Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories

CONVERGENCE OF THE EILENBERG-MOORE S P ECTRAL SEQUENCE FOR GENERALIZED COHOMOLOGY THEORIES TILMAN BAUER A B S T R A C T . W e prove th at th e Morava- K -theory-based Eilenberg-Moore spectral sequence h as good convergence properties wheneve r the base space is a p -local finite Postnikov system with vanishing ( n + 1 ) st homotopy group. 1. I N T R O D U C T I O N Let F → E → B be a fibration of topological spaces. There are three classi- cal spec tr a l sequences allowing one to derive the singular homology of any one of these three spaces from the singular homolog y of the other two. If H ∗ B and H ∗ F are known (the latter as a module over π 1 B ), the Serre spectra l sequence H ∗ ( B ; H ∗ ( F ) ) = ⇒ H ∗ ( E ) is a first quadrant spectral sequence which always con- verges. Slightly less known is the bar spectral sequence ( also known as the Rothen- berg–Steenrod spectral sequence [RS65]), which a llows one to compute H ∗ ( B ) whenever F → E → B is a principal fibration. In this case, H ∗ ( F ) is a ring, H ∗ ( E ) is a module over H ∗ ( F ) , a nd the natura l filtra tion of the bar construction on F gives a spectra l sequence T or H ∗ ( F ; F p ) ( H ∗ ( E ; F p ) , F p ) = ⇒ H ∗ ( B ; F p ) which again is a first-quadrant spectral sequence with good convergence properties. This spec- tral sequence e x ists and converges with any coefficients (not just field coefficient), but for the d escription of the E 2 -term as a T or group one needs a K ¨ unneth isomo r- phism. In fac t, one ca n replace H ∗ ( − ; F p ) by any genera lized homology theory having K ¨ unneth isomorphisms and still obtain a strongly convergent right half plane spectral sequence. This paper is a bout the dua l of the bar spectra l sequence, the Eilenberg-Moore spectral sequence (EM SS) Cotor K ∗ ( B ) ∗ ∗ ( K ∗ ( E ) , K ∗ ) = ⇒ K ∗ ( F ) . Historically preda ting the bar spectra l sequence, the E M SS is a c tua lly much harder to understand b e cause it is a sec ond-quadrant ( or left ha lf-plane, for nonconnec- tive theories K ) spectral sequence; in general, it does not converge to its target in any sense. Our main result is: Theorem 1.1. Let p be an odd prime, K ( n ) the n t h Morava K - t heory at the prime p , and E 1 → B ← E 2 be a diagram of spaces such that π ∗ B is a finite (g raded) p - group and π n + 1 ( B ) = 0 . Let F = holim ( E 1 → B ← E 2 ) . Then the K ( n ) - based Eilenberg-Moore spectral sequence, E 2 ∗ ∗ = Cotor K ( n ) ∗ ( B ) ∗ ∗ ( K ( n ) ∗ ( E 1 ) , K ( n ) ∗ ( E 2 ) ) = ⇒ K ( n ) ∗ F Date : March 26, 2008. 2000 Mathematics Subject Classificati on. 55T20,57T35,55N20. Key words and phrases. Eilenberg-Moor e spectral sequen ce, Morava K -t heory , cobar constructio n, spectral sequen ce co nvergence. 1 2 TILMAN BAUER Ind-converges for any E 1 , E 2 . In particular , if E 1 , E 2 are of th e homotop y t ype of finite CW-complexes, the above spect ral sequence conv erges pro-constantly to K ( n ) ∗ F, w h ich has to be a finite K ( n ) ∗ -module. Before discussing this result, a few words a bout the history of the probl em are in order . The case of K ∗ = H ∗ ( − ; F p ) has been studied extensively [EM66, Smi70, Dwy74, Dwy75], and the convergence issues arising here are the same a s for any connective theory K . Roughly , the question of convergence only depends on π 1 ( B ) and its action on the homology of F . Now let K be nonconnective and possessing K ¨ unneth isomorphisms. Thus K is one of Morava ’s extraordinary K -theories K ( n ) or an ex tension of it, which is the c ase I am interested in in this work. The question of c onvergence becomes much more intricate; in particula r , nonconvergence can occur even for simply con- nected base space s. As an example, consider the path-loop fibra tion K ( Z / 2, 1 ) → ∗ → K ( Z /2 , 2 ) a nd K = K ( 1 ) = K U /2, mod-2 ordinary K -theory . In this case K ( 1 ) ∗ ( K ( Z /2, 2 ) ) = K ( 1 ) ∗ , but K ( 1 ) ∗ ( K ( Z /2, 1 ) ) is nontrivial, so that there is no chance for the (trivial) EMS S to converge. Theorem 1.1 says that this nonvanishing of π n + 1 is in fac t the only obstruction to convergence if the ba se space has totally finite homotopy groups which are p -groups. Note that the condition that the ho- motopy groups are p - groups is not too restrictive b e cause we can always replace the fibration under consideration by its Z / p -localiz a tion, and the EM SS will never know the difference (although the fiber might change dra stically). Previous work on the K ( n ) -based EM SS includes work by T amaki [T am94], where he shows convergence when the base spa ce is of the f orm Ω n − 1 Σ n X (he mistakenly claims strong convergence), and work by J eanneret and Osse [J O99], where the authors show convergence whenever the base space has certain homo- logical global finiteness properties, f or ex a mple, if the base spa ce is the classifying space of a polynomial p - compact gro up. The term “ Ind-convergence” in Theorem 1.1 requires explanation. W e first re- call the classical notion of p ro-convergence (c a lled strong convergence in [B ou87, Shi96], but different f rom Cartan-Eilenberg’s and Boardman’s notion of str ong convergence [CE99, Boa 9 9]). Associated to F = holim ( E 1 → B ← E 2 ) , there is a tower of K ( n ) -module spectra T • ( E 1 , E 2 ) = T ot • K ( n ) [ C B ( E 1 , E 2 ) ] coming f rom the two-sided cobar constr uction of E 1 and E 2 over B , a nd a map K ( n ) [ F ] → T • ( E 1 , E 2 ) . The spectral sequence always converges conditionally to π ∗ ( holim T • ) , which may be d ifferent from K ( n ) ∗ ( F ) . W e say that the spectral sequence is pro- constantly convergent if K ( n ) ∗ F → π ∗ T • is a pro-isomorphis m from the constant object K ( n ) ∗ F to this natura l target. This in particular implies that the spe c tral se- quence converges in a very strong sense, namely , only finitely many d ifferentials live at any bid e gree, a nd in E ∞ the filtration is finite in every total de gree. W e consider the E MSS not as one spectr a l sequence, but a s a whole directed system of spectral sequences, one for eac h pair of finite sub-CW -complexes of E 1 and E 2 . Similarly , the target K ( n ) ∗ F can be thought of as the directed system of K ( n ) ∗ F ′ where F ′ runs through all finite sub-CW -complexes of F . W e call the spec- tral sequence Ind-convergent if the comparison map K ( n ) ∗ F → π ∗ T • ( E 1 , E 2 ) is an isomorphism in the ca tegory of ind-pro-abelian groups. In particula r , if E 1 and E 2 are finite CW -complexes, then T • ( E 1 , E 2 ) is ind-constant, thus K ( n ) ∗ F also has to be ind-constant, which can only happ e n if K ( n ) ∗ ( F ) is finite. Furthermore, in this case, since K ( n ) ∗ F is pro-constant, T • also has to be isomorphic to a pro-constant tower , and we get the specialization mentioned in the theorem. W e also mention that Ind-convergence is a good enough notion to allow for a comparison theorem of spectral sequences: CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 3 Corollary 1.2. Let ( E ′ 1 → B ′ ← E ′ 2 ) → ( E 1 → B ← E 2 ) be a ma p of diagrams such that the associat ed spectral sequences ( E ∗ ) ′ , E ∗ converge Ind-constant ly to K ∗ ( F ′ ) and K ∗ ( F ) , respectively . Then th ere is an induced ind-m ap ( E ∗ ) ′ → E ∗ , and if for any s, ( E s ) ′ → E s is an ind-isomo rp hism, then K ∗ ( F ) ∼ = K ∗ ( F ′ ) . W e get an induced map be c ause eve r y finite sub-CW -c omplex of E ′ i maps to another finite sub-CW -complex of E i . The result follows f rom the fact that we get an ind-isomorphism K ∗ ( F ) → K ∗ ( F ′ ) , and the ordinary group K ∗ ( F ) is simply the colimit of the ind-group K ∗ ( F ) . The heuristic reason for introducing this slightly unwieldy notion of Ind-con- vergence is that the natura l ta rget of the spectr a l sequence, holim T • , does not commute with infinite colimits or even infinite coproducts because it is a n inverse limit. However , the fiber of a colimit of total spaces over a fixed base space is just the colimit of the total spa ces of the individual fibrations, a nd thus we have to “ train” the spectral sequence to commute with colimits. This is achieved by passing to the ind-ca tegory . Theorem 1.1 is proved by means of a series of results of possibly independent interest. W e collect the main steps here. Let K = K ( n ) be the n th Morava K -theory . Theorem 1.3 . For odd p , the K ( n ) -based Eilenberg-Moore spectral sequence Ind-converges for the path -loop fibration on K ( Z / p , m ) whenever m 6 = n + 1 . This result is proved in Section 6 by a complete computation of the spectral sequence, a ided by the computations of K ( n ) ∗ ( K ( Z / p , m ) ) for all m , n in [RW80]. It is likely that the result also holds f or p = 2 , but extra care is need ed due the non-commutativity of K ( n ) in this case. T o pass from a contractible total space to more genera l cases, in Section 4 we prove: Theorem 1 .4. Let B be a spac e such th at ∗ → B ← ∗ has an Ind-co nvergent EMSS. Then so has E 1 → B ← E 2 for any E 1 , E 2 . W e call a space B with this property Ind-conv ergent . Finally , in Section 5 we show how to pa ss to more complicated base spaces than just K ( Z / p , m ) : Theorem 1.5. Let F → Y → X be a fibration with F and X Ind-convergent. T hen so is Y . Proof of Theorem 1.1. By Theorem 1.4, it suffices to consider the ca se E 1 = E 2 = ∗ . Since B has finite homotopy groups which are p -groups, it has a finite Postni kov decomposition B = B k   . . .   B 1   / / K ( Z / p , n 3 ) B 0 = K ( Z / p , n 1 ) / / K ( Z / p , n 2 ) , where none of the n i equal n + 1. By induction, Theorem 1.5 and Theorem 1.3, we conclude that B is Ind-convergent.  4 TILMAN BAUER Section 2 is a n e x position of the c onstruction of the Eilenberg-Moore spectral sequence in the generality we need, and Section 3 deals with the structure of ind- pro-objects. 2. T H E G E N E R A L I Z E D B O U S FI E L D A N D E I L E N B E R G - M O O R E S P E C T R A L S E Q U E N C E S Associated with any cosimplicial spa ce C • there is a tower of spa ces, called the T ot-tower , { T ot s C • } s ≥ 0 , T ot s C • = map ( ∆ • ≤ s , C • ) , where ∆ • denotes the cosimplicial space whose s th space ∆ s = | ∆ [ s ] | is the stan- dard s -simplex, ∆ • ≤ s denotes the s -skeleton of ∆ • , and “map” is the mapping space functor . This tower is the target of a ca nonical map from T ot C • = holim s T ot s C • = map ( ∆ • , C • ) . A similar construction ca n b e made for cosimplicial spectra ; these are conn ected by homotopy equivalences T ot s Ω ∞ C • ≃ Ω ∞ T ot s C • , for cosimplicial spectra C • , but in general, T ot s Σ ∞ C • 6≃ Σ ∞ T ot s C • . More generally , let K be any spe c trum, and define K [ X ] to be the K -module spe c- trum K ∧ Σ ∞ ( X + ) on the space X . There is a natura l map of towers of spectra { K [ T ot s C • ] } s Φ − → { T ot s ( K [ C • ]) } s which is rarely a homotopy equivalence. For both towers there is a spectra l sequence abutting to their respective homo- topy inverse limits; it is, however , only the one on the right hand side whose E 1 - and E 2 -terms ha ve a c onvenient formulation. For we hav e a cofibra tion sequence of spectra (2.1) Σ − s N s ( K [ C • ]) → T ot s ( K [ C • ]) → T ot s − 1 ( K [ C • ]) where the normalization N s is the fiber of C s → M s − 1 ( K [ C • ]) , the la tte r being the cosimplicial matching space [GJ99, Chapter s VII.4, VIII.1]. This yields E 2 s , t = π s K t ( C • ) . This spectral sequence is the K- based Bousfield spect ral seq uence or homol- ogy spectral sequence of a cosimplicial space [ B ou87]. The spectral sequence belonging to the tower in the source of Φ , however , has a more accessible target, namely π ∗ holim s K [ T ot s C • ] . There is a map of towers (2.2) { K [ T ot C • ] } P − → { K [ T ot s C • ] } s , where the left tower is constant, a nd thus we get a comparison map from K ∗ T ot C • to the target of either spectra l sequence. Bousfield [Bou87] studied when the maps P and Φ a re pro-isomorphisms in the case of K = H F p , which would imply that the associated spectral sequences a ll converge to K ∗ ( T ot C • ) . V arious criteria were given for convergence [Bou87, The- orems 3. 2 , 3.4 , 3.6] , which generalize to the case of connective homology theories K . However , convergence for periodic theories remained an intricate problem. 2.1. Forms of convergence. Let us ex a mine the notion of convergence in the Bous- field spectr a l sequence more closely . W e d efine a decreasing filtration F • K ∗ ( T ot C • ) by F s K ∗ ( T ot C • ) = ker  K ∗ ( T ot C • ) Φ ∗ ◦ P ∗ − − − → π ∗ T ot s ( K [ C • ])  . Recall from [ CE99, Boa99] that the spectra l sequence is called strongly convergent to K ∗ ( T ot C • ) if two conditions are satisfied: CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 5 (1) The natural map F s / F s + 1 K ∗ ( T ot C • ) → E s , ∗ ∞ is an isomorphism and (2) the filtration F • K ∗ ( T ot C • ) is complete Hausdorff, i. e. lim F s = lim 1 F s = 0. It is called co mpletely convergent if it is strongly convergent and additionally , lim s 1 π ∗ T ot s ( K [ C • ]) = 0. Remark 2.3. Assuming stron g convergence, complete c onvergence is equivalent to K [ T ot C • ] ≃ T ot K [ C • ] : In fac t, strong convergence means that K ∗ ( T ot C • ) ∼ = lim s π ∗ T ot s ( K [ C • ]) ; the Milnor exac t sequence 0 → lim s 1 π t + 1 T ot s K [ C • ] → π t T ot ( K [ C • ]) → lim s π t T ot s ( K [ C • ] → 0 thus shows that the lim 1 -term vanishes if and only if K [ − ] commutes with T ot for C • . The litmus test for the usability of any notion of converg ence is whether it im- plies the spec tr al sequence comparison theorem. For strong convergence, this was proved in [Boa99, Theorem 5.3]: Theorem 2.4 (Boardman) . Let f : C • → D • be a map of cosimp licial spaces such that the Bousfield sp ectral sequ ences for C • and D • converge strongly. If f induces an isomorph ism on any E s -term ( 1 ≤ s ≤ ∞ ) then f also induces an isomorphism K ∗ ( T ot C • ) → K ∗ ( T ot D • ) . There is a different and stronger version of the term strong convergence in the context of the Bousfield or Eilenberg-Moore spectral sequence. In [Bou87, Shi96], the Bousfield spectra l sequence associated to a homology theory K a nd a cosim- plicial space C • is called strongly convergent if the tower map { K t ( T ot C • ) } s Φ ∗ ◦ P ∗ − − − → { π t T ot s ( K [ C • ]) } s is a pro-isomorphism for each t , where the tower on the left hand side is constant. Explicitly , this means that f or every t ∈ Z and s ≥ 0, there is an N ( s , t ) ∈ N a nd a map in the following diagram, making both triangles c ommute: K t ( T ot C • ) Φ ∗ ◦ P ∗ / / π t T ot s + N ( s , t ) K [ C • ] v v m m m m m m   K t ( T ot C • ) Φ ∗ ◦ P ∗ / / π t T ot s K [ C • ] W e will call this kind of convergence p ro-constant convergence . If the function N ( s , t ) can be chosen to be constant (say ≡ N ), we call the tower map a n N - isomorphism and the spectral sequence N -convergent . In our applications, K is a periodic homology theory , which means that if the spectral sequence is pro- constantly convergent, we can choose N ( s , t ) to be independent of t ; v iz , take N ( s ) = max { N ( s , t ) | 0 ≤ t < period of K } . Recall the following well-known lemma, which is a generaliza tion of [Bou87, Lemma 3.5]: Lemma 2 .5. Th e Bousfield spectral sequence E ∗ ∗ ∗ associated t o a tower o f spectra C • is pro-constantly convergent if a nd only if 6 TILMAN BAUER (1) For each s , t , there is an N = N ( s , t ) > 0 such that E N s , t = E ∞ s , t in the spectral sequence and (2) For each k there is an N ( k ) such that E ∞ s , s + k = 0 for s ≤ N ( k ) . Moreover , the spectral sequence is N -convergent if and only if N ( s , t ) and N ( k ) above can be chosen to be constant with value N . Lemma 2.6. Pro-constant convergence implies complete convergence. Proof. Le t Y → X • be a tower of spectra under Y such that the associated spectral sequence is pro-constantly convergent. By Lemma 2 . 5, this implies that for each s , t , there is an n such that E s , t n = E s , t ∞ . Thus the derived E ∞ term R E ∞ = lim 1 r Z r is zero, and by [Boa99], the spectra l sequence is completely convergent.  Remark 2.7. The Bousfield spectral sequence a ssociated to a tower of spectra C • is pro-constantly (resp. N -) convergent if and only if (1) Φ : K [ T ot C • ] → T ot K [ C • ] is a homotopy equivalence; a nd (2) The tower π t T ot K [ C • ] is pro- (resp. N -) constant for each t . The reason for this, as in Remark 2 .3, is that f or pro-constant towers, the derived functor of the inverse limit is trivial. While c omplete convergence is a perfectly fine property to ensure that the spec- tral sequence dete r mines its target uniquely up to filtration, it has the technical disadvantage that the leve lwise cofiber of two completely convergent towers need not be completely convergent, making it all but unusable for inductive arguments. On the other hand, the tower Five Lemma [BK72] implies that pro-constant con- vergence is preserved under taking levelwise cofibers. In general, there is no Five Lemma for N - isomorphism s. 2.2. The Eile nberg-Moore s pectral s e quence as a specia l case of the Bousfiel d spectral seque nce. Let F / /   E 1 p 1   E 2 p 2 / / B be a pullback diagra m of spaces, where a t least one of E 1 → B and E 2 → B is a fibration. Denote b y C • = C • B ( E 1 , E 2 ) the coba r construction for E 1 → B ← E 2 ; that is, C s = E 1 × B s × E 2 with the usual coboundary and codegenera c y ma ps. W e abb reviate C B ( E ) f or C B ( E , ∗ ) . The Bousfield spectra l sequence associated to C • is called the Eilenberg-Moore spectral sequence. The convergence question in this spe cial case becomes a little bit simpler , but is still hard to ta ckle. In fa ct, one readily sees that T ot 0 C • = E 1 × E 2 , whereas T ot i C • ≃ F f or all i ≥ 1. Thus the map P of (2.2) is always an isomorphism in positive degrees. In Bousfield’s terms, in this case “ pro-convergence” implies pro-constant convergence. The description of the E 1 -term becomes significantly sim pler than in the generic Bousfield spectral sequence since one ca n avoid the computation of the normaliza- tion of the cosimplicial spectrum K [ C • ] a s in (2.1). Lemma 2.8. The cosimplicial spectrum K [ C • ] is codegeneracy-free , i.e. it is the right Kan ext ension of a dia g ram Z : Λ → { Spectra } , where Λ is the category with objects [ n ] , n ≥ 0 , and injective m onotonic map s as mo rphisms. This diag ram is defined by Z n = K ∧ ( ( E 1 ) + ∧ B ∧ n ∧ ( E 2 ) + ) . CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 7 Proof. The cofa c e maps in Z • are defined by d 0 ( e 1 , b 1 , . . . , b n , e 2 ) = ( e 1 , p 1 ( e 1 ) , b 1 , . . . , b n , e 2 ) , d n + 1 ( e 1 , b 1 , . . . , b n , e 2 ) = ( e 1 , b 1 , . . . , b n , p 2 ( e 2 ) , e 2 ) and d i ( e 1 , b 1 , . . . , b n , e 2 ) = ( e 1 , b 1 , . . . , b i , b i , . . . , b n , e 2 ) for 1 ≤ i ≤ n . Let I : Λ → ∆ denote the inclusion functor . Then the right Ka n extension over I of any Λ -diagra m of spectra Z • can explicitly be d e scribed as RKan I Z • n = _ [ n ] ։ [ k ] Z k where the wedge runs over all surjections [ n ] → [ k ] (of which there are ( n k ) ). For every such φ : [ n ] → [ k ] , there is a map φ ∗ : B ∧ k → ( B + ) ∧ k = B n + given on the i th coordinate ( i = 1, . . . , n ) by φ ∗ i ( b 1 , . . . , b k ) = ( ∗ ; φ ( i ) = φ ( i − 1 ) b φ ( i − 1 ) ; otherwise Now K [ C n ] = K [ E 1 × B n × E 2 ] = K ∧  ( E 1 ) + ∧ ( B + ) ∧ n ∧ ( E 2 ) +  ( φ ∗ ) ← − − _ φ : [ n ] ։ [ k ] K ∧  ( E 1 ) + ∧ B ∧ k ∧ ( E 2 ) +  is an isomorphism compatible with the cosimplicial structure maps.  This implies that E 1 r , s = K r − s ( ( E 1 ) + ∧ B ∧ n ∧ ( E 2 ) + ) . The E 2 -term a lso has a convenient description, at least if K ∗ is a graded field or , more generally , if K ∗ ( B × X ) ∼ = K ∗ ( B ) ⊗ K ∗ K ∗ ( X ) for a ll X . It is given by E 2 r , s = Cotor K ∗ B r , s ( K ∗ E 1 , K ∗ E 2 ) . 2.3. The Eilenb e rg-Moore spectral sequen c e for parametriz ed spectra. It is cru- cial f or the Eilenberg-Moore spectral sequence that the ba se spa ce is indee d a space and not, say , a spectrum. That is, we ca nnot e x pect a functorial spectr a l sequence that takes as input a dia gram of spectra E 1 → B ← E 2 and which computes some- thing that in the case of suspension spectra of spaces is the suspensio n spectrum of the homotopy pullback. The technical reason is that the d iagonal on B is needed for the cobar construction. However , for E 1 and E 2 we only need the coaction E i → E i × B , never the diagonal. In this section, I will set up a convenient category of K -module spe c tra over a space B and show that the Eilenberg-Moore spectra l sequence can be genera lized for pairs of such K -module spe c tra over B . Let B be a pointed space. The objects of the category ( T op / B ) ∗ of sectioned spaces over B are of the form ( X p X − → B , s X ) , where s X : B → X is a section of p X . The maps are given by maps f : X → Y over B compatible with the sections. In the cate gory ( T op / B ) ∗ one can define fiberwise homotopical c onstructio ns such as cofibers, fibers, suspensions, smash products etc, which we will denote by adding a subscript B to the usua l symbol, e. g. ∧ B . The category ( T op / B ) ∗ is also complete and cocomplete. For details about these construction, consult e.g. [Smi70, MS 06]. The ca tegory Sp B is the cate gory of spectra over the space B . An object is a se- quence X i x − → B in ( T op / B ) ∗ together with maps Σ B X i → X i + 1 . If X ∈ ( T op / B ) ∗ 8 TILMAN BAUER is a sectioned space, the fiberwise suspension spec trum Σ ∞ B X is an object in Sp B . As for spaces, all homotopical constructions such as (co-)fibers, smash products, (co-)limits work in this fiber wise setting. In [MS06], this is treated in a “brave new” way , but for our purposes, the naive notions of spectra are enough. If X is a pointed space (not over B ), we denote by X B the object ( X × B → B , s ) , where s ( b ) = ( ∗ , b ) . Similarly , if E is any spectrum, we denote by E B the spectrum over B whose n th space is ( E n ) B . This is indeed a spectrum over B by mea ns of the structure maps Σ B ( E n ) B = ( Σ E n ) B → ( E n + 1 ) B . In fact, the functor X → X B is right adjoint to the forgetful functor U : T op / B → T op (forgetting the map to B ). and E → E B is right adjoint to the U : Sp / B → Sp sending ( E → B , s ) to { E n / s ( B ) } , which is an ordinary spectrum. In particular , for an ordinary ring spectrum K , we have the notion of a fiber wise K -module spectrum, i.e. a spectrum X ∈ S p B with a homotopy a ssociative and unital action K B ∧ B X → X . W e denote the category of K -bimodule spectra over B by Mod K / B . There is a fiberwise smash p roduct ∧ K , B : Mod K / B × Mod K / B → Mod K / B . Now let K be a ring spectrum and E 1 , E 2 ∈ Mod K / B . W e define the cobar construction C B ( E 1 , E 2 ) to be the cosimplicial K -module spectrum C B ( E 1 , E 2 ) n = ( U ( E 1 ) ∧ K K [ B n ] ∧ K U ( E 2 ) ) , where the middle fa ce a nd degener a cy maps are induced by the diagonal of B and projections , and the a djunction counits E i → ( U ( E i ) ) B induce the remaining structure maps U ( E i ) → U ( U ( E i ) B ) = U ( E i ) ∧ K K [ B ] . If E 1 → B ← E 2 is a dia gram of spaces, we have that C n ( K [ E 1 ] ∨ B , K [ E 2 ] ∨ B ) ∼ = K [ E 1 ] ∧ K K [ B n ] ∼ = K [ E 2 ] = K [ C n B ( E 1 , E 2 ) ] , thus we recover our original cobar construction for spaces. 3. P R O - O B J E C T S A N D I N D - P R O - O B J E C T S In this section, we will study ind-pro-objects in K -module spectra a nd their homotopical properties. W e start with a motivating example illustrating our need to introduce ind-structur es. Consider a n infinite set of fibrations E i → B with the same base space and inclusions E i → E i + 1 , and let E = S i E i . Then obviously , the fiber of E → B is the union of the various fibers F i of E i → B . T hus, ta king fibers commutes with filtered colimits. However , if we study the c onstruction of the E ilenberg-Moore spectral sequence, we find that while K ( n ) ( X ) = T ot n K [ C B ( X )] still commutes with filtered colimits (T ot n is a finite limit), K ( X ) = holim n K ( n ) ( X ) = T ot K [ C B ( X )] might not because the inverse limit has no reason to commu te with colimits. Thus, in a way , K ( X ) is not “ the correct target” of the spectral sequence. T o offset this deficit, we could think of X as the directed system of all compact subobjects of X , i.e., all finite sub-CW - complexes of X if we assume X to be a CW - complex. Ap- plying K to this system, we obtain a functor that now commutes with all filtered colimits and thus represents a be tte r target for the spectral sequence, which is now CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 9 really a filtered diagram of spectr al sequences. Thus, instead of looking a t towers as objects in the pro-category Pro − Mod K , we a re now looking at ind-objects in this pro-category , that is, objects in T = Ind − Pro − Mod K . Let C be any category . The category Pro −C has as objects pa irs ( I , X : I → C ) where I is a cofiltered small category and X is a functor . The morphisms are given by Hom Pro −C ( ( I , X ) , ( J , Y ) ) = lim J colim I Hom C ( X , Y ) . It is useful to think of this as saying that giving a morphism is giving for every j ∈ J a map X ( i ) → Y ( j ) for some i ∈ I , although this ignores the fa c t that these have to be compatible in some way . Dually , the category Ind −C has a s objects p a irs ( I , X ) as above b ut with I a filtered category; the morphisms are given by Hom Ind − C ( ( I , X ) , ( J , Y )) = lim I colim J Hom C ( X , Y ) . Recall [AM69, Appendix] that any map X → Y in Ind −C or Pro −C can be represented by a level map, that is, there is a filtered cate gory I (or , without loss of generality , a directed set I ), f unctors X ′ , Y ′ : I → C , a natural transformation X ′ → Y ′ and isomorphism s X ∼ = X ′ , Y ∼ = Y ′ in Ind − C (resp. Pro −C ) such that X → Y is the composite X ∼ = X ′ → Y ′ ∼ = Y . S imilarly , any finite, loopless diagra m D → Ind −C or D → Pro −C is isomorphic to a diagram of levelwise maps. However , an obj ect X ∈ Ind − Pro −C is not necessarily isomorphi c to a doubly indexed system X t s , where t runs through an inverse system T and s runs through a d irect system S . Any such X is isomorphic to a diagra m S → Pro −C , for a di- rected set S , but since S is not finite, we c a nnot replace this diagram by a levelwise diagram. Instead, X can always be represented by a functor Q S , α → C , where S is a d irected set, α : S op → { inverse sets } is a functor , and Q S , α is a poset with objects pairs ( s , t ) ( s ∈ S , t ∈ α ( s ) ) and ( s , t ) ≤ ( s ′ , t ′ ) if s ≤ s ′ and t ≤ α ( s < s ′ ) ( t ′ ) ∈ α ( s ) . Remark 3 .1. For later reference, we note that pro-pro-objects c an be described in a similar way , giving an inverse set S , a functor α : S op → { inverse sets } , and a functor X : Q S , α → C . In this case, unlike for ind-pro-objects, the poset Q S , α is again an inverse set, a nd thus X can also be interpreted as an object in Pro −C . W e denote this tautological “reinterpretation functor ” by D : Pro − Pro −C → Pro −C . (The letter is supposed to remind one of the diagonal of a double tower .) W e call a na tural tra nsformation of functors X → Y : Q S , α in Ind − Pro −C a levelwise map . The fact that any ind- oder pro-map is isomorphic to a levelwise map now easily genera lizes to Lemma 3.2. Every map in Ind − Pro − C is isomorph ic to a levelwise map .  In our applications, the category C will be either T op / B , the category of topo- logical spaces over a base space B , or Mod K / B , the homotopy K -module spectr a over B . Recall from Section 2. 3 that there are forgetful functors U : T op / B → T op (forgetting the map to B ) and U : Mod K / B → Mod K (sending ( X → B , s ) to X / s ( B ) ). If K is a homology theory or π ∗ , we write K ∗ ( X ) for K ∗ ( U ( X ) ) . The category T op / B carries a model structure, where a map f is a weak equiva- lence, fibration, or cofibration if the underlying map U ( f ) in T op is a weak equiv- alence, Serre fibration, or Ser re cofibration. However , as is pointed out in great detail in [MS 06 , Ch. 6], this model structur e has ba d properties: for ex a mple, the fibrant replacement functor does not in genera l commute with cofibers, e v e n up to wea k equivalence. However , there is a model structure, which M ay a nd Sig- urdsson call the qf-structur e, which has the same weak e quivalences, but different cofibrations a nd fibrations [M S06, Thm 6.2.5]. This allows us to equip the ca tegory 10 TILMAN BAUER of spectra over B with a good model structure as well [MS06, Thm 12.3.1 0] whos e fibrant objects are the Ω -spectra over B . Similarly , the cate gory of K -module spec- tra over B carrie s a model structure [MS 06 , Thm 14 .1.7]. In these two structures, the weak equivalences a re the stable homotopy equivalences after applying U . The reader should be wa rned that constructing these model structur es is a sub- stantial amount of work. However , for our purposes, it is enough to know that there exists a model structure with the right weak equivalences. W e can use this model structure as a black box. Definition. A map X → Y in Ind −C or in Ind − Pro −C is an essentially levelwise weak equivalence if it is isomorphic to a levelwise map which is a weak equivalence on every level. It was proven in [Isa04] that a composite of essentially levelwise weak equiv- alences is aga in a n essentially leve lwise weak equivalence (in fact, that the ind- objects in any proper model c a tegory carry a model str uc ture where the wea k equivalences are the essentially levelwise weak equivalences). For the following results, we need K to be a field. In this case, we have that [ X , Y ] K ∼ = Hom K ∗ ( π ∗ X , π ∗ Y ) , where the left hand side means homotopy classes of K -module maps. Lemma 3.3. Let (3.4) X f / / p   Y ~ ~ p   X ′ f ′ / / Y ′ , be a commutative d iag ram in Mod K / B , where the d otted arrow denotes a map π ∗ Y → π ∗ X ′ such that the dia g ram commutes on homot opy groups. (1) Assume X ′ is cofibrant and fibrant and Y is fibrant in Mod K , and that f ′ is a fibration in Mod K / B . Then there exists ˜ X ∈ M od K / B and a commutativ e diagram X f / /   A A A Y   ˜ X = = | | |      X ′ f ′ / / Y ′ , where ˜ X → Y is a fibration inducing a surjective m ap in homotop y. (2) If we hav e a map of diag rams D 1 → D of th e form (3.4) , given by X 1 → X , Y 1 → Y , etc., and ˜ X as in (1) , there is a completion ˜ X 1 of D 1 as in (1) a nd a m ap ˜ X 1 → ˜ X ma k ing everything commute: X f / /   Y p   ˜ X > > } } } } } } } }   X ′ f ′ / / Y ′ X 1 f 1 / /   ? ? ? ? ?   Y 1 p 1     > > > > > ˜ X 1 A A A A ? ? ~ ~ ~ ~ ~ ~ ~   X ′ 1 f ′ 1 / / " " D D D Y ′ 1 ! ! C C C C CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 11 (3) Dually , assume Y is cofibrant and fibrant and X ′ is cofibrant in Mod K , and that f is a cofibration in Mod K / B . Th en there exists ˜ Y ′ ∈ M od K / B and a c o mmu- tative diagram X f / /   Y   } } { { { ˜ Y ′ A A X ′ f ′ / / > > ~ ~ Y ′ , where X ′ → ˜ Y ′ is a cofibration inducing an injective map in homotop y . (4) If we hav e a ma p of diagrams D 1 → D as in (4) and ˜ X as in (3) , there is a completion ˜ X 1 of D 1 as in (3) and a map ˜ X 1 → ˜ X ma k ing everything commute: X f / / p   Y   ˜ Y ′   X ′ f ′ / / > > ~ ~ ~ ~ ~ ~ ~ Y ′ X 1 f 1 / /   ? ? ? ? ? p   Y 1     > > > > > ˜ Y ′ 1   A A A A X ′ 1 f ′ 1 / / " " F F F ? ?        Y ′ 1 " " E E E E Proof. Consider the case (1). Let V = coker ( π ∗ X → π ∗ Y ) . Since Y is fibrant, we can find a K -module map g : M → Y for some cofibra nt K -module M realizing a section of π ∗ Y → V = π ∗ M . L e t φ : Y → X ′ be a map realizing the dotted arrow , making the tria ngles homotopy commutative. Note that we may not be able to choose φ as a ma p over B . The maps p ◦ g and f ′ ◦ φ ◦ g : M → Y ′ are homotopic by assumption . Choose a homotopy H : I ⊗ M → Y ′ such that H 0 = p ◦ g a nd H 1 = f ′ ◦ φ ◦ g . Here I denotes the unit interval [ 0, 1 ] . Since f ′ is a fibration, we have a lift in the diagra m M i 1   φ ◦ g / / X ′ f ′     I ⊗ M / / ˜ H ; ; x x x x x Y ′ . W e now have a a map M → Y × Y ′ X ′ given by ( ˜ H 0 , g ) , and it is a map over B in a unique way . Now factor the map X ∨ M ( f , g ) − − → Y a s a trivial cofibration followed by a fibra- tion, X ∨ M / / ∼ / / ˜ X / / / / Y . The map X ∨ M → X ′ given b y ( p , ∗ ) f actors through ˜ X because X ′ is fibrant, a nd the commutativity of the dia gram is verified. Note tha t the construction of ˜ X is by no means functorial. Assertion (2) is a partial substitute for this deficiency . T o show (2 ), consider the f ollowing diagram: X 1 f 1 / /   Y 1   ˜ X × X ′ X ′ 1 / / Y × Y ′ Y ′ 1 12 TILMAN BAUER I claim that this diagra m satisfies the assumptions of (1). Since ˜ X → Y , X ′ → Y ′ , and X ′ 1 → Y ′ 1 are fibra tions, so is the fiber product, thus the bottom map is a fibration. Moreover , ˜ X × X ′ X ′ 1 is fibrant as a fiber product of fibrant spa ces. W e need to produce a map π ∗ Y 1 → π ∗ ( ˜ X × X ′ X ′ 1 ) = π ∗ ˜ X × π ∗ X ′ π ∗ X ′ 1 making the resulting triangles commute in homotopy . A map π ∗ Y 1 → π ∗ X ′ 1 is given b y φ 1 ; furthermore, since π ∗ ˜ X → π ∗ Y is surjective, we can find a section π ∗ Y → π ∗ ˜ X whose composite with π ∗ ˜ X → π ∗ X ′ is φ . By the commutativity of π ∗ Y 1 φ 1 / /   π ∗ X ′ 1   π ∗ Y φ / / π ∗ X ′ , we obtain a well-defined diagonal map making everything commute in homotopy . For the assertion (3), we app ly a dual constr uction. Let V = ker ( π ∗ X ′ → π ∗ Y ′ ) . Since X ′ is cofibrant, we ca n find a K - module map g : X ′ → M for some fibrant K -module M realizing a retraction of π ∗ M = V → π ∗ X ′ . Let φ : Y → X ′ be a s before. The maps g ◦ p and g ◦ f ◦ φ ◦ g : X → M are homotopic by assumption. Choose a homotopy H : X → hom ( I , M ) such that H 0 = g ◦ p and H 1 = g ◦ f ◦ φ ◦ g . Since f is a cofibration, we have a lift in the diagram X / /     hom ( I , M ) i ∗ 1   X ′ g ◦ φ / / H 7 7 p p p p p p hom ( { 1 } , M ) ∼ = M W e now have a a map X ′ ∪ X Y → M given by ( g , ˜ H 0 ) . Now Y ′ × M is a K -module over B by means of the map Y ′ × M → Y ′ → B . Fa c tor X ′ ( f ′ , g ) − − − → Y ′ × M a s a cofibration followed by a trivia l fibration X ′ / / / / ˜ Y ′ ∼ / / / / Y ′ × M . The map Y → Y ′ × M given by ( p , ∗ ) fa c tors through ˜ Y ′ because Y is cofibra nt, and the commutativity of the dia gra m is again verified. The proof for (4) is not dual to (2). ( T he d ual assertion would be that given ( D 1 , φ 1 ) , we ca n find φ in D c ompatible with φ 1 ). It is in fact easier . Consider the diagram X 1 / /   Y 1   X ′ 1 / / ˜ Y ′ × Y ′ Y ′ 1 . This diagram satisfies the c onditions of (4) be cause the original map φ still works. The resulting ˜ Y ′ 1 maps down to ˜ Y and to Y ′ 1 . A lso π ∗ ˜ Y ′ 1 → π ∗ Y ′ 1 is injective because π ∗ ˜ Y ′ 1 → π ∗ ( ˜ Y ′ × Y ′ Y ′ 1 ) = π ∗ ˜ Y ′ × π ∗ Y ′ π ∗ Y ′ 1 is injective by construction and π ∗ ˜ Y ′ → π ∗ Y ′ is injective by assumption.  Proposition 3 . 5. T he ind-weak equivalences in Ind − Mod K / B are exactly the essen- tially levelwise weak equivalences. Similarly , th e ind-pro-weak equivalences in Ind − Pro − Mod K / B are th e essentially levelwise weak equivalences. CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 13 By the two-out-of-three property for e ssentially levelwise wea k equivalences, we have the liberty to produce any composition of levelwise weak equivalences and ind-(pro-)isomorphisms in the proof of the proposition. Proof. Le t f : X → Y be an ind-weak equivalence in Ind − Mod K / B . W e may assume f is given by a levelwise map f s : X s → Y s . The condition that f is an ind- weak equivalence means tha t for every s there is an s ′ > s and a ma p φ : π ∗ Y s → π ∗ X s ′ such that the d ia gram π ∗ X s / /   π ∗ Y s   φ z z v v v v v v v v v π ∗ X s ′ / / π ∗ Y s ′ commutes. By passing to a cofinal subsystem (inducing an ind-isomorphism), we may assume that there are no t between s and s ′ . W e will procede in two steps, first factoring X → Y as X → ˜ X → Y where the first map is an ind-isomorphism and the second map is a levelwise map that is surjective in homotopy , and then f a ctoring ˜ X → Y as ˜ X → ˜ Y → Y such that ˜ Y → Y is an ind-isomorphism and ˜ X → ˜ Y is a levelwise weak equivalence. First, using functorial cofibrant-fibrant replacement, we may assume that all X s and Y s are cofibrant and fibra nt in Mod K / B . By another functorial f actorization, we may assume that X s → Y s is a fibration in M od K for a ll s . All these operations induce levelwise weak equivalences. Applying Lemma 3.3(1) inductively , we obtain a diagram X s / /   ˜ X s   } } | | | | | | | | ˜ f s / / Y s   X s ′ / /   ˜ X s ′   } } | | | | | | | | ˜ f s ′ / / Y s ′   X s ′′ / /   ˜ X s ′′   ~ ~ } } } } } } } } } ˜ f s ′′ / / Y s ′′   . . . . . . . . . with maps ˜ f which are surjective in π ∗ . Thus, we have found an ind-isomorphism X → ˜ X and a levelwise surjective map ˜ X → Y . Now let us assume that X → Y is a levelwise cofibration (in M od K ) of fibrant- cofibrant objects in Mod K / B , which is levelwise surjective in π ∗ . Then, a rguing a s before but using Le mma 3. 3(3), we get a fa ctorization of X → Y as a map X → ˜ Y which is a levelwise isomorphism in π ∗ , followed by an ind-isomorphism ˜ Y → Y . The proof f or Ind − Pro − Mod K / B is very similar . W ithout loss of ge ner ality by Lemma 3.2, let f : X → Y be a levelwise map, where X , Y : Q S , α → Mod K / B are functors. W e assume the we have prepared f b y levelwise cofibrant/fibrant replacement as before, so that Lemma 3.3 is applica ble when we need it. Since f is assumed to be an ind-pro-weak equivalence, this means that for eve ry s there is an s ′ > s such that for ev e ry t ′ ∈ α ( s ′ ) there is a t ∈ α ( s ) , t < α ( s < s ′ ) ( t ′ ) , 14 TILMAN BAUER and a map φ : π ∗ Y t s → π ∗ X t ′ s ′ such that the d ia gram π ∗ X t s / /   π ∗ Y t s   φ { { x x x x x x x x π ∗ X t ′ s ′ / / π ∗ Y t ′ s ′ commutes. As before, by pa ssing to a cofinal subsystem, we may assume that s ′ is a direct successor of s . In the first step, we apply Lemma 3 . 3(1) and (2) to produce a factorization X t s → ˜ X t s → X t ′ s ′ with X t ′ s → Y t s surjective in homotopy . W e cannot simply do this f or every s and t because the construction of ˜ X in Lemma 3.3 is not functorial. Fix s a nd assume ˜ X t s has been constr ucted for some t ′ < t ′ 1 . Consider the diagram X t s f / /   Y t s   ˜ X t s > > ~ ~ ~ ~ ~ ~ ~ ~   X t ′ s ′ f ′ / / Y t ′ s ′ X t 1 s f 1 / /   ? ? ? ?      Y t s     > > > > ˜ X t 1 s   @ @ > > } } } }      X t ′ 1 s ′ f ′ 1 / / B B B Y t ′ 1 s ′ A A A By Lemma 3.3(2), we ca n find ˜ X t 1 s and maps as indicated in the diagram. Pro- ceeding inductively , we obtain a pro-object ˜ X s for all s ∈ S . For va rying s , these assemble to an ind-pro-object by means of the maps ˜ X t s → X t ′ s ′ → ˜ X t ′ s ′ . Furthermore, the commutative diagram X t 1 s / /   ˜ X t 1 s   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ X t ′ s ′ / / ˜ X t ′ s ′ shows tha t X → ˜ X is an ind-pro-isomorphism . By constr uction, ˜ X t s → Y t s is surjec- tive in homotopy . W e lea ve the dual construction of ˜ Y → Y and of a levelwise weak equivalence ˜ X → ˜ Y to the reader .  4. I N D E P E N D E N C E O F T H E T O TA L S P A C E The aim of this section is to prove Theorem 1.4. Fix a multiplicative homol ogy theory K . For the results of this section, K d oes not need to be a field (i.e. have K ¨ unneth isomorphisms for any two spaces). W e define the categories T = Ind − Pro − Mod K and A = Ind − Pro − Mod K ∗ . For a CW -complex X , let F ( X ) be the directed set of finite subcomplexes of X . Denote by K fib : ( T op × T op ) / B → T the functor with K fib ( X 1 → B ← X 2 ) = { K [ F ′ ] } F ′ ∈ F ( holim ( X 1 → B ← X 2 ) ) CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 15 where K [ F ′ ] ∈ Pr o − Mod K as an object indexed over the one-point category , or , according to taste, as a constant tower . Similarly , define a functor K : ( T op × T op ) / B → T by K ( X 1 → B ← X 2 ) = { T ot s K [ C B ( X ′ 1 , X ′ 2 ) ] } s ≥ 0, X ′ 1 ∈ F ( X 1 ) , X ′ 2 ∈ F ( X 2 ) . W e write K fib ∗ = π ∗ and K ∗ = π ∗ K : K fib : ( T op × T op ) / B → A . The map Φ : K [ holim X 1 → B ← X 2 ] → { T ot s K [ C B ( X 1 , X 2 ) ] } s extends to a natural tra nsformation Φ : K fib → K as follows: if F ′ is a finite subcomplex of holim ( X 1 → B ← X 2 ) then its images in X 1 and X 2 are finite subcomplexes, and we get a compar ison map K [ F ′ ] → { T ot s K [ C B ( im ( F ′ → X 1 ) , im ( F ′ → X 2 ) ) ] } s ≥ 0 These assemble to a map in T . Definition. W e call a diagram X 1 → B ← X 2 of spaces ind-p ro-constantly conver- gent , or , more br ie fly , Ind- c onvergent , if π ∗ Φ ( X 1 → B ← X 2 ) is an isomorphism in A . W e call a ma p X → B Ind-convergent if X → B ← Y is Ind-convergent for every Y → B , and we call a space B Ind-convergent if ev e ry homotopy pullback diagram X 1 → B ← X 2 is Ind-convergent. Remark 4 .1. By saying tha t π ∗ Φ is an isomorphism, we really mean that Φ induces an isomorphism after applying the grade d group va lued functor ∏ i π i , which a priori is stronger than requirin g that for every k , π k induces an isomorphism. However , since in the context of this paper (if not this section) all homotopy groups are homotopy groups of K -modules for periodic theories K , the two notions coin- cide. Note that if X 1 and X 2 are finite CW -complexes, K ( X 1 → B ← X 2 ) is ind- constant, but K fib ( X 1 → B ← X 2 ) is ind-constant if and only if K ∗ ( F ) is finite. Thus Ind-convergence of B implies in particular that the fiber of a fibration with total space a finite C W -complex has finite K -homol ogy . In this situation, Ind- convergence is the same as pro-constant convergence. Example 4.2. T his example shows that Ind-convergence is weaker than pro-con- stant convergence. Let B = S 1 , E i = S 1 , a nd p i : E i → B multiplication by i . Thus, F i is the discrete spa ce with i points. The H Z -based Eilenberg-Moore spectral se- quence for E i → B is i -convergent; more spe c ifically , E 2 ∗ ∗ = Z [ x ] ⊗ ^ ( y ) , where x is in bidegree ( − 1, 1 ) and y is in bidegree ( 0, 1 ) , a nd we have differentials d i ( y ) = x i . This shows that f or E = ∐ i E i , E → B cannot be pro-constantly convergent because there are differentials of arbitrary length. However , E → B is Ind- convergent because ∐ i ≤ n E i → B , which is n -convergent, constitutes a cofinal subsystem of the finite sub-CW -complexes of E . Theorem 4. 3 . Let Y → B ← ∗ be an Ind-c o nvergent map for some Y → B, where K ∗ is a graded field. Then Y → B is Ind-co nv ergent. A c ohomological version of this theorem ( in terms of pro-constant convergence) was proven in [Hod75, JO9 9] under some cohomo logical finiteness conditions on K ∗ ( X ) . More strongly , [Sey78] claims that the c ohomological finiteness condition is not necessary if B is a finite-dimensional CW -complex, but the proof seems to 16 TILMAN BAUER contain mistakes. Our formulation does not require any such restriction ; however we need the ra ther strong assumption of Ind-convergence to begin with. Theorem 1.4 is an immediate corollary . Fix a ma p Y → B , and abbreviate K ( X → B ← Y ) as K ( X ) and K fib ( X → B ← Y ) as K fib ( X ) . Lemma 4.4 . K fib ∗ ( X ) and K ∗ ( X ) are h omology theories o n the cat egory ( T op / B ) ∗ of sectioned spaces over B with values in A in th e sense of Dold [ Dol71] . This means: in addition to the usual ax iom s for a h omology theory h o n ( T op / B ) ∗ (long exac t sequence, excision), the following t wo axioms are satisfied: (CYL) For any X → B , p : X × [ 0, 1 ] → B, h ∗ ( X × { 0 } ) → h ∗ ( X × [ 0, 1 ] ) is an isomorphism; (EXC) I f A / /   X 1   X 2 / / X is a pushout squar e in T op and p : X → B is a map then we ha v e an exact Mayer- V ietoris sequence. Remark 4 .5. The additional axioms have an analog for G -spaces. In general, if a G - equivariant map X → Y is also a homotopy equivalence, it need not induce a n isomorphism on some G -equivariant homology theory because the homotopy in- verse need not be G -equivariant. It does so, however , if the homology theory only depends on the homo t opy fixed p oint or orbits for some subgroup; f or e x ample, Borel homo logy satisfies the analogue of (CYL) a nd (EXC). Proof of Lemma 4.4. Let us first consider K fib . The functor X 7 → K ∗ ( F ) = colim K fib ∗ ( X ) is a homology theory by [Dol71, 3 .4]; fur thermore a sequence in Ind − Mod K ∗ is exact if and only if its colimit sequence is exact. This shows that we have long ex a ct sequences. T o verify the wedge axiom, we show that K fib maps filtered hocolim its to colimits. Note that holim ( hocolim i X i → B ← Y ) ≃ hocolim i holim ( X i → B ← Y ) , so that we only need to see that { K ∗ F } F ′ ∈ F ( hocolim i F i ) ∼ = colim i { K ∗ ( F ′ i ) } F ′ i ∈ F ( F i ) ∼ = { K ∗ ( F ′ i ) } i , F ′ i ∈ F ( F i ) where the colimit is taken in Ind − Mod K ∗ and the last isomorphism is its defini- tion in any ind-category . Since any finite sub-CW -complex of a hocolim is a lready contained in an F i , the two indexing systems are mutually cofinal, and the iso- morphism is shown. Axiom ( CYL) is clearly sa tisfied because the hom otopy fiber s of X and of X × [ 0, 1 ] are homotopy equivalent; finally , ( E XC) is satisfied because the homotopy pullbac k functor sends pushout squares in a total space to pushout squares. Now consider K . The functor X 7 → π ∗ ( T ot s K [ C B ( X , Y ) ] ) is a homology theory by construction (as an iteration of taking cofibers and smash products with B a nd Y of the suspension spectrum of X), a nd levelwise exa ct sequences induce exact se - quences in Pr o − M od K ∗ , thus X 7 → { π ∗ ( T ot s K [ C B ( X , Y ) ] ) } s is a hom ology theory with values in Pro − Mod K ∗ (not satisfying the wed ge axiom!) and thus induces a homology theory on finite C W -complexes over B . CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 17 Now let U → V be a map in ( T op / B ) ∗ , we ma y assume an inclusion of CW - complexes. For any V ′ ∈ F ( V ) , let U ′ = U ∩ V ′ ∈ F ( U ) . Then the sequence K ∗ ( U ) → K ∗ ( V ) → K ∗ ( V / B U ) has the level representation {K ∗ ( U ′ ) → K ∗ ( V ′ ) → K ∗ ( V ′ / B U ′ ) } V ′ ∈ F ( V ) . Since this sequence is levelwise exact by the above, it is exact in A . The wedge axiom is also satisfied: K ( hocolim X i ) = {K ( X ′ ) } X ′ ∈ F ( hocolim ( X i ) = {K ( X ′ ) } i , X ′ ∈ F ( X i ) = colim K ( X i ) . The a xioms ( CYL) and (EXC) are satisfied levelwise in Pro − Mod K whenever the total spaces are finite CW -complexes, and a n argument similar to the one above for exact sequences shows that they hold e ssentially levelwise in A .  The added value of the a xioms ( CYL) and (EXC) is that a natural transformation of hom ology theories satisfying the axioms is a natural isomorphism if and only if it is an isomorphism on points [Dol71, Theorem 4.1] : Theorem 4.6. Let Φ : h → h ′ : ( T op / B ) ∗ → A be a nat ural transformation of hom o lo g y theories, where A is some abelian c ategory . Then Φ ( X → B ) is an isomorphism for all X → B iff Φ ( ∗ → B ) is an isomorphism for all points in B.  Proof of Theorem 4.3. Putting h = K fib , h ′ = K in Theorem 4.6, using Lemma 4.4, we obtain that Φ ( X → B ) is an isomorphism.  5. T R A N S I T I V I T Y O F C O N V E R G E N C E The principal a im of this section is to prove Theorem 1. 5. This will follow rather easily from the following, more general result. Theorem 5 .1. Let F 1 → X π 1 − → B 1 , F 2 → X π 2 − → B 2 be two fi brations of connected spaces. Denote by F the fiber of F 1 → B 2 , which is th e sam e as t he fiber of F 2 → B 1 . Let K be a hom ology theory which is a fi eld, and assume tha t X → B i are Ind-convergent for i = 1, 2 . Then t he fibration F 1 → B 2 is Ind-c onvergent if and only if the fibration F 2 → B 1 is Ind-conv ergent. The following corollary is the special c a se of B = B 1 , B 2 = X and explains why I call this result a transitivity property . Corollary 5 .2. Let F → X → B be a fibration sequence such th at X → B a nd F → X are Ind-convergent. Then B is Ind-co nvergent.  W e are now in a position to derive Theorem 1.5 from Theorem 5.1. Corollary 5.3 (Theorem 1.5) . Let F → Y → X be a fibration with F and X Ind- convergent. Then so is Y . Proof. Consider the d iagram Ω X / /   ∗ / /   X F / /   Y / / X   Y Y / / ∗ Since X is Ind-convergent, b oth upper rows are Ind-convergent, and the middle vertical row is 0-c onvergent because the fibration is trivial. By Theor em 5.1, F → Y is Ind-convergent (to K ∗ ( Ω X ) ). Applying Corollary 5.2 to Ω X → F → Y , we find that Y is Ind-convergent.  18 TILMAN BAUER The technical heart of Theorem 5.1, and in fa ct the raison d’ ˆ etre for a ll of Sec- tion 3, is the following lemma. Lemma 5.4. Let f : X → Y be a map in Ind − Pro − Mod K / B such th at π ∗ f : π ∗ X → π ∗ Y is an isomorphism in A . Then T ot • C B ( X ) → T ot • C B ( Y ) also induces an isomo r- phism in A . Remark 5 .5. By the Five Lemma, it is obvious that f induces an isomorphism in Pro − Ind − Pro − Mod K ∗ , but it is not obvious that it lif ts to Ind − Pro − Mod K ∗ . Proof. By assumption, f : X → Y is an ind-pro-weak equivalence in the c a tegory Ind − Pro − Mod K / B . By Propositio n 3.5, we may assume that there is a d irected set S , a functor α : S op → { inverse sets } , functors M , N : Q S , α → Mod K / B , and a commutative diagram X K ∧ f / / Ind − Pro − ∼ = α X   Y Ind − Pro − ∼ = α Y   M ˜ f / / N where the vertical maps α X and α Y are ind-pro -isomorphisms and ˜ f is a levelwise weak equivalence. Now note that the functor T ot • C B : Mod K / B → Pro − Mod K X 7 → { T ot s C B ( X ) } s ≥ 0 extends to a f unctor Ind − Pro − Mod K / B Ind − Pro − T ot • C B − − − − − − − − − − → Ind − Pro − Pro − Mod K Ind − D − − − − → Ind − Pro − Mod K , where D : Pro − Pro −C → Pro − C is the ta utological functor of Remark 3. 1. W e thus obtain isomorphisms T ot • C B ( X ) ∼ = T ot • C B ( M ) and T ot • C B ( Y ) ∼ = T ot • C B ( N ) as well as a levelwise weak equivalence C B ( ˜ f ) : C B ( M ) → C B ( N ) . This induces a levelwise weak e quivalence of total towers in e very de gree and thus an ind-pro- isomorphism π ∗ T ot • C B ( X ) ∼ = Ind − Pro π ∗ T ot • C B ( M ) ∼ = T ot • C B ( N ) ∼ = Ind − Pro π ∗ T ot • C B ( Y ) .  Consider the diagram of fibra tions (5.6) F / /   F 1   / / B 2 F 2 / /   X π 2 / / π 1   B 2 ⊤   B 1 B 1 ⊤ / / ∗ T o compare the two E ilenberg-Moore spectra l sequences abutting to the K - homo- logy of F , we construct a bicosimplicial space C s t ( X ) = X × B s 1 × B t 2 CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 19 W e ha v e T ot C • t ( X ) = F 1 × B t 2 and thus T ot T ot C • • ( X ) = F ; on the other hand we have T ot C s • ( X ) = B s 1 × F 2 . W e denote the “horizontal” total space T ot { C s t } s by T ot h C • • and the “vertica l” total space by T ot v C • • . Lemma 5.7. Given a d iagram as in (5. 6) wit h X → B 1 Ind-convergent, there is an isomorphism K ( F 1 → B 2 ) ∼ = − → K ( X → B 1 × B 2 ) . Proof. Since X → B 1 is Ind-convergent, the map { K [ F ] } F ∈ F ( F 1 ) = K fib ( X → B 1 ) → K ( X → B 1 ) = { T ot • K [ C B 1 ( X ′ ) ] } X ′ ∈ F ( X ) . is an ind-pro-weak equivalence as well as a map over B 2 . By Lemma 5 .4, it induces an ind-pro-weak equivalence K ( F 1 → B 2 ) = { T ot • K [ C B 2 ( F ′ ) ] } F ′ ∈ F ( F 1 ) → { T ot • C B 2 ( T ot • C B 1 ( X ′ ) ) } X ′ ∈ F ( X ) ∼ = { T ot • T ot • h K [ C • • ( X ′ ) ] } X ′ ∈ F ( X ) . Since the diagonal N ֒ → N × N is cofinal, the right hand side is { T ot • T ot • h K [ C • • ( X ′ ) ] } ∼ = { diag T ot • T ot • h K [ C • • ( X ′ ) ] } ∼ = K ( X → B 1 × B 2 ) .  Proof of Theorem 5.1. From Diagra m 5.6, we obtain a diagram in T from the various comparison maps K fib ( F 2 → B 1 ) convergence of F 2 → B 1 / / K ( F 2 → B 1 ) Lemma 5.7 for X → B 2 ∼   K fib ( X → B 1 × B 2 ) K ( X → B 1 × B 2 ) K fib ( F 1 → B 2 ) convergence of F 1 → B 2 / / K ( F 1 → B 2 ) . Lemma 5.7 for X → B 1 ∼ O O The two-out-of-three property in this diagram finishes the proof of the theorem.  6. T H E H O P F C O R I N G F O R M O R A VA K - T H E O R Y O F E I L E N B E R G - M A C L A N E S PA C E S In this section, we completely a nalyze the K ( n ) -based Eilenberg-Moore spectral sequence for path-loop fibrations on mod- p Eilenberg-Mac Lane space s, where p is the same prime as the characteristic of K ( n ) ∗ . As n ≥ 1 is fixed in this section, we will abb reviate K ( n ) by K . Through out this section, p is assumed to be odd ( an assumption made in the crucial input [RW80], and a lso necessary to ascertain that K ( n ) is a homotopy commutative ring spe c trum). It is likely that the convergence result also holds for p = 2. In order to understand the K -based Eilenberg-Moore spectral sequence for the Eilenberg-Mac Lane spaces H i = K ( Z / p , i ) , it will be necessar y to understand the algebra structure of K ∗ ( H ∗ ) quite well. Ravenel and W ilson study the structure of K ∗ ( H ∗ ) in [R W8 0], and all the necessary information can be extr a cted fro m that 20 TILMAN BAUER paper . However , they use the bar spectral sequence , which is known to c onverge, to compute K ∗ H i + 1 from K ∗ H i . In our case, we are interested in the Eilenberg- Moore spectral sequence, and from a knowledge of K ∗ H i + 1 and K ∗ H i we want to conclude that the spectral sequence converges for i < n . Since we know what the answer should be, we could guess what the differentials would ha v e to be, and that guess is in fa ct correct, but unfortunately p rovides no proof for convergence. Thus we need to study E ∗ H i for other homology theories E 6 = K , for which we know the EMSS converges, and then compare it to the K -based EMSS, deriving the differentials there in a r igid way . By the various multiplicative properties of the EM SS a s described below , we can restrict ourselves to computing the E -ba sed EMSS for H 1 and E = k ( n ) , connective Morava K -theory , to derive the d ifferentials in all other ca ses for K . Since K is a graded field, we ha v e a K ¨ unneth isomorphism K ∗ ( H r × H s ) ∼ = K ∗ ( H r ) ⊗ K ∗ K ∗ ( H s ) for all r and s , a nd thus K ∗ ( H r ) is a coalgebra , a nd therefore the gadget K ∗ ( H ∗ ) obtains the structure of a Hop f ring , i. e., it is a ring object in the category of coalge- bras. For a survey on this kind of algebra ic structure, consult [ W il00] . The impor- tant data here are operations Ψ , + , ∗ , ◦ , where Ψ : K s ( H r ) → ( K ∗ ( H r ) ⊗ K ∗ K ∗ ( H r ) ) s ; + : K s ( H r ) ⊗ K s ( H r ) → K s ( H r ) ; ∗ : K s ′ ( H r ) ⊗ K s ′′ ( H r ) → K s ′ + s ′′ ( H r ) ; ◦ : K s ′ ( H r ′ ) ⊗ K s ′′ ( H r ′′ ) → K s ′ + s ′′ ( H r ′ + r ′′ ) . The coproduct and addition are the usual maps in homology; the ∗ -product is the “additive” product c oming from the (infinite) loop space structure of H r ; a nd the ◦ -product is the “multiplicative” product coming from the ring spectrum ma p H r × H r ′ → H r + r ′ . As usual, ∗ distributes over + , but there is a second layer of distributivity; namely , in the Sweedler notation Ψ ( a ) = ∑ ( a ) a ′ ⊗ a ′′ , a ◦ ( b ∗ c ) = ∑ ( a ) ( a ′ ◦ b ) ∗ ( a ′′ ◦ c ) . By convention, we give ◦ operator precede nce over ∗ , so that we could write the summand in the above f ormula without parentheses. Both products are Ψ -comodule maps. There are, of course, a number of other structural maps corresponding to units, counits, a nd c oinverses. W e denote by [ 1 ] : π 0 ( H 0 ) → K 0 ( H 0 ) the image of the unit under the Hur e wicz homomorphism, which is the unit for the ◦ -product. Similarly , d e note b y [ 0 ] r ∈ K 0 ( H r ) a re the units for the ∗ -products in degree r . Notation. In our computations, we will need to dea l with algebras, coalgebr as, and Hopf algebra s over F p or K ∗ . W e ad opt the following standard notation: P ( x ) : is the Hopf algebra whose underlying a lgebra is the polynomial a lge- bra on x and whose underlying coalgebr a is the divide d polynomial coal- gebra. W e denote the standa rd a dditive generators by x i , as usual. P k ( x ) : is the quo tient of P ( x ) whose underlying algebra is the truncated poly- nomial algebra P ( x ) / ( x p k ) . Γ ( x ) : is the Hopf algebra dual of P ( x ) : its underlying algebra is the d ivided polynomial algebra, and its underlying coalgebra is the tensor coalgebra. W e d enote the standard additive generators by x i . Γ k ( x ) : is the sub-Hopf a lgebra of Γ ( x ) on the generators x i (0 ≤ i < p k ). V ( x ) : is the exterior Hopf a lgebra on a primitive generator . CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 21 R u ( x ) : is the Hopf a lgebra P ( x ) / ( x p − ux ) (where u is a unit) with x primi- tive. W e will use the convention that x ( i ) = x ∗ p i and x ( i ) = x p i . If x ∈ K s ( H r ) , we will write | x | = ( s , r ) . The following lemma is basic multiplicative homological algebra : Lemma 6.1. If A is a (graded ) commutat iv e algebra ov er a field k of characteristic p , T or ∗ ∗ ( A ) = def T or A ∗ ∗ ( k , k ) is a c ommutative and cocommut a tive Hopf algebra. In p artic- ular , • T or ∗ ∗ ( V ( y ) ) ∼ = Γ ( σ y ) ; • T or ∗ ∗ ( P ( x ) ) ∼ = V ( σ x ) ; • T or ∗ ∗ ( P n ( x )) ∼ = V ( σ x ) ⊗ Γ ( φ x ) . Here and in the following, σ x denotes the suspension, i. e. the element in T or 1 which is represe nted in the bar resolution by [ a ] , and φ x denotes t h e “t ranspotence” element in T or 2 , wh ich is represented in th e bar resolution by any of th e classes [ x i | x j ] with i , j ≥ 1 , i + j = p n (up to units). Dually , if C is a cocomm utative coalgebra over k , Cotor ∗ ∗ ( C ) = def Cotor C ∗ ∗ ( k , k ) is a commutativ e a nd cocommutat iv e Hop f algebra as well, with • Cotor ∗ ∗ ( V ( y ) ) ∼ = P ( σ y ) ; • Cotor ∗ ∗ ( Γ ( x ) ) ∼ = V ( σ x ) ; • Cotor ∗ ∗ ( Γ n ( x )) ∼ = V ( σ x ) ⊗ P ( φ x ) . Again, σ x denotes the dual of the suspension and φ x denotes the “cotranspot ence” element in Cotor 2 , wh ich is re p resen ted in th e bar r esolution by ∑ i + j = n i , j ≥ 1 u i [ x i | x j ] for som e units u i we do not care a bout. 6.1. Getting starte d: K ∗ H 1 → K ∗ H 0 . W e start our computation by studying the differentials of the EMSS for the path-loop fibration on H 1 = K ( Z / p , 1 ) in connec- tive Morava K -theory k . W e have, as coalgebr a s, E ∗ ( H 0 ) = E ∗ [ Z / p ] for any E (6.2) k ev ∗ ( H 1 ) = Γ n ( a ) (6.3) k odd ∗ ( H 1 ) = h y 1 , y 2 , . . . i / ( v n ) with | y i | = 2 i − 1 (6.4) The result for k is easily computed with the Atiyah-Hirz e bruch spectral sequence. Since k is not a gra ded field, k ∗ ( H 1 ) need not be a coalgebra, and indeed is not. But we can compute the cobar spectral sequence modulo the Serre class of v n -torsion groups, so that we have a K ¨ unneth isomorphism aga in and an isomorphism of Hopf algebras E 2 s , t = Cotor s , t ( Γ n ( x )) = V ( σ a ) ⊗ P ( φ a ) (Lemma 6.1) The only way this ca n converge is to have a d ifferential d 2 p − 1 ( v p n σ a ) = ( φ a ) ( 1 ) . Inverting v n , we derive the same differential in the K -ba sed EMSS , a nd we thus have convergence there, too , with E ∞ s , t = E 2 p s , t = P 1 ( φ a ) , where v − 1 n φ a represents [ 1 ] − [ 0 ] ∈ K 0 ( H 0 ) , and we hav e a ∗ -multiplicative ex- tension ( v − 1 n φ a ) ( 1 ) 99K v − 1 n φ a . In fact, we have 2 p -convergence since there are no longer differentials a nd the filtration in E ∞ is bounded by 2 p . 22 TILMAN BAUER 6.2. Morava K - theory of Ei lenberg-Ma c Lane spa c es a .k.a. Automorphisms of Siegel domains. Unfortunately , we will need to juggl e around with multi-indices quite a bit. A multi-index I is an n -tuple ( i 0 , i 1 , . . . , i n − 1 ) with i ν ∈ { 0, 1 } . For such multi-indices we will need some operators, functions, and constructions, which we a ssemble in the following definition. The reader is a dvised to skip it and refer back to it when the notation is used. B eware that our s I is what in [RW80] and [W il84] would be called s − 1 I , and our usage of multi-indices differs from [RW80] but agrees with [W il84]. Definition. • Constructions . Denote by ∆ k the index ( δ l k ) l , where 0 ≤ l ≤ n − 1. Denote by ∆ [ k ] the index ( 1, . . . , 1, 0, . . . , 0 ) ( k copies of ones) and ∇ [ k ] for the index ( 0 , . . . , 0, 1, . . . , 1 ) ( k copies of ones). • Operations . For a multi-index I = ( i 0 , i 1 , . . . , i n − 1 ) , let s ( I ) d enote the shift ( i 1 , i 2 , . . . , i n − 1 , 0 ) . Also denote by c I the cyclic per mutation ( i 1 , i 2 , . . . , i n − 1 , i 0 ) . • Functions. Let t ǫ ( I ) + 1 (the number of tr ailing ǫ ) denote the smallest k with i n − k 6 = ǫ , or ∞ if I = ( 1 − ǫ , . . . , 1 − ǫ ) . Denote by l ǫ ( I ) (the number of leading ǫ ) the smallest k such that i k = 0. W e write a I = a ◦ i 0 ( 0 ) ◦ a ◦ i 1 ( 1 ) ◦ a ◦ i n − 1 ( n − 1 ) where the a ( i ) ∈ K ∗ ( H 1 ) are defined as in (6.3). Thus, | a I | = ( ∑ ν i ν p ν , ∑ I ) . Theorem 6.5 (Ravenel-W ilson) . In t erms o f the classes d efined above, we have an iso- morphism of K ∗ -algebras K ∗ ( H ∗ ) ∼ = O I i 0 = 0 P t 1 ( I ) + 1 ( a I ) ⊗ R u ( a ∆ [ n ] ) , for some unit u ∈ K × ∗ , where I = ( i 0 , . . . , i n − 1 ) runs through all multi-indices. The coproduct is completely determined by stating t hat (6.6) Ψ ( a i ) = i ∑ j = 0 a i − j ⊗ a j . The cla sses a I for i 0 = 1 do not appear as ge ner a tors in Theorem 6 .5, but they a re nonzero and thus can be expressed in terms of the generators. This computation is a reformulation of Ravenel-W ilson’s. Lemma 6.7. Let m = l 1 ( I ) , I 6 = ∆ [ n ] . Then a I = ( − 1 ) m ∑ I  a c m I  ( m ) Corollary 6.8. As K ∗ -modules, K ∗ ( H ∗ ) = O I P 1 ( a I ) = M ∗ ∗ . The coalg ebra structure is given by ( 6.6) and the fact t hat Ψ is an algebra m orphism with respect to ◦ and also with respect to the above algebra structure for the multiplication ∗ . In particular , if Ψ p : K ∗ ( H r ) → ( K ∗ ( H r ) ) ⊗ p denotes p -fold comultiplication, then Ψ p ( a I ) = ( 0; if i 0 = 1  a s I  ⊗ p + decomposables ; otherwise. CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 23 Proof. Using Lemma 6.7, it is elementary to see that M ∗ ∗ ∼ = K ∗ ( H ∗ ) a s K ∗ -modules. The Hopf ring K ∗ ( H ∗ ) is generated a s an K ∗ -algebra by primitives and one group-like element a ∆ [ 0 ] = def [ 1 ] − [ 0 ] 0 . T o prove the c laims about the coalgebra structure, we only have to notice that a I is primitive when i 0 = 1. Clearly , a ( 0 ) is primitive. If x is primitive a nd y is any other element in the augmentation ideal, then Ψ ( x ◦ y ) = Ψ ( x ) ◦ Ψ ( y ) = ∑ ( y )  x ◦ y ′ ⊗ [ 0 ] ◦ y ′′ + [ 0 ] ◦ y ′ ⊗ x ◦ y ′′  = x ◦ y ⊗ [ 0 ] + [ 0 ] ⊗ x ◦ y , and thus x ◦ y is also primitive. This shows that a ll elements of the form a I with i 0 = 1 are primitive. Conversely , if i 0 = 0, then Ψ p ( a I ) =  a s I  ⊗ p (mod ∗ ), and thus a I is not primitive.  Corollary 6. 9. Cho ose a basis of K ∗ ( H ∗ ) containing the generators a I in M ∗ ∗ . In the dual basis, let x I be the dual of a I . Denote by e H t he connected component of 0 in H . Then modulo phanto ms, (6.10) K ∗ ( e H ∗ ) ∼ = O i 0 = 1 P t 0 ( I ) + 1 ( x I ) . Proof. For a Hopf algebra A with a chosen basis, denote by P ( A ) its primitives a nd by Q ( A ) its indecomposables as a submodule (using the basis). First note that P ( M ∗ ∗ ) ⊂ Q ( M ∗ ∗ ) . (This is not true in K ∗ ( H ∗ ) !) Now K ∗ ( e H ∗ ) is precisely the sub-co-Hopf ring generated by the indecompos- ables, i. e., the duals of the ∗ -pr imitives. Thus the classes x I with i 0 = 1 generate K ∗ ( e H ∗ ) . The algebra structure follows by inspection of the c oproduct.  In any commutative and cocommutative Hopf algebra over F p with suitable finiteness hypothesis, there a re Frobenius and V erschiebung maps corresponding to the p -fold product and coproduct. In the case of a Hopf ring, they interact with the circle pro duct in a simple way [R W80, Section 7]. W e recall the definitions and basic properties for the reader ’s convenience. Definition. For a graded algebra A over F p , d efine the Fr obenius homomorphism F : A → A to be the p th p ower map x → x ( 1 ) . Similarly , for a grade d coalgebra C over F p which, as a graded vector space, is the colimit of coalgebras C n of finite type, define the V erschiebung V : C → C to be the continuous dual of the Frobenius on the pro-finite type algebra C ∨ . For H-spaces X of finite type, the filtration C n of K ∗ ( X ) is by de finition given by the skeletal filtration of X . The spa ces H k are of finite type. Lemma 6.1 1 ([RW80, L e mma 7.1 ]) . In a commutative and cocommutative Hopf ring A , (1) V and F are Hopf algebra maps multiplying resp. dividing the degree by p ; (2) V F ( x ) = F V ( x ) = [ p ] ◦ x ; (3) V ( x ◦ y ) = V ( x ) ◦ V ( y ) ; (4) Ψ p ( x ) ≡ V ( x ) ⊗ · · · ⊗ V ( x ) ( mod asymm etric terms ) ; (5) F ( V ( x ) ◦ y ) = x ◦ F ( y ) . The following results are dual to the pairing of bar spe c tr al sequences intro- duced in [TW80] or , in the case of the Eilenberg-Ma c Lane spectrum, [ R W80, Se c - tion 1]. 24 TILMAN BAUER Proposition 6.12 ( Module structures on EMSS) . Given a map of spaces X × T µ − → Y and a field spectrum K . Let E n ( X ) denote the K -based EMSS for t he p a th-loop fibration on X , similarly for Y . Then there are homomorphisms E n ( X ) ⊗ K ∗ K ∗ ( T ) ˜ µ − → E n ( Y ) such that d n ( ˜ µ ( x , η ) ) = ˜ µ ( d n ( x ) , η ) for x ∈ E n ( X ) , η ∈ K ∗ ( T ) , and if x is a permanent cyc le repr esenting a class ξ ∈ K ∗ ( Ω X ) then ˜ µ ( x , η ) is also permanent cycle and represents the class Ω µ ∗ ( ξ , η ) ∈ Ω Y . In the cobar resolution, ˜ µ is giv en by [ x 1 | · · · | x s ] ⊗ η 7→ ∑ ( η ) [ µ ∗ x 1 , η ( 1 ) ) | · · · | µ ∗ ( x s , η ( s ) ) ] , where the sum is given by Ψ s ( η ) = ∑ ( η ) η ( 1 ) ⊗ · · · ⊗ η ( s ) . Proof. Denote by X • (resp. Y • ) the cobar construction C X ( ∗ , ∗ ) (resp. C Y ( ∗ , ∗ ) ). Then the canonical ma p X s × T id × diag s − − − − − → X s × T s → ( X × T ) s µ s − → Y s induces a map of cosimplicial spaces. W e thus obtain a map of total towers ( T ot s X • ) × T → T ot s ( X • × T ) → T ot s Y • which, on homotopy inverse limits, agrees with the standard map ( Ω X ) × T Ω µ − → Ω Y given by ( Ω µ ) ( γ , η ) ( t ) = µ ( γ ( t ) , η ) . Applying K [ − ] , we obtain the f ollowing diagram: K [ ( T ot s X • ) × T ] / /   K [ T ot s Y • ]   T ot s K [ X • × T ] / / T ot s K [ Y • ] . Since π ∗ T ot s K [ X • × T ] ∼ = π ∗ T ot s K [ X • ] ⊗ K ∗ K ∗ ( T ) , we obtain a commutative square K ∗ ( T ot s X • ) ⊗ K ∗ K ∗ ( T ) / /   K ∗ ( T ot s Y • )   D 1 s ∗ ( X ) ⊗ K ∗ K ∗ ( T ) / / D 1 s ∗ ( Y ) and hence a map of spe c tr al sequences compatible with the filtration. The descrip- tion of ˜ µ in the cobar complex follows directly from this constructio n, as the s - fold diagonal on X is used in defining ˜ µ .  Let K be a commutative ring spectrum, and denote by r E n s , t the E -based E MSS for the path-loop fibration on H r ; we write E ∗ ∗ ∗ for the collection of all such spectral sequences for r ≥ 0. Corollary 6.13. There is a homom orphism ◦ : r E m s , t ⊗ K t ′ ( H r ′ ) → r + r ′ E m s , t + t ′ compatible with th e circle product ◦ : K t − s H r ⊗ K t ′ H r ′ → K t + t ′ − s H r + r ′ CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 25 For x ∈ E n and η ∈ K ∗ H ∗ , d n ( x ◦ η ) = d n ( x ) ◦ η . If x is a p ermanent cycle represen ting a class ξ ∈ K ∗ H ∗ , then x ◦ y is also a permanent cycle, and it represents ξ ◦ y. Proof. This is a special case of Prop. 6.1 2 for C = H r , X = H ′ r , D = H r + r ′ , and µ = ◦ .  The following proposition follows from the multiplicative pairing and the rela- tion between Frobenius and V erschiebung stated in Lemma 6.11(5): Proposition 6 .14. Let x ∈ r E n ∗ ∗ and η ∈ K ∗ ( H r ′ ) . Assume that d n ( x ) = z ( 1 ) for some permanent cycle z ∈ E n ∗ ∗ repr esenting a class ζ ∈ K ∗ ( H r ) . Then there is an m ≥ n and a permanent cycle represen ting ζ ◦ V ( η ) such that x ◦ η is an ( m − 1 ) -cyc le and d m ( x ◦ η ) = t ( 1 ) . Proposition 6.15 (Determination of the EMSS for H ∗ ) . W e have: (1) As K ∗ -Hopf algebras, T or ∗ ∗ ( K ∗ e H ∗ ) ∼ = O i 0 = 1 ( V ( σ x I ) ⊗ Γ ( φ x I )) . (2) Dually , E 2 ∗ ∗ = Cotor ∗ ∗ ( K ∗ H ∗ ) is g iven by E 2 ∗ ∗ ∼ = O i 0 = 1  V  σ a I  ⊗  φ a I  . (3) The cotranspot ence φ a I is repre sented in the cobar co mplex by the c la ss φ a I ≡ p − 1 ∑ l = 1 u l   a s − t 0 ( I ) I  ∗ l      a s − t 0 ( I ) I  ∗ ( p − l )  for units u l ∈ F × p , modulo classes th at are more tha n p times decomposable. Furthermore, setting m = l 0 ( I ) , we have tha t φ ( a ( 0 ) ) ◦ a I = ( 0; if m = 0 φ  a s m ( I ) + ∆ n − m  ; otherwise or , equivalently, φ ( a I ) = φ ( a ( 0 ) ) ◦  a s − ( t 0 ( I ) + 1 ) I  (4) The only sources of differentials are the factors of the form σ a I with i 0 = 1 : (a) If l = ∆ [ m ] for some m ≥ 1 th en d 2 p m − 1 ( σ a I ) = φ ( a I ) ( m ) (b) O therwise, let m = l 1 ( I ) be t he number of leading ones in I and m ′ = l 0 ( s m I ) be the number of zeroes following the m ones in I . Then d 2 p m − 1 ( σ a I ) =  φ a c m + k − 1 s I + ∆ n − m ′  ( m ) (5) The spectral sequence collapses at E 2 p n with E 2 p n ∗ ∗ = E ∞ ∗ ∗ = O i 0 = 1 P t 1 ( s − t 0 ( I ) I ) φ ( a I ) . (6) In E ∞ ∗ ∗ , a I is repre sented by φ a s l 0 ( I ) I + ∆ n − l 0 ( I ) if i 0 = 0 . 26 TILMAN BAUER Proof. (1 and 2): This is a routin e calculation using Corollary 6 .9, using the basic building blocks from Lemma 6.1. (3): If R = P k ( x ) is divide d power algebra on x , then the transpotence element in the cobar complex is given by a ny one of the homologo us representatives [ x l | x p k − l ] . Dually , the cotranspotence is given by the sum of the d uals of these classes. Thus (6.16) φ ( a I ) = p t 0 ( I ) + 1 − 1 ∑ k = 1   x k I  ∨      x p t 0 ( I ) − k I  ∨  . Now if k = k 0 + k 1 p + · · · + k r p r with 0 ≤ k i < p , the d ual of x k I is decomposable as  x k I  ∨ = 1 k 0 ! k 1 ! · · · k r !  a I  ∗ k 0 ∗  a s − 1 I  ∗ k 1 ∗ · · · ∗  a s − r I  ∗ k r . If we de note by Q ( k ) = k 0 + · · · + k r the sum of the p -digits of k , the dual of x k I is thus de composable into Q ( k ) factors. In order for the k - summand in (6.16) to have p or less factors, we therefore need that Q ( k ) + Q ( p t 0 ( I ) + 1 − k ) ≤ p . T his is the case if and only if k = l p t 0 ( I ) for 1 ≤ l ≤ p − 1, proving the first formula of part (3). For the second formula, first note that φ ( a ( 0 ) ) = ∑ p − 1 l = 1 u l  a ∗ l ( n − 1 )   a ∗ ( p − l ) ( n − 1 )  mod- ulo ( p + 1 ) - decomposables. Now f or I with i 0 = 0,  a ∗ l ( n − 1 )   a ∗ ( p − l ) ( n − 1 )  ◦ a I =   a ( n − 1 ) ◦ a s I  ∗ l      a ( n − 1 ) ◦ a s I  ∗ ( n − l )  The sum of the right hand sides can be written (up to ( p + 1 ) - decomposables) as φ ( a I ′ ) , where I ′ is the multi-index s I + ∆ n − 1 , shifted to the left such that i ′ 0 = 1. This amount is given by the number m − 1 in the statement. The last formula of (3) follows by reindexing. (6): By the computation of the EM S S for H 1 , φ ( a 0 ) represents the class [ 1 ] − [ 0 ] = a ∆ [ 0 ] ∈ K 0 ( H 0 ) . Thus a I = ([ 1 ] − [ 0 ] ) ◦ a I b J is represented by ( φ a 0 ) ◦ a I = φ a s l 0 ( I ) I + ∆ n − l 0 ( I ) for i 0 = 0. (4): W e have computed that d 2 p − 1 ( σ a ( 0 ) ) = ( φ a ( 0 ) ) ( 1 ) . By (6), the class φ a ( 0 ) repre- sents [ 1 ] − [ 0 ] . W e are thus in the situation of Prop. 6.14, a nd there is a j a nd a per ma nent cycle t representing ( [ 1 ] − [ 0 ] ) ◦ a s I = a s I such that d j ( σ a I ) = d j ( σ a ( 0 ) ◦ a I − ∆ 0 ) = t ( 1 ) . W e now a pply Lemma 6.7 to a s I to get: a s I = ±  a c m − 1 s I  ( m − 1 ) , where m = l 1 ( I ) . W e will study which class represents a c m − 1 s I . W rite I = m z }| { 1 · · · 1 m ′ z }| { 0 · · · 0 I ′ , where m ≥ 1, m ′ ≥ 0, and I ′ is either empty or star ts with a 1 , and if m ′ = 0 then I is empty . Then c m − 1 s I = m ′ z }| { 0 · · · 0 I ′ 0 m − 1 z }| { 1 · · · 1 , CONVERGENCE OF T HE EI LENBERG-MOORE SPECTR AL SEQUENCE 27 and thus (6) tells us that a c m − 1 s I is represented by φ ( a J ) with J =          I ′ 0 m z }| { 1 · · · 1 m ′ − 1 z }| { 0 · · · 0; I ′ nonempty m z }| { 1 · · · 1 m ′ z }| { 0 · · · 0; I ′ empty. This d etermines all the differentials. For ( 5), note that any I with i 0 = 1 is of the form J as above, with uniquely determined numbers m , m ′ and subindices I ′ . This means that all the classes φ ( a I ) are torsion (of order p t 1 ( s − t 0 ( I ) I ) ) in E ∞ and that a ll the classes σ ( a I ) support d ifferentials.  Theorem 6.17 . Let n > 0 and t 6 = n + 1 be integers. Then the Eilenberg-Moore spectral sequence E 2 r , s = Cotor K ( n ) ∗ ( e H t ) r , s ( K ( n ) ∗ , K ( n ) ∗ ) = ⇒ K ( n ) ∗ ( H t − 1 ) converges 2 p n -constantly. Proof. From Prop. 6.15(5) we know that E 2 p n ∗ ∗ = E ∞ ∗ ∗ = O i 0 = 1 P t 1 ( s − t 0 ( I ) I ) φ ( a I ) . and from Ravenel-W ilson’s computation (Theorem 6.5), we know that K ∗ ( H ∗ ) ∼ = O i 0 = 0 P t 1 ( I ) + 1 ( a I ) ⊗ R ( a ∆ [ n ] ) , Furthermore, we know by Prop. 6.15(6) that the comparison map from the latter to the target of the spectral sequence is such that a I (for I 6 = ∆ [ n ] ) is represented by φ a s l 0 ( I ) I + ∆ n − l 0 ( I ) . Since these classes have the same multiplicative order , we observ e that the comparison map is an isomorphism. Note that the spe ctral sequence does not converge for t = n + 1 (the class a ∆ [ n ] ∈ K ∗ ( H n ) has no representative in E ∞ ), which is not surprising since K ∗ ( H n + 1 ) = 0. However , for t > n + 1, the EMSS again converges for trivial reasons : source and target are trivia l.  Proof of Theorem 1.3. In Theorem 6.17 we p roved pro-constant c onvergence, and since K ( n ) ∗ ( H r ) is a finite K ( n ) ∗ -module for a ll r by the calculations of Ravenel and W ilson, { K ( n ) ∗ ( X ) } X ∈ F ( H r ) is ind-constant. Thus we ha ve Ind-convergence.  R E F E R E N C E S [AM69] M. A rtin and B. Mazur . Etale homotopy . Lecture Notes in Mathem atics, No. 100. Springer- V e rlag, Berlin, 1969. [BK72] A. K. Bousfield and D. M. Kan. H omotopy limits, completions and loca lizations . Springer-V erlag, Berlin, 1972. L ecture Notes in Mathe m atics, V ol. 304. [Boa99] J. M ichael Boardman. Cond itionally convergent spectral sequences. In Homotop y invaria nt al- gebraic str u ctures (Baltimor e, MD, 1998) , volume 239 of Contemp. Math. , pages 49–84. Amer . Math. Soc., Providence, RI, 1999. [Bou87] A. K. Bousfield. On the homology spectral sequence of a cosimplicial space. Amer . J. Math. , 109(2):361–394 , 1987. [CE99] Hen ri Cartan and S amuel Eilenberg. H omological algebra . Princeton Landmarks in M athemat- ics. Princeton Univ e rsity Press, Princeton, NJ, 1999. W ith an appendix by David A. Buchs- baum, Reprint of the 1956 origi nal. [Dol71] Albrecht Dold. Chern clas ses in general cohomology . In Symposia Mathematica, V ol. V (IN- DAM, Rome, 1969/70) , pages 385–410. Academic Press, London, 1971. [Dwy74] W . G. Dwyer . Strong convergence of the Eilenb erg-Moor e spectral sequence. T opology , 13:255– 265, 1974. 28 TILMAN BAUER [Dwy75] W illiam G. Dwyer . Exotic convergence of the Eilenbe rg-Moore spectral sequen ce. Illinois J. Math. , 19(4):607–617, 1975. [EM66] Samuel Eilenberg and John C. Moore. Homology and fibrations. I. Coalgebras, cotensor pro- duct and its d erived functors. Comment. Math. Helv . , 40:199–236, 1966. [GJ99] Pa ul G. Goerss and John F . Jardine. Simplicial homotopy theory . B irkh ¨ auser V erlag, Basel, 1999. [Hod75] Luke Hodgkin. Th e equivariant K ¨ unneth t heorem in K -theorem. In T opics in K -theory . T wo independent contributions , pages 1–101. Lecture Notes in Math., V ol. 496. Springer , Berlin, 1975. [Isa04] Daniel C. Isaksen. Strict model structures for pro-c ategories. In Catego rical decomposition tech- niques in algebraic topolo gy (Isle of Skye, 2001) , volume 215 of Progr . Math. , pages 179–198. Birkh ¨ auser , Basel, 2004. [JO99] A . Jeanneret an d A. Osse. The Eilenberg-Moore spectral sequen ce in K -theory . T opology , 38(5):1049–107 3, 1999. [MS06] J. P . May and J. Sigur d sson. Parametrized homotopy theory , volume 132 of Mathematical Surveys and M onographs . American Mathematical Society , Providence, RI, 2006. [RS65] M. Rothen b erg and N. E . Steenrod. The cohomolo gy of classifying spaces of H -spaces. Bull. Amer . M ath. Soc. , 71:872–875, 1965 . [R W80] Douglas C. Ravenel and W . Stephen W ilson. The Morava K -the ories of Eilenberg-Mac Lane spaces and the Conn er-Floyd conjecture. Amer . J. M ath. , 102(4):691–748, 1980. [Sey78] R. M. Seymour . On the convergence of the E ilenb erg-Moor e spectral sequence. Proc. London Math. Soc. (3) , 36(1):141–162, 1978. [Shi96] Brooke E. Shipley . Conv e rgence of the homology spectral sequence of a cosimplicial space. Amer . J. Math. , 118(1):179–207, 1996. [Smi70] Larry Smith . Lectures on the Eilenberg-Moore spectral seq uence . Lecture Notes in Mathem atics, V ol. 134. Springer-V erlag, Berlin, 1970. [T am94] Dai T amaki. A dual Rothenb e rg-Steenrod spectral sequence. T opology , 33(4):631–662, 1994. [TW80] Robert W . Thomason and W . Stephen W ilson. Hopf rings in the bar spectral sequence. Quart. J. Math. Oxford Ser . (2) , 31(124):507–511, 1980. [W il84] W . Stephen W ilson. T he Hopf ring for Morava K -theory . P ubl. Res. Inst. Math. S ci. , 20(5):1025– 1036, 1984. [W il00] W . Stephe n W ilson. Hopf rings in algebraic topology . E xpo. Math. , 18(5):369–388, 2000. M AT H E M AT I S C H E S I N S T I T U T D E R U N I V E R S I T ¨ A T M ¨ U N S T E R , E I N S T E I N S T R . 6 2 , 4 8 1 4 9 M ¨ U N S T E R , G E R M A N Y E-mail address : tbauer@math .uni-mue nster.d e