well-defined. It remains only to check that φ respects j . Again, let [ x ] ∈ π t − s T ot s Π X b e represented by x : ∆[ s + 1] + ∧ S t − s → Π X . Recall that j ([ x ]) is obtained by the map π t − s T ot s +1 Π X → π t − s − 1 F ( s + 2 ) , giv en by lifting a horn ∆[ s + 1] + ∧ Λ s − t i, + → ∆[ s + 1] + ∧ S s − t → Π X to ∆[ s + 2] + ∧ ∆ s − t + → Π X and then restricting to the i th face to obtain ∆[ s + 2] + ∧ S t − s − 1 → Π X , which by adjunction is in π t − s − 1 F ( s + 2 ) . Lift ∆[ s + 2] + ∧ ∆ s − t + → Π X as above to a map ∆ + ∧ ∆ s − t + → Π X ( t + 1 ) . By construction, it maps down to φ ( x ) : ∆ + ∧ S t − s → Π X ( t ) , while the restriction to ∆ + ∧ S t − s − 1 is in the fiber G ( t + 1) . There fore, φ com mutes with j . T o p rove that φ is injective o n E 2 , let x ∈ HL E s,t 2 be represented by x : ∆[ s + 1] + ∧ S t − s → Π X , and let φ ( x ) : ∆ + ∧ S t − s → Π X ( t ) factor through G ( t ) . Suppose that φ ( x ) is homotopic to the constant map in G ( t ) , and let y : ∆ 1 + ∧ ∆ + ∧ S t − s → G ( t ) → Π X ( t ) be such a homotopy . It is p ossible to find an extension 4 SPECTRAL SEQUENCES FOR COSIMPLICIAL SP A CES 18 ∆ 1 + ∧ ∆[ s ] + ∧ S t − s Π X ∆ 1 + ∧ ∆ + ∧ S t − s Π X ( t ) y giving a ho motopy between x an d the constant map by d efinition o f th e coskeleton Π X ( t ) . Ther efore, φ is injective on E 2 . T o show that φ is surjec ti ve on E 2 , let y : S t − s → G ( t ) be represented by ∆ + ∧ S t − s y − − − − → Π X ( t )   y   y ∗ − − − − → Π X ( t − 1 ) . A lift x f or the diagram ∆ n [ s − 1 ] + ∧ S t − s ∗ ∆ n [ s ] + ∧ S t − s ∆ n [ s + 1 ] + ∧ S t − s X i n ∆ n + ∧ S t − s X i n ( t ) ∗ X i n ( t − 1) y x exists by definition for n ≤ s . For n > s , the to p quadrilatera l mean s that the map ∆ n [ s ] + ∧ S t − s → X i n is a diagram of t − s -spheres in Ω s X i n , and finding a lift x is the same as sayin g that this class is 0 in π t − s Ω s X i n . But, the map y in the diagr am says that this class is 0 in π t − s Ω s X i n ( t ) . On ce again, by definition o f the coskeleton functor , it f ollows that it is already 0 in π t − s Ω s X i n . No w , arguing as above, one may inductively ch oose exten - sions in the diagr am to c reate an element x of HL E s,t 2 such th at φ ( x ) = y . Therefo re, φ is a lso surjective on E 2 . 5 THE ˇ CECH APPR O XIMA TION 19 Remark 4.8 . This theor em does not app ear to be n e w . Ind eed, it app ears that Thomaso n was aware of it: see [ 17 , pag e 542, paragrap h 3]. Howe ver , we know o f n o reference for a pro of. I t in fact holds in the g reater genera lity of presheaves o f simplicial sets on small categories. Corollary 4 .9. S uppose that X is a le vel Kan cosimplicial pointed spa ce. If one of the spectral sequences HL E 1 X ⇀ π t − s holim ∆ X P E 2 X ⇀ π t − s holim ∆ X . conver ges, then both do, and the filtrations HL Q s and P Q s coincide on π ∗ holim ∆ X . 5 The ˇ Cech A ppr oximation Let C be a Grothendieck site with termin al object U . Den ote by Pre ( C ) the category of presheaves on C , a nd write sPre ( C ) fo r the category o f simplicial presheaves. W e use the following closed model cate gor y structur e on simplicial p resheaves, called the local mode l categor y structure. The cofibrations are the pointwise cofibra- tions. Thus, X → Y is a co fibration if and only if X ( V ) → Y ( V ) is a monom orphism for e very object V o f C . For an object V of C , there is a site with terminal object C /V . Each presheaf on C restricts to a presheaf on C /V . For a simplicial presheaf X , an object V of C , a nd a basepoint x ∈ X ( V ) 0 , ther e are p resheaves o f hom otopy groups π p k ( X | V , x ) : ( f : W → V ) 7→ π k ( | X ( W ) | , f ∗ ( x )) , where | X ( W ) | d enotes the geometric realization of the simplicial set X ( W ) . Let π k ( X | V , x ) be the associated homoto py sheaf. Call w : X → Y a weak equ iv alence if it induces an isomorphism of homotopy shea ves π k ( X | V , x ) ≃ → π k ( Y | V , w ( x )) for all choices of V , all basepoints x of X ( V ) , an d all k ≥ 0 . The fibrations are all maps having th e right lifting prop erty with resp ect to all cofibratio ns that ar e simulta- neously wea k equ i valences (the acyclic cofibrations). That this is a simplicial clo sed model category is due to Joyal; for a pro of, see Jard ine’ s paper [ 12 ]. Refer to these classes of morphisms more specifically as glo bal fibrations , glo bal cofibratio ns , and local weak equivalences . Theorem 5.1 (Du gger-Hollander-Isaksen [ 8 ]) . If X is a simp licial pr esheaf, and if V • → V is a hype r cover in C , then let X V • denote th e cosimplicial spa ce associated to V • . Ther e is a canonica l augmen tation X ( V ) → X V • . The simplicial pr esheaf X is globally fibrant if and only if X ( V ) → holim ∆ X V • is a weak equivalence for all hyper covers V • → V in C . 5 THE ˇ CECH APPR O XIMA TION 20 There ar e other types of mor phisms, namely p ointwise weak equi valences and pointwise fibrations. A poin twise weak equiv alence is a morph ism f : X → Y such that X ( V ) ∼ → Y ( V ) is a weak equiv alence of simplicial sets for all objects V of C . T wo poin twise weak equiv alent shea ves are local weak equiv alent, and tw o loca l weak equiv alent fibrant p resheaves are pointwise weak equi valent. A po intwise fibration is a morph ism f : X → Y such th at e very f : X ( V ) → Y ( V ) is a fibration of simplicial sets. Say th at X is pointwise Kan o r p ointwise fibrant i f X → ∗ is a pointwise fibration Note that if a simplicial presheaf is pointed, then the homotopy pr esheav es and sheaves ab ove may be defined globally . Let F be a f unctor from simplicial sets to simplicial sets such that F ( ∅ ) = ∅ , or from pointed simplicial sets to p ointed simplicial sets such that F ( ∗ ) = ∗ . If X is a simplicial presheaf, d enote by F X the p ointwise application of F to X , so that ( F X )( V ) = F ( X ( V )) for all V . For instance, belo w , cos k n X will be the pointwise n - coskeleton of X . If F preserves weak equiv alences of simplicial sets, then it preserves local weak equivalences of simplicial presheaves. This is the case, for instance, f or the coskeleta functors and for the Ex functor . In particu lar , one may always replace X with the pointwise weakly equiv alent presheaf Ex ∞ X , which is pointwise Kan. Definition 5.2. If X is a p resheaf of pointed spaces, and if X → H X is a poin ted fibrant reso lution of X , then H X is called the hypercohomolo gy space of X . Hy per- cohomo logy sets, group s, and ab elian groups are defined by H s ( U, X ) = π s Γ( U, H X ) . Definition 5.3. Let X be a pr esheaf of po inted simplicial sets on C , and let X → H X be a p ointed fibrant resolu tion. Fina lly , let X ( n ) be a po inted fibrant resolutio n of cosk n H X so that X ( n ) → X ( n − 1) is a globa l fibration for all n ≥ 0 . Th en, the Brown-Gersten spectral sequence associat ed to X is the ˜ E 2 -spectral sequence (see Section 2 ) associated to the tower of fibrations Γ( U, X ( n )) → Γ( U, X ( n − 1)) . That is, define the Brown-Gersten spectral sequence as BG E s,t 2 X = ˜ E s,t 2 { Γ( U, X ( ∗ )) } ⇀ π t − s lim ← Γ( U, X ( n )) . This is also called the descent spectral sequence . Lemma 5.4 ([ 9 ]) . Su ppose that th e sit e C is locally of finite cohomological dimension . Then, the natural morphism Γ( U, H X ) → lim ← Γ( U, X ( n )) is a weak e quivalence for all o bjects U whenever X is loca lly weak equivalen t to a pr o duct of a constant pointed space and a connected space. In the situation of the lemma, write the Brown-Gersten spectral sequence as BG E 2 X ⇀ H t − s ( U, X ) . 5 THE ˇ CECH APPR O XIMA TION 21 For instance, the lemma h olds f or presh eav es o f Qu illen K -theo ry spaces on the ´ etale sites of schemes of finite ´ etale cohomo logical d imension. The theorem of Du gger-Hollander-Isaksen, Theorem 5.1 , says that th e natural map Γ( U, X ( n )) → T ot ∞ Π X ( n ) U • is an isomorphism wh enever U • → U is a hyper cover , and X ( n ) U • denotes the cosim- plicial space obtained by ev aluating X ( n ) at each level o f U • . Definition 5.5. Let U • be a hyper cover of an object U of C . Let X be a simplicial presheaf. Then, let X U • denote the cosimplicial space one gets b y e valuating X at U • . The ˇ Cech hypercohomology of X on U • is defined as ˇ H s ( U • , X ) = π s T ot ∞ H c X U • . Definition 5 .6. Th e ˇ Cech hyper cohomo logy spectral sequence associated to X an d U • → U is the total s pace spectral sequence associated to H c X U • : U • E 1 X = T E 1 X U • . By Equa tion ( 8 ), th e E 2 -terms for the corresponding sp ectral sequence are n aturally identified with E s,t 2 ≃ ˇ H s ( U • , π p t X ) , as desired. There is a natural mo rphism from ˇ Cech h ypercoh omolog y to hy percoho mology induced by a natural morphism T ot ∞ H c X U • → Γ ( U , H X ) . (11) This morp hism is the composition of T ot ∞ H c X U • → T ot ∞ H c Π ∗ X U • = T ot ∞ Π ∗ X U • → T ot ∞ Π ∗ ( H X ) U • with the in verse of the local weak equi valence (Theorem 5.1 ) Γ( U, H X ) ∼ → T ot ∞ Π ∗ ( H X ) U • . Theorem 5 .7. Let X be a p ointed simplicia l pr esheaf, and let U • → U be a hypercover . Ther e is a natural morphism U • E 2 X → BG E 2 X fr om the ˇ Cech hypercohomology spectral sequ ence to the Br own-Gersten spectral se- quence, which o n E 2 -terms is the natural map ˇ H s ( U • , π p t X ) → H s ( U, π t X ) , and which r espects the morph ism on ab utments of Equatio n ( 11 ) . 6 DIFFERENTI ALS IN THE HOLIM SPECTRAL SEQUENCE 22 Pr o of. One ma y assume, possibly by ap plying the E x ∞ that X is pointwise Kan . Therefo re, X U • is a level Kan cosimplicial po inted space. Then, the space Π X U • is fibrant, so that there is a natural morph ism H c X U • → Π X U • making the diagram X U • Π X U • H c X U • commutative. This ind uces a morphism of total space spectral sequences U • E 2 X → T E 2 H c X U • → HL E 2 X U • (12) There is a morph ism HL E 2 X U • → HL E 2 ( H X ) U • (13) induced by X → H X . Theorem 4.7 furn ishes an isomorph ism HL E 2 ( H X ) U • ≃ → P E 2 ( H X ) U • . (14) Again, the fibrant replacement cosk n H X → X ( n ) indu ces a morphism P E 2 ( H X ) U • → ˜ E s,t 2 { T ot ∞ Π X ( n ) U • } . (15) Finally , the result of Dugger-Hollander-Isaksen [ 8 ] says that the natural morphism BG E 2 X → ˜ E s,t 2 { T ot ∞ Π X ( n ) U • } (16) is in fact an isomorphism. The theore m f ollows by taking the comp osition of the morph isms of E quations ( 12 ), ( 13 ), ( 14 ), and ( 15 ), and the in verse of the morphism of Equation ( 16 ). 6 Differ entials in the Holim Spectral Sequence Lemma 6. 1 ([ 4 , Pro position X.6.3 .i]) . T ake X to be a fibrant poin ted c osimplicial space. Ther e are equivalences F ( s ) ≃ Map ∗ ( S s , N X s ) , wh er e F ( s ) is th e fiber of T ot s X → T ot s − 1 X , and N X s is the fiber of the fibration X s → M s − 1 X . Sketch o f pr oof. Let β : ∆ n → F ( s ) . By adjun ction, this is a mo rphism ∆[ s ] + ∧ ∆ n + β − → X such that th e restrictio n to ∆[ s − 1 ] + ∧ ∆ n + factors thr ough the base-po int. In particular, th e lev el s p icture is a map of pointed spaces ∆ s + ∧ ∆ n + → X s such that 6 DIFFERENTI ALS IN THE HOLIM SPECTRAL SEQUENCE 23 the restriction to the s − 1 -skeleton on th e left ∆ s [ s − 1] + ∧ ∆ n + factors through the base- point of X s . Therefore, th is is a morph ism S s ∧ ∆ n + → X . That is, β d etermines an n - simplex of Map ∗ ( S s , X s ) . Howe ver , the degeneracy s i β : ∆ s − 1 [ s ] + ∧ ∆ n + → X s − 1 factors thr ough ∆ s − 1 [ s − 1] + ∧ ∆ n + and so is trivial. Therefo re the n -simplex of Map ∗ ( S s , X s ) actually lives in Map ∗ ( S s , N X s ) . Conversely , suppo se th at γ is an n -simplex of Map ∗ ( S s , N X s ) . Again , b y ad junction, vie w this as a map S s ∧ ∆ n + → N X s → X s . Extend this as f ollows in to a map ∆[ s ] + ∧ ∆ n + → X . The lift o f γ to ∆ s + ∧ ∆ n + is the s -level of this morp hism. Sin ce λ factors throug h ∗ for any degeneracy on ∆ s , let ∆ k [ s ] + ∧ ∆ n + → ∗ → X k for k < s . If k > s , use the following diagram: ∆ s [ s ] + ∧ ∆ n + β − − − − → X s ∂   y ∂   y ∆ k [ s ] + ∧ ∆ n + ∂ ◦ β ◦ s − − − − → X k s   y s   y ∆ s [ s ] + ∧ ∆ n + β − − − − → X s , (17) where the face map ∂ (resp. th e degeneracy map s ) is some c omposition of face (resp. degeneracy) m aps such that the vertical com positions are the identity . This defines the extension inductively on the faces of ∆ k [ s ] . But, one need on ly define it up to faces, since k > s . This gi ves us an n -simplex of T ot s X . Clearly , the se constru ctions are mutually in verse. It is n ot hard, using this identification, to show that the h omotopy groups of F ( s ) are the groups of the normalized cochain complexes associated to X . That is π t − s F ( s ) ≃ π t − s Map ∗ ( S s , N X s ) ≃ π t X s ∩ ker s 0 ∩ · · · ∩ ker s s − 1 , where k er s i is ker ( s i : π t X s → π t X s − 1 ) . See [ 4 , proposition X.6.3]. The differentials in the homoto py limit spectral sequence are those of the sp ectral sequence associated to a tower of fibrations. In particu lar , the y arise fro m trying to lift simplices to higher and higher le vels of the tower . The recipe is as follows : let · · · → X s +1 → X s → X s − 1 → · · · be a tower of fib rations, a nd let δ : S t → X s − 1 be a homotopy class. Writing S t as the quotien t o f ∆ t by its bound ary , we get a map ∆ t → X s − 1 that is trivial on the bound ary ∂ ∆ t . W e can o bviously th en map any horn Λ t i to X s by sending it to the base-poin t. T hen, there is a lifting problem Λ t i − − − − → X s   y   y ∆ t − − − − → X s − 1 . Since the map on the righ t is a fibration, we do get a lift to a m ap ∆ t → X s . Ho wever , it may no longer map the boundary of ∆ t to the base-po int o f X s . Instead, the i th face 6 DIFFERENTI ALS IN THE HOLIM SPECTRAL SEQUENCE 24 determines a map ∆ t − 1 → X s that is tri vial on the boundary . The correspo nding class in π n − 1 F ( s ) is the obstruction to lifting the homoto py class o f δ to a homotopy class on X s . Below , the tower of fibrations is the total space to wer associated to a fibrant cosim- plicial space (see Section 3 ). Remark 6.2 . W e briefly indicate h ow to show that d 1 is homotopic to the coch ain differential on the n ormalized cocha in com plex for π t X . Let β this time represent a class in π t − s Map ∗ ( S s , N X s ) . T his determines a co rrespon ding map S s ∧ S t − s → X s . Exten ding this, one g ets a m ap ∆[ s ] + ∧ S t − s → X . Lift this, u sing some ho rn Λ t − s i , to a ma p ∆[ s + 1] + ∧ ∆ t − s + → X . On ∆[ s ] + ∧ ∆ t − s + this agrees with β . Restricting to the i th face of ∆ t − s , one gets a map ∆[ s + 1] + ∧ S t − s − 1 → X . Th en, look at just ∆ s +1 + ∧ S t − s − 1 → X s +1 , and check that o ne land s in the fiber . T he reason that this agree s with the alternating sum d = Σ( − 1 ) i d i is that th e simplices ∆ s +1 + ∧ ∆ t − s + → X s +1 giv e a homoto py be tween d ( β ) and d 1 ( β ) . Construction 6. 3. Let δ : S t → X s represent a class [ δ ] o f π s π t X , where t − s ≥ 0 , and supp ose that δ factor s as δ : S t − s → Map ∗ ( S s , N X s ) . T hat is, suppose that δ represents [ δ ] in the n ormalized chain complex. Then , as in the proof of Lemma 6.1 , extend δ to a map δ ′ : ∆[ s ] + ∧ ∆ t − s + → X . Now , the restriction of this map to ∆[ s ] + ∧ ∂ ∆ t − s + factors through th e base-point. Choosing a horn Λ t − s i ⊂ ∂ ∆ t − s and the map ∆[ s + 1] + ∧  Λ t − s i  → ∗ → X , one gets, using the fact that T ot s +1 X → T ot s X is a fibration, a lift to a map γ : ∆[ s + 1] + ∧ ∆ t − s + → X such that the following diagram commutes ∆[ s + 1 ] + ∧ ∆ t − s + γ − − − − → X x      ∆[ s ] + ∧ ∆ t − s + δ ′ − − − − → X . The fact that δ represents a coh omotopy class implies, by Remar k 6.2 , th at th e restric- tion of γ to ∆ s +1 + ∧ ∂ i ∆ t − s + → X s +1 is contractib le in Map ∗ ( S s +1 , N X s +1 ) . Ther efore, on e can replace γ by a hom otopic map γ ′ such that the r estriction to γ ′ : ∆[ s + 1 ] + ∧ ∂ ∆ t − s + → X factors throug h the base- point. T hus, o ne can repeat th e pr ocess, using a tr i vial map ∆[ s + 2 ] + ∧  Λ t − s j  + → ∗ → X , to get a lift ǫ : ∆[ s + 2] + ∧ ∆ t − s + → X , using som e horn Λ t − s j . The d ifferential d 2 ([ δ ]) is the class of ǫ : ∆ s +2 + ∧ ∂ j ∆ t − s + → X s +2 in π s +2 π t − s − 1 X . Note the follo wing , wh ich will b e an aid to mak ing t he extensions described abov e. W e c laim th at to extend ∆[ s ] + ∧ ∆ t − s + → X to ∆[ s + 1] + ∧ ∆ t − s + it is sufficient to describe the extension of ∆ s +1 [ s ] + ∧ ∆ t − s + → X s +1 to ∆ s +1 + ∧ ∆ t − s + . Ind eed, make the sam e argum ent as in th e pr oof of Lem ma 6.1 , especially the argument using Diagram ( 17 ). 7 DESCRIPTION OF D 2 FOR THE ˇ CECH SPECTRAL SEQUENCE 25 7 Description of d 2 f or the ˇ Cech Spectral Sequ ence The descrip tions of the d ifferentials ab ove translate in to the f ollowing th eorem in th e setting of the ˇ Cech approxim ation. Theorem 7.1. Let X be a pr eshea f of p ointed simplicial sets on the site C , and let U • : V A → U I → U be a 1 -hypercover in C . Thus, for α ∈ A , V α ij → U ij is a covering morphism, wher e U ij = U i × U U j . Then, for t > 0 , the differ e ntials d 2 : ˇ H 0 ( U • , π t X ) → ˇ H 2 ( U • , π t +1 X ) can be described as follows. An elemen t [ δ ] of ˇ H 0 ( U • , π t X ) is repr esented by a map δ : S t → X 0 U • = Y i ∈ I X ( U i ) such that δ i is homo topic to δ j on V α ij for every i, j, α . Pic k a specific based ho motopy y α ij : δ i → δ j on V α ij . This data determines a ma p ∂ ∆ 2 + ∧ ∆ t + → X 2 U • such th at on each face of ∆ 2 , the compon ent in X ( V αβ γ ij k ) is one of the homo topies y α ij , y β j k , or y γ ki . Then , let Λ t n ⊆ ∆ t be a horn, and let ∆ 2 + ∧ ∆ t + → X 2 U • be a fill of the horn. Then, the res triction to ∆ 2 + ∧ ∂ n ∆ t + is in the class of d 2 ([ δ ]) . Pr o of. The proof follows immediately f rom Construction 6.3 and Definition 5.6 . 8 Divisibil ity Theor em In this sectio n, the ˇ Cech spectr al sequence is used to study the dif feren tials d α 2 in the Brown-Gersten spe ctral sequence of twisted algebraic K -th eory . T o a class α ∈ H 2 ( U ´ et , G m ) , one associates a stack Pro j α of α - twisted c oherent sheaves locally free and of finite ran k. This is a stack of exact categories. Th e pointwise K - theory of this s tack will b e written K α , where K α ( V ) = K Q ( Pro j α V ) = B Q ( Pro j α V ) , for V → U , and wh ere B Q ( Pro j α V ) is the classifying space of Quillen ’ s category Q ( Pro j α V ) [ 15 ]. Th e homotopy presheaves are K α k : V 7→ π k +1 K α ( V ) , 8 DIVISI BILITY THEOREM 26 and the associated sheaves are K α k . The differentials of the Brown-Gersten spectral se- quence associated to K α are written d α r . The idea o f an α -twisted sh eaf was apparently created by Girau d [ 10 ], and was re vived in the thesis of C ˘ ald ˘ araru [ 7 ]. For a mod ern geometric approach, see Lieblich [ 14 ]. In the con text o f K -th eory , see [ 1 ]. Let K α, ´ et = H K α be a fibrant replacement f or K ´ et in the local model categor y of p resheaves of space s on U ´ et . When U is of finite coho mological dimension , the Brown-Gersten spe ctral sequence of K α , E s,t 2 = H s ( U ´ et , π t K α ) = H s ( U ´ et , K α t − 1 ) ⇒ π t − s K α, ´ et ( U ) , conv erges stron gly to the homotopy of K α, ´ et . Definition 8.1 . In [ 1 ], n atural isomorp hisms K α k ≃ → K k are given. It follows th at, for U geometrica lly connected, H 0 ( U ´ et , K α 0 ) ≃ → Z . T he ´ etale ind ex of α , denoted by e ti ( α ) , is defined to be the smallest integer n ∈ H 0 ( U ´ et , K α 0 ) such that d α k ( n ) = 0 for all k ≥ 2 . In other words, eti ( α ) is the gener ator of E 0 , 1 ∞ ⊆ H 0 ( U ´ et , K α 0 ) = H 0 ( U ´ et , π 1 K α ) . Equiv alently , eti ( α ) is the positive generato r of the imag e o f the rank map K α, ´ et 0 ( U ) → Z . The per iod of α , denoted by per ( α ) , is th e order of α in H 2 ( U ´ et , G m ) . Remark 8.2 . In general, if X is a presheaf of s paces, then the differentials leaving H 0 ( U, π 0 X ) in the Brown-Gersten spectral seq uence for X are all zer o. Th is is why we use B Q ( Pro j α ) instead of Ω B Q ( Pro j α ) as our mod el for K -the ory . Of cour se, one can stabilize th e argumen t an d work with p resheaves of s pectr a. But, that argume nt requires another lev el of complexity which we wished to a void. A key ingred ient of our m ain th eorem is the following lemma , wh ich allows us to identify the cr itical lift in the description o f d 2 in the ˇ Cech spectral sequ ence for twisted K - theory . First, som e notation. Let E be an exact category , let M i , i = 0 , 1 , 2 , be objects of E , and let θ ij : M i → M j be isomorp hisms for θ 01 , θ 12 , and θ 20 . Then, the M i determine loop s in B QE , that is, elements of π 1 B QE . Recall here th at the ba se-point o f B QE is the zero object of E . The isomorphism s θ ij determine homoto pies of the loops. W e are thus in the position of h aving a map ∂ ∆ 2 + ∧ ∆ 1 + → B QE . Use th e horn Λ 1 0 ⊆ ∆ 1 to create a lif t ∆ 2 + ∧ ∆ 1 + → B QE . Then, ∆ 2 + ∧ ∂ 0 ∆ 1 + → B QE is an element of π 2 B QE , say σ . Lemma 8.3. The element σ ∈ π 2 B QE is the s ame as th e class of π 2 B QE cano nically associated to the automo rphism θ 20 ◦ θ 12 ◦ θ 01 of M 0 . Pr o of. Each face ∂ n ∆ 2 + ∧ ∆ 1 , i.e. each homotopy θ ij , correspon ds to m ap θ ij : S 2 −  D 1 ∨ D 1  → B QE , where the r estriction o f θ ij to the fir st (r esp. second) b ounda ry disc is M i (resp. M j ). Let X = S 2 −  D 1 ∨ D 1  . Then, to gether, θ 01 and θ 12 induce a map f rom the con- nected sum o f X with itself glu ed along M 1 . This connected sum is itself ho motopy equiv alent to X . T hrowing θ 20 into the pictu re, we get two maps X → B QE which 8 DIVISI BILITY THEOREM 27 on one disc boundary ar e M 0 and on the second ar e M 2 . Thu s, th ey indu ce a map f rom the connected sum of X with itself along the figure eight S 1 ∨ S 1 . But, this connected sum is homotopy equiv alent t o S 2 . The homotopy class of this map is σ . Now we check that σ agrees with the automo rphism of M 0 above. Recall tha t the category QE consists of the sam e objects as E b ut has as mo rphisms from L to N the co llection of d iagrams L և M ֒ → N , where և and ֒ → denote admissible surjectio ns a nd injection s in E , mo dulo isomorph isms of such diagrams which are equalities on L and N . An element M of E gi ves rise to an element r M in K 0 ( E ) . As a based loop in B QE , this is co nstructed as the com position of i M : 0 և 0 ֒ → M with the in verse of q M : 0 և M ֒ → M . See [ 16 ]. As explain ed in [ 16 , pp. 43-4 5], an exact sequ ence 0 → L ֒ → M ։ N → 0 in E cor responds to a choice of ho motopy b etween r M and r L · r N . This homotopy is constructed explicitly in ibid. as a map from the 2 - sphere with a wedge sum o f three discs removed, wh ere the boundar ies of these discs are the corr ectly oriented r ∗ . In particular, g i ven an isomorphism θ : L ≃ → M , we get a map a θ : S 2 −  D 1 ∨ D 1 ∨ D 1  → B QE with three discs removed. Since θ is an isomorp hism, o ne of those discs can be filled in in B QE , and we get a map X → B QE . So, the map from the pun ctured S 2 is just a proof of the equality [ L ] = [ M ] in K 0 ( E ) . When θ : L ≃ → L , the map a θ is a map from X to B Q E whose restrictions to the two boun dary discs is the same. Thus, a θ induces a map S 2 → B QE . Th is is th e element of π 2 B QE associated to θ . In our situation o f the M i and th e θ ij , we have maps a θ ij : X → B QE . It is easy to see, lookin g at the construction of th ese maps that a θ 12 ◦ θ 01 : X → B Q E is homoto pic to gluing th e two m aps a θ 01 and a θ 12 along their co mmon b oundar y circle M 1 . Similarly , gluing a θ 20 to a θ 12 ◦ θ 01 along M 2 is homoto pic to a = a θ 20 ◦ θ 12 ◦ θ 01 . As identified above, since a agrees on its two bound ary circles, a induces a map S 2 → B QE that is homotopic to σ . But, this map is als o the map associated to the automorp hism θ 20 ◦ θ 12 ◦ θ 01 of M 0 , so the lemma follows. Construction 8.4 . W e describ e how to reconstru ct th e class α ∈ H 2 ( U ´ et , G m ) f rom the stack Pro j α . The rank 1 objects of Pro j α form a G m -gerbe, Pic α . Recall from [ 10 , Sec tion IV .3.4 ] or [ 5 , Theor em 5.2.8] the fo llowing pr ocedure. First, on e takes a cover U of U such that there is an object of L i ∈ Pic α U i for all i . Seco nd, choose isomorph isms σ i : Aut ( L i ) ≃ → G m | U i , Third, pick isomorp hisms θ α ij : L i ≃ → L j on a suitable refinement 1 -hype rcover V A → U I → U . The composition θ δ ki ◦ θ β j k ◦ θ α ij 8 DIVISI BILITY THEOREM 28 is an element of Aut( L i )( Z αβ δ ij k ) , where Z αβ δ ij k = V α ij × U j V β j k × U k V δ ik . Then, σ i ( θ δ ki ◦ θ β j k ◦ θ α ij ) ∈ G m ( Z αβ δ ij k ) defines a 2 -cocycle in G m which is in the same cohom ology class as α . Theorem 8.5. Su ppose that U is geometrically conn ected and quasi-sepa rated. Let α ∈ H 2 ( U ´ et , G m ) . Th en, d α 2 ([1]) = α , th r ou gh the natural iso morphism H 2 ( U ´ et , K α 1 ) ≃ → H 2 ( U ´ et , K 1 ) ≃ → H 2 ( U ´ et , G m ) . Pr o of. Let α ∈ H 2 ( U ´ et , G m ) be define d by a ˇ Cech cocycle ˇ α ∈ ˇ H 2 ( U • , G m ) for a hyperc over V → U → U . Using Theorem 5.7 , in order to prove the theorem, it su ffices to pr ove that d 2 ([1]) = ˇ α in the ˇ Cech app roximatio n spectr al sequen ce associated to U • . (T he existence of su ch a hyp ercover is whe re the quasi-separate d hypothesis is used; see [ 2 , Theorem V .7. 4.1].) Let Z αβ δ ij k = V α ij × U j V β j k × U k V δ ik . W e m ay represent [1] ∈ ˇ H 0 ( U • , K α 0 ) by an α - twisted line bundle L i on each U i of U • . A homotopy from L i to L j is just an isomorph ism θ α ij : L i | V α ij ≃ → L j | V α ij , where such an isomo rphism exists, up to possibly refining the hy percover U • . Then, by Theore m 7.1 and Lemma 8.3 , the class of the automorp hism θ δ ki ◦ θ β j k ◦ θ α ij of L i | Z αβ δ ijk in K α 1 ( Z αβ δ ij k ) is the Z αβ δ ij k -compo nent of d α 2 ([1]) . The α -twisted line bundles L i on U i induce pointwise weak equiv alences of K - theory p resheaves φ i : K α | U i ≃ → K | U i by tensor product with L − 1 i . As shown in [ 1 ], these local morphisms patch to create natural isomorphisms of K -th eory shea ves φ i : K α k ≃ → K k , and of K -co homolog y grou ps, in particular of H 2 ( U ´ et , K α 1 ) ≃ → H 2 ( U ´ et , K 1 ) . Define σ i by fixing an isomorp hism σ i : L i ⊗ L − 1 i ≃ → G m | U i , 8 DIVISI BILITY THEOREM 29 possibly refining U • . The n, the diagram Aut( L i ) σ i, ∗ − − − − → G m | U i   y   y K α 1 ( U i ) φ i − − − − → K 1 ( U i ) is commutative, wher e σ i, ∗ is the natural isomorphism induced by σ i . The diagram an d Construction 8.4 imply t hat d α 2 ([1]) ma ps to the image of ˇ α in the map ˇ H 2 ( U • , G m ) → ˇ H 2 ( U • , K 1 ) . Remark 8.6 . There is a more so phisticated version of this theor em which uses the K - theory rin g spectrum K . Then the K α are mod ule spectra over this ring spe ctrum. In this situation, the descent spectral sequence for K α is a module over the de scent spectral s equ ence for K , and thus, if x ∈ H s ( U ´ et , K α t ) , we may write x = 1 α ∪ y , with y ∈ H s ( U ´ et , K t ) and 1 ∈ H 0 ( U ´ et , K α 0 ) the generator . Thus, d α 2 ( x ) = d α 2 (1 α ) ∪ x ± 1 α ∪ d 2 ( y ) = α ∪ x ± 1 α ∪ d 2 ( y ) . So, all of the differentials in the E 2 -page are determined by the theorem and the differ - entials of the descent spectral sequence for K . Theorem 8.7 ( Divisibility ) . Let U be g eometrica lly conne cted and qu asi-separated. If α ∈ H 2 ( U ´ et , G m ) , then per ( α ) | eti ( α ) . Pr o of. This follows immediately from Theore m 8.5 . Indeed , since d α 2 ([1]) = α , it fol- lows that p er ( α ) gener ates E 0 , 1 3 ⊆ H 0 ( U ´ et , Z ) in the Brown-Gersten spectral sequ ence. Therefo re, the gen erator of E 0 , 1 ∞ belongs to the subgr oup per ( α ) · Z . I n other word s, per ( α ) divides eti ( α ) . Remark 8.8 . In [ 1 3 ], there is an Atiyah-Hirzeb ruch spectral sequ ence in ´ etale mo ti vic cohomo logy H p − q ( X ´ et , Z ( − q )) ⇒ K ´ et − p − q ( A ) , where A re presents a class α ∈ Br( X ) . Kahn and Le vine show that d 2 ([1]) = α in H 3 ( X ´ et , Z (1)) = Br( X ) . Similarly , in [ 3 ], in the Atiyah-Hirzeb ruch spectral sequen ce H p ( X, Z ( q / 2)) ⇒ K U p − q ( X ) α conv erging to twisted top ological K -theor y , it is shown tha t d 3 ([1]) = α in H 3 ( X, Z ) . 9 INDEX AND SPECTRAL IND EX 30 9 Index and Spectral Index Let k be a field o f finite etale co homolo gical dimension . Then, the main result of [ 1 ] is that eti ( α ) | per ( α ) [ d 2 ] , (18) for all α ∈ Br( k ) h a vin g period not divisible by a f ew small p rimes. Defin e e ( k ) to be the smallest integer such that eti ( α ) | per ( α ) e ( k ) for all α ∈ B r( k ) who se period is no t divisible by the characteristic of k . Su ch an integer exists b y [ 1 , Th eorem 6 .10]. Moreover, Theorem 8.7 says th at e ( k ) ≥ 1 . Recall, in the notation of the introduction , th e group F α = K α, ´ et 0 ( k ) (0) / K α 0 ( k ) (0) . Its order is ind ( α ) eti ( α ) . Definition 9.1. Say that the field k has pr operty B n if F α is of order at most per ( α ) n for all α ∈ Br( k ) . Proposition 9.2. Let k be a field o f finite etale co homological dimension d , a nd let e ( k ) be the inte ger defined a bove. Then, the p eriod-index conjectur e ho lds for k if k has pr operty B d − 1 − e ( k ) . If the period -index con jectur e holds for k , th en k has pr op erty B d − 2 . Pr o of. Fix α ∈ Br( k ) . Then, eti ( α ) | per ( α ) e ( k ) . I f the order of F α is at m ost per ( α ) d − 1 − e ( k ) , then, in d ( α ) | eti ( α ) d − 1 − e ( k ) , whence the fir st statemen t. T o p rove the second statement, it suffices to note that ind ( α ) eti ( α ) is at most per ( α ) d − 1 per ( α ) , by the pe riod- index conjecture and Theorem 8.7 . Question 9.3. If the perio d-index conjecture h old for k do es it imply that k ha s property B d − 1 − e ( k ) ? Remark 9.4 . Th e proposition is interesting b ecause it re veals a connection between the period- index conjectur e and ´ etale descen t for (twisted) K-theory . Remark 9.5 . W e do not know the in teger e ( k ) fo r a ny fields besides 2 -dim ensional fields where the pe riod-ind ex conjectur e is known, in which case e ( k ) = 1 . Our guess is th at it is as b ig as possible, namely a t lea st [ d 2 ] . Some evidence f or this may be that the field s such as k = C (( t 1 )) · · · (( t d )) h a ve ind ( α ) | per ( α ) [ d 2 ] , so it is na tural to ask whether eti ( α ) = ind ( α ) fo r these fields. This is a line of inq uiry we a re currently pursuing . Refer ences [1] Benjamin Antieau, Cohomo logical obstruction theory for Brauer classes and the period-in dex pr oblem , J. K-Theo ry (2010) , A vailable on CJO 13 Dec 201 0 doi:10.1017/ is010011030jk t136 . 1 , 1 , 8 , 8.1 , 8.5 , 9 , 9 REFERENCES 31 [2] M. Artin, A. Grothen dieck, and J. L. V erdier (eds.), Th ´ eorie des topos et coho- mologie ´ etale des sc h ´ emas. 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Thomason , Algebraic K -th eory and ´ etale cohomology , Ann. Sci. ´ Ecole Norm. Sup. (4) 18 (1985) , n o. 3, 437–552 . MR82610 2 4 .8 Benjamin Antieau [ antieau@mat h.ucla.edu ] Departmen t of Math ematics University of Californ ia Los Angeles Box 95155 5 Los Angeles, CA 90095 -1555 USA