Commuting unbounded homotopy limits with Morava K-theory
aa r X i v : . [ m a t h . A T ] M a r COMMUTING UNBOUNDED HOMOTOPY LIMITS WITHMORAVA K-THEORY
GABRIEL ANGELINI-KNOLL ˚ AND ANDREW SALCH : Institut f¨ur Mathematik, Freie Universit¨atBerlin, Germany ˚ Department of Mathematics, Wayne State UniversityDetroit, Michigan, U.S.A. : Abstract.
This paper provides conditions for Morava K -theory to commutewith certain homotopy limits. These conditions extend previous work on thisquestion by allowing for homotopy limits of sequences of spectra that arenot uniformly bounded below. As an application, we prove the K p n q -localtriviality (for sufficiently large n ) of the algebraic K -theory of algebras overtruncated Brown-Peterson spectra, building on work of Bruner and Rognesand extending a classical theorem of Mitchell on K p n q -local triviality of thealgebraic K-theory spectrum of the integers for large enough n . Contents
1. Introduction 11.1. Conventions 41.2. Organization 41.3. Acknowledgements 52. When does killing homotopy induce an injection in homology? 53. Morava K-theory of homotopy limits 94. A higher chromatic height analogue of Mitchell’s theorem 18Appendix A. Review of Margolis homology. 26A.1. Basics of Margolis homology. 27A.2. The relationships between Margolis homology and Morava K -theory. 30References 311. Introduction
Given a generalized homology theory E ˚ and a sequence ¨ ¨ ¨ Ñ X Ñ X Ñ X E-mail address : [email protected],[email protected] . of spectra, one often needs to know, for computations, whether there is an isomor-phism lim i E ˚ p X i q – E ˚ p holim i X i q . This cannot be true in full generality. For example, the limit of the sequence ¨ ¨ ¨ Ñ S { p Ñ S { p is the p -complete sphere ˆ S p and therefore H ˚ p holim i S { p i ; Q q – H ˚ p ˆ S p ; Q q – Q p whereas H ˚ p S { p i ; Q q – i ě i H ˚ p S { p i ; Q q –
0, andlim i H ˚ p S { p i ; Q q also vanishes, so we do not even have hope of recovering H ˚ p holim i S { p i ; Q q from a “Milnor sequence.”This motivates the question: what conditions on E ˚ and the sequence ¨ ¨ ¨ Ñ X Ñ X Ñ X allow us to commute the homotopy limit with E ˚ ? There are known results alongthese lines, most famously a commonly-used result of Adams from [1], but the usualhypothesis is that the spectra X i are uniformly bounded below and the homologytheory E ˚ is connective. In this paper, we remove each of these assumptions, undersome reasonable additional hypotheses. Our particular focus is on the case where E ˚ is a Morava K -theory K p n q ˚ . This paper is written with a view towards filtered spectra that arise when study-ing topological periodic cyclic homology, in particular, the Greenlees filtration (28)on topological periodic cyclic homology
T P p R q : “ T HH p R q t T , is not uniformly bounded below. Nevertheless, these filtered spectra often have niceenough homological properties to apply the main result of this paper.Following the red-shift program of Ausoni and Rognes [6], we are most interestedin the chromatic complexity of topological periodic cyclic homology. Therefore, ageneralized homology theory of primary interest is Morava K-theory K p n q ˚ . Calcu-lating Morava K -theory of topological periodic cyclic homology using the Greenleesfiltration requires that one be able to commute a non-bounded-below generalized ho-mology theory (Morava K-theory) with a non-uniformly-bounded-below homotopylimit, so existing results on generalized homology of limits, like Adams’ theoremfrom [1] reproduced as Theorem 3.2 below, do not suffice.Classically, the vanishing of the Margolis homology H p H ˚ p X ; F p q , Q n q of abounded-below spectrum X with finite-type mod p homology implies that X has is K p n q ˚ -acyclic. However, even if each spectrum X i is bounded below, has finite-typehomology, and H p H ˚ p X ; F p q , Q n q vanishes, it is not always the case that holim i X i has trivial K p n q -homology. Our main theorem establishes sufficient conditions forthis homotopy limit to indeed have trivial K p n q -homology. Given a spectrum X , weadopt the notation that X ă N is its Postnikov truncation constructed by attaching The easiest case of a Morava K -theory—in particular, the only case which is bounded below—is K p q ˚ , which coincides with rational homology. So the above example with the rational homol-ogy of the p -complete sphere shows that some hypotheses are needed in order to commute Morava K -theory with a sequential limit. OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 3 cells to kill all the homotopy groups of X in degrees ě N . Our main result maythen be summarized as follows. Theorem 1.1. (Theorem 3.7) Suppose M is an integer and ¨ ¨ ¨ Ñ Y Ñ Y Ñ Y is a sequence of bounded below finite-type spectra which are H F p -nilpotently com-pete, such that the largest grading degree of a comodule primitive in H ˚ p Y i ; F p q isstrictly less than M , the homology groups H ˚ p Y i ; F p q and H ˚ p Y ă Mi ; F p q are finitelygenerated for each i , and the limit lim i Ñ8 H p H ˚ p Y i ; F p q , Q n q of the Margolis homologies vanishes. Then the n -th Morava K-theory of holim i Y i is trivial. That is, we have an isomorphism K p n q ˚ p holim i Y i q – . This result relies on another result which resolves the question of when Postnikovtruncations X Ñ X ă N induce injections in homology. Theorem 1.2. (Theorem 2.6) Let
M, N be integers with N ě M , and let X bea bounded below H F p -nilpotent spectrum with the grading degrees of the comoduleprimitives in H ˚ p X ; F p q bounded above by M . Then the map H ˚ p X ; F p q Ñ H ˚ p X ă N ; F p q , induced by the canonical map X Ñ X ă N , is injective. As the main application in the present paper, we prove a higher chromatic heightanalogue of Mitchell’s theorem for algebraic K-theory of truncated Brown-Petersonspectra, building on work of Bruner and Rognes [12]. In particular, Mitchell provesin [28] that K p m q ˚ p K p Z qq “ m ě
2, and consquently the same vanishing of Morava K-theory occurs for any H Z -algebra. Let BP x n y denote the p -completion of the truncated Brown-Petersonspectrum. As an application of the main result of this paper, we prove the followingresult, building on [12, Prop. 6.1]. Theorem 1.3. (Theorem 4.10) There are isomorphisms K p m q ˚ p K p BP x n yqq – for m ě n ` , which holds at all primes when n “ , and holds at p “ , when n “ . Consequently, the same vanishing of Morava K-theory holds for any BP x n y -algebra. In particular, we note that this recovers the vanishing of K p m q ˚ p K p BP x yqq for m ě K p m q ˚ p K p BP x yqq for m ě K p i q ˚ K p E n q vanishes for i ě n `
2, where E n is This result also holds for arbitrary p and n if certain plausible-looking conditions on E -ringspectra can be shown to hold. See Remark 4.3 for explanation. COMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY the n -th Lubin-Tate E -theory. Also, Land, Meier, and Tamme [20] recently provedseveral results of this flavor, for example, they show that if R is a K p q -acyclic E -algebra then K p n q ˚ p K p R qq “ n ě T P p y p n qqr k s for the Greenlees filtration oftopological periodic cyclic homology of y p n q (see (28)). Conjecture 1.4.
There are isomorphismslim k Ñ´8 K p m q ˚ p T P p y p n qqr k sq – K p m q ˚ p T P p y p n qq for 1 ď m ď n and all 1 ď n ă 8 .This conjecture is resolved by the main theorem in the present paper, givenabove as Theorem 1.1, together with calculations of the first author and Quigleywhich verify that the hypotheses (on comodule primitives and finite-typeness ofhomology) of that main theorem are satisfied. The first author and Quigley planto update [3] with these stronger results.1.1. Conventions.
When κ is a cardinal number and A an object in some cate-gory, we will write A κ for the κ -fold categorical product and A š κ for the κ -foldcategorical coproduct. If our category is additive, then we write A ‘ κ for the κ -foldcategorical biproduct of A with itself. Let R be a commutative ring. In this paper,the term “finite type” is used in the sense common in algebraic topology: a graded R -module V is finite type if, for each integer n , the grading degree n summand V n is a finitely generated R -module. In particular, if R is a field, then we say V is finite type if, for each integer n , the grading degree n summand V n is a finitedimensional vector space.We will write P p x , . . . , x n q for a polynomial algebra with generators x , . . . , x n over F p , E p x , . . . , x n q for an exterior algebra over F p with generators x , . . . , x n and we write P k p x , . . . , x n q for the truncated polynomial algebra with generators x , . . . , x n , where P k p x q is often also denoted F p r x s{ x k in the literature. Givencategories C and D , we will write D C for the category of functors from C to D .Given a ring R , a non-zero-divisor r P R , and a left R -module M , we say that M is r -power-torsion if, for each m P M , there exists some integer n such that r n m “
0. We say that M is simple r -torsion if rm “ m P M .1.2. Organization.
In Section 2, we give sufficient conditions for the canonicalmap X Ñ X ă N to induce an injection on homology. This is a key result usedto prove the main theorem in Section 3 and we believe it is a useful contributionto the literature in its own right. In Section 3, we prove the main theorem aboutwhen we may commute Morava K-theory with a non-uniformly bounded below In other words, M is r -power-torsion if the zeroth local cohomology H p r q p M q – colim n hom R p R {p r n q , M q vanishes. Some references say that M is “ r -torsion” instead of “ r -power-torsion,” but then again, some references say that M is “ r -torsion” when M is what we call simple r -torsion; for the sake of clarity, we prefer to only use the unambiguous terms “ r -power-torsion”and “simple r -torsion.”. OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 5 sequential limits. In Section 4, we give our main application, which is a proof of ahigher chromatic height analogue of Mitchell’s theorem about vanishing of MoravaK-theory of algebraic K-theory of the integers. Finally, we give some supplementaryresults and background on Margolis homology in Appendix A.1.3.
Acknowledgements.
The first author would like to thank J.D. Quigley fordiscussions related to this project and B. Dundas and E. Peterson for expressinginterest in the results in this paper. The authors would also like to thank D. Chuafor a careful reading of the paper that lead to several improvements. This paperbegan as a batch of rough notes by the second author about the technical questionof when one can commute a sequential homotopy limit with a smash product, in“bad” cases where the hypotheses of Adams’ theorem are not satisfied. The firstauthor joined the paper later and since then several of the proofs in Section 3have been significantly reworked. Also, Section 4 containing the applications totruncated Brown-Peterson spectra are a primary contribution of the second authorleading to the paper’s present form. The second author is therefore grateful to thefirst author and to J.D. Quigley for their patience in waiting for these results to bemade publicly available.2.
When does killing homotopy induce an injection in homology?
This section proves a general result, Theorem 2.6, which will be used to proveTheorem 3.7. However, it takes some work to prove and is useful in its own right.The question is when, given a spectrum X and an integer N , the map X Ñ X ă N given by attaching cells to kill all the homotopy groups of X in degrees ě N inducesan injection in mod p homology groups. Theorem 2.6 gives some practical sufficientconditions for H ˚ p X ; F p q Ñ H ˚ p X ă N ; F p q to be injective. A simple example wherethe hypotheses, and therefore the result, hold can be constructed by letting X “ S , M “ N “ F p ã Ñ p A {{ E p qq ˚ is an inclusion. A simple example where thehypotheses are not satisfied is the case where X “ S and M “ N “
0, where ofcourse X ă N is the 0 spectrum and the result cannot hold.The notation A ˚ in the statement of Lemma 2.1 refers to the dual mod p Steenrodalgebra, as usual, but the lemma and its proof both work for any graded Hopfalgebra over a field.
Lemma 2.1.
Let p be a prime number, let I be a set, and for each i P I , let M i be a graded A ˚ -comodule. Then, for each integer t and nonnegative integer s , thenatural map of F p -vector spaces ž i P I Cotor s,tA ˚ p F p , M i q Ñ Cotor s,tA ˚ ˜ F p , ž i P I M i ¸ is an isomorphism.Proof. This result follows by observing that each step in the construction of thecobar complex of A ˚ , with coefficients in an A ˚ -comodule M , commutes with co-products in the variable M . (cid:3) COMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY
Definition 2.2.
Let p be a prime number and let M be an integer. We will say X satisfies condition H p M q if it is bounded below and H F p -nilpotently complete andthe A ˚ -comodule primitives of H ˚ p X ; F p q are trivial in grading degrees ě M . We suppress the prime p from the notation H p M q because it will always be clearfrom the context. Remark 2.3.
Given a bounded below H F p -nilpotent complete spectrum X , wenote that the 0-line of the E -page of the Adams spectral sequenceExt ˚ , ˚ Comod p A ˚ q p F p , H ˚ p X qq Ñ π ˚ X is isomorphic to Hom Comod p A ˚ q p F p , H ˚ X q , which is isomorphic to the sub- A ˚ -comodule of A ˚ -comodule primitives in H ˚ X .So, if X is bounded below and H F p -nilpotently complete, then condition H p M q isequivalent to the following statements being true for all integers j ě M :(1) the Hurewicz map π j p X q Ñ H j p X ; F p q is zero, and(2) for all r ě
2, the H F p -Adams spectral sequence for X does not have anynonzero d r -differentials supported in bidegree p , j q (i.e., s “ t “ j ).It is the above two conditions that actually play a role in the proofs in this section,but the form of condition H p M q given in Definition 2.2 is a relatively familiar andeasily-checked condition which is equivalent (for bounded below H F p -nilpotentlycomplete spectra) to the two properties given above.In Theorem 6.6 of [10], Bousfield proved that, if E is a connective ring spectrumwith π p E q – F p and X is a connective spectrum, then the E -nilpotent completionˆ X E of X is weakly equivalent to the Bousfield localization L E X . As a special case,we have: Lemma 2.4.
Let X be a connective spectrum. Then the H F p -nilpotent completionmap X Ñ ˆ X H F p is an H F p -local equivalence. That is, the induced map of spectra X ^ H F p Ñ ˆ X H F p ^ H F p is a weak equivalence. Lemma 2.5.
Let p be a prime number and let I be a set. Let X satisfy conditionH p M q for some integer M . For each i P I , let f i : S n Ñ X be a map of spectra whichinduces the zero map in mod p homology, and let f : p š i P I S n q ˆ H F p Ñ ˆ X H F p – ÝÑ X denote the map given by H F p -nilpotent completion and the universal property ofthe coproduct applied to the set of maps t f i : i P I u . Suppose that the map π n ˜ž i P I S n ¸ b Z F p Ñ π n p X q b Z F p (1) is injective. Finally, suppose that n ă M . Then the following statements are true: (1) the map H ˚ p X ; F p q Ñ H ˚ p cof f ; F p q is injective, and (2) cof f satisfies condition H p M q . Readers who are not used to thinking about nilpotent completion may be relieved to knowthat a bounded-below spectrum is H F p -nilpotently complete if and only if its homotopy groups areExt- p -complete, by [10, Prop. 2.5]. In particular, if all the homotopy groups of a bounded-belowspectrum are p -adically complete, then that spectrum is also H F p -nilpotently complete. OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 7
Proof.
The H F p -nilpotent completion map š I k S n Ñ `š I k S n ˘ ˆ H F p is an H F p -localequivalence, by Lemma 2.4, so we have a natural isomorphism ž I k H ˚ p S n ; F p q – ÝÑ H ˚ ¨˝˜ž I k S n ¸ ˆ H F p ; F p ˛‚ . Since each f i induces the zero map in mod p homology, so does the map š i P I k f i ,and consequently so does the composite ž I k S n š i P Ik f i ÝÑ ž I k X ∇ ÝÑ X, (2)i.e., the map š I k S Ñ X given by the universal property of the coproduct. (Thesymbol ∇ in (2) stands for the fold map, given again by the universal property ofthe coproduct.) We have the commutative square `š I k S n ˘ ^ H F p / / – (cid:15) (cid:15) X ^ H F p – (cid:15) (cid:15) `š I k S n ˘ ˆ H F p ^ H F pf ^ H F p / / ˆ X H F p ^ H F p in which the vertical maps are isomorphisms in SHC by Lemma 2.4. The tophorizontal map was already shown to be zero, so the bottom horizontal map is aswell. So f induces the zero map in mod p homology. So we have the short exactsequence of A ˚ -comodules0 Ñ H ˚ p X ; F p q Ñ H ˚ p cof f ; F p q Ñ ž i P I H ˚ p Σ S n ; F p q Ñ H ˚ p X ; F p q Ñ H ˚ p cof f ; F p q , as well as aninduced long exact sequence0 / / Cotor ,jA ˚ p F p , H ˚ p X ; F p qq a / / Cotor ,jA ˚ p F p , H ˚ p cof f ; F p qq BECDGF b (cid:15) (cid:15) Cotor ,jA ˚ p F p , š i P I H ˚ p Σ S n ; F p qq / / . . . . (4)Condition H p M q gives us that the left-hand term in (4) vanishes for j ě M , whilethe right-hand term in (4) vanishes for j ě n `
1, so cof f satisfies conditionH p max t M, n ` uq . When n ă M , then clearly max t M, n ` u “ M . (cid:3) Theorem 2.6.
Let p be a prime number, let M, N be integers with N ě M , andlet X satisfy condition H p M q . Let X ă N be X with cells attached to kill all thehomotopy groups of X in degrees ě N . Then the map H ˚ p X ; F p q Ñ H ˚ p X ă N ; F p q ,induced by the canonical map X Ñ X ă N , is injective.Proof. This is a proof by induction. Suppose k is an integer, k ě N , and that wehave already constructed a sequence of maps X “ X p N q Ñ X p N ` q Ñ ¨ ¨ ¨ Ñ X p k q (5)satisfying the properties: COMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY (1) for each j ą N , the map X p j ´ q Ñ X p j q induces an isomorphism in π i for all i ă j ´
1, and(2) for each j ą N , π j ´ p X p j qq{ p –
0, and(3) for each j ą N , the induced map H ˚ p X p j ´ q ; F p q Ñ H ˚ p X p j q ; F p q isinjective, and(4) for each j ě N , X p j q satisfies condition H p j q .Note that if X p j q satisfies condition H p j q , then it is H F p -nilpotently complete.Consequently, the group π j p X p j qq is Ext- p -complete by [10, Thm. 2.5,6.6]. Thevanishing mod p of an Ext- p -complete abelian group implies the vanishing of thatabelian group, as a special case of [19, Thm. A.6(d)]; so π j ´ p X p j qq – j ą N . We want to construct a map X p k q Ñ X p k ` q which extends the sequence(5) one step to the right and satisfies the same four properties listed above. Clearly,if this can be done, then the third property implies that the map H ˚ p X ; F p q Ñ H ˚ p hocolim k X p k q ; F p q is injective, as desired. The first two of the four properties listed above also implythat the map π j p X q Ñ π j p hocolim k X p k qq is an isomorphism for j ă N . Propertiestwo and four then imply that π j p hocolim k X p k qq vanishes for all j ě N .Those observations are enough to show that hocolim k X p k q agrees with any ofthe usual “attach cells to kill homotopy in degrees ą N ” constructions applied to X ,for the following reason: if A N : SHC Ñ SHC is any functor and η N : id SHC Ñ A j any natural transformation such that, for all spectra X , A N p X q has homotopyconcentrated in degrees ă N and η N p X q : X Ñ A j p X q induces an isomorphismin homotopy in degrees ă j , then applying A N to the maps η N p X q and X Ñ hocolim k X p k q yields a commutative diagram X η N p X q (cid:15) (cid:15) / / hocolim k X p k q η N p hocolim k X p k qq– (cid:15) (cid:15) A N p X q / / A N p hocolim k X p k qq by naturality of η N . The bottom horizontal map induces an isomorphism π j p A N p X qq Ñ π j p A N p hocolim k X p k qqq for j ă N , and both sides vanish if j ě N , so the bottom horizontal map isan isomorphism in SHC. So A n p X q – hocolim k X p k q . So although we will notconstruct a spectrum X ă N by simply attaching cells as one does classically, ourspectrum hocolim k X p k q will indeed be isomorphic to the result of that classicalconstruction.We construct X p k ` q , and the map X p k q Ñ X p k ` q , as follows: choose a set I k of elements of π k p X p k qq with the property that the image of I k in π k p X p k qq{ p is a minimal set of generators for the group π k p X p k qq{ p . Then we apply Lemma2.5, letting the I, X, n, and M in the statement of Lemma 2.5 be I k , X p k q , k, and k `
1, respectively. We check that the hypotheses of Lemma 2.5 are satisfied: ‚ for each i P I k , the map f i : S k Ñ X p k q is zero in mod p homology bythe j “ k case of the fourth inductive hypothesis (see Remark 2.3 Item 1),above, This is because each X p i q is H F p -nilpotently complete and has the property that π i ´ X p i q{ p is trivial, and so π i ´ X p i q is trivial; this is a typical argument about Ext- p -completeness. OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 9 ‚ the map (1) is an isomorphism, by the minimality hypothesis on I k as a setof generators for π k p X p k qq b Z F p , ‚ and X p k q satisfies condition H p k q by the fourth inductive hypothesis.Now we define X p k ` q to be the cofiber of the map f : ˜ž i P I k S k ¸ ˆ H F p ÝÑ p X p k qq ˆ H F p – ÝÑ X p k q given by H F p -nilpotent completion and the universal property of the coproductapplied to the maps t f i : i P I k u , just as in the statement of Lemma 2.5. Byconstruction, the first and second inductive hypotheses then hold for j “ k ` j “ k `
1, and that X p k ` q satisfies condition H p k ` q so the fourth hypothesisis also true when k is replaced by k `
1. This completes the induction.The resulting map X Ñ hocolim k X p k q is an isomorphism in homotopy in de-grees ă N , and injective in mod p homology, as desired. We still need to show that π j p hocolim k X p k qq vanishes for all j ě N . By construction, π j p hocolim k X p k qq{ p – π j p X p j ` qq{ p is trivial for all j ě N . Since X p j ` q is also H F p -nilpotentlycomplete, the group π j p X p j ` qq is Ext- p -complete by Theorems 2.5 and 6.6of [10]. The vanishing mod p of an Ext- p -complete abelian group implies thevanishing of that abelian group, as a special case of Theorem A.6(d) of [19]; so π j p X p j ` qq – π j p hocolim k X p k qq – j ě N , as desired. (cid:3) Morava K-theory of homotopy limits
Recall that, for each prime number p and positive integer n , we have the homo-topy fiber sequence Σ p p n ´ q k p n q Ñ k p n q Ñ H F p , and the composite map H F p Ñ Σ p n ´ k p n q Ñ Σ p n ´ H F p (6)is the cohomology operation Q n , which satisfies Q n “
0. This implies some use-ful relationships between Morava K -theories and Margolis homology of E p Q n q -modules, which we summarize in an appendix to this paper, appendix A, containingvarious results which are basically well-known but which are not well-documentedin the literature. That appendix does not logically depend on anything earlier inthe paper. The reader who is not already familiar with Margolis homology and itstopological applications can consult appendix A for a crash course. Definition 3.1.
We say that an E p Q n q -module is Q n -acyclic if the Margolis Q n -homology H p M ; Q n q vanishes. We say that a morphism of E p Q n q -modules is a Q n -equivalence if it induces an isomorphism in Q n -Margolis homology. Recall the following useful result of Adams, Theorem III.15.2 of [1]:
Theorem 3.2.
Suppose that R is a subring of Q , E is a bounded-below spectrumsuch that H r p E q is a finitely generated R -module for all r , and t Y i u i P I is a setof spectra such that π r p Y i q is an R -module for all r . Suppose that there exists a uniform lower bound for π ˚ p Y i q , i.e., there exists an integer N such that π n p Y i q – for all n ă N . Then the canonical map of spectra E ^ ź i P I X i Ñ ź i P I p E ^ X i q is a weak equivalence. As an easy corollary of Theorem 3.2:
Corollary 3.3.
Let ¨ ¨ ¨ Ñ Y Ñ Y Ñ Y be a sequence of morphisms of spectra. Suppose that there exists a uniform lowerbound on π ˚ p Y i q . Then the canonical map of spectra H F p ^ holim i X i Ñ holim i p H F p ^ X i q is a weak equivalence.Consequently, for each nonnegative integer n , we have a short exact sequence Ñ R lim i H n ` p X i ; F p q Ñ H n p holim i X i ; F p q Ñ lim i H n p X i ; F p q Ñ . The dual action of Q n on mod p homology is compatible with the comparisonmap (3.2) , so that the direct sum of the sequences (3.3) , Ñ Σ ´ R lim i H ˚ p X i ; F p q Ñ H ˚ p holim i X i ; F p q Ñ lim i H ˚ p X i ; F p q Ñ , is a short exact sequence of graded E p Q n q -modules.Proof. We have the commutative diagram in the stable homotopy category E ^ holim i X i / / (cid:15) (cid:15) E ^ ś i X i id ´ T / / (cid:15) (cid:15) E ^ ś i X i (cid:15) (cid:15) holim i p E ^ X i q / / ś i p E ^ X i q id ´ T / / ś i p E ^ X i q whose rows are homotopy fiber sequences. The middle and right-hand verticalmaps in (3) are weak equivalence, by the special case R “ Z , E “ H F p of Theorem3.2. The map of long exact sequences induced in π ˚ by (3), together with the FiveLemma, now imply that the left-hand vertical map in (3) is also a weak equivalence.The sequence (3.3) is simply the Milnor exact sequence for π ˚ holim i p H F p ^ X i q . (cid:3) Lemma 3.4.
Let n ą , let p be a prime, and let X be a bounded spectrum (i.e., X is bounded above and bounded below) such that H ˚ p X ; F p q is finite type. Thenthe E -page of the Adams spectral sequence E s,t – Ext s,t gr Comod p A ˚ q p F p , H ˚ p k p n q ^ X ; F p qq ñ π t ´ s ´ p k p n q ^ X q ˆ H F p ¯ (7) d r : E s,tr Ñ E s ` r,t ` r ´ r is v n -power-torsion. That is, every bihomogeneous element x in that E -page sat-isfies v m x “ for some m . OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 11
Proof.
Since X is bounded, it is in particular bounded below, so k p n q ^ X is alsobounded below. Since k p n q is a ring spectrum with π p k p n qq – F p , π ˚ p k p n q ^ X q isa F p -vector space, hence is automatically p -adically complete. So k p n q ^ X is H F p -nilpotently complete, so the E -page of spectral sequence (7) is the associatedgraded of a filtration on k p n q ˚ p X q . Since X is bounded, it is K p n q -acyclic, so v ´ n p k p n q ˚ p X qq must vanish. The associated graded of a filtered F p r v n s -module M can have fewer v n -torsion-free elements than M , but it cannot have more, soinverting v n on the E -page of spectral sequence (7) must yield zero. (cid:3) In fact, it will be useful to know more information about the bound on theAdams filtration of the v n -torsion in the E -page of the Adams spectral sequencefor k p n q homology of a bounded spectrum. First we prove a lemma that we expectto be well known, but we could not find in the literature. Lemma 3.5.
Let M be a finite type ˆ Z p -module. Then the graded k p n q ˚ -module k p n q ˚ p HM q is simple v n -torsion.Proof. First, note that a finite type ˆ Z p -module M is of the form M “ ˆ Z n p ‘ p Z { p Z q ‘ n ‘ p Z { p Z q ‘ n ‘ ¨ ¨ ¨ ‘ p Z { p k Z q ‘ n k so it suffices to show that k p n q ˚ p H Z p q is simple v n torsion and that k p n q ˚ ˚p H Z { p j Z q is simple v n -torsion for all positive integers j and n and primes p .First, note that there is an isomorphism of modules over the Steenrod algebra H ˚ p H Z p q – p A {{ E p Q qq ˚ , which is free over E p Q n q for any positive integer n .Consequently, the Margolis homology H p H ˚ p H Z p q , Q n q vanishes. The E -page ofthe Adams spectral sequence converging to k p n q ˚ p H ˆ Z p q is isomorphic toExt ˚ , ˚ E p Q n q p F p , H ˚ p H ˆ Z p qq by Proposition A.7, from the appendix on Margolis homology, so above the zeroline these groups can be identified with Margolis homology, which vanishes as wejust concluded. Therefore, the Adams spectral sequence collapses to the zero line,which is simple v n -torsion. Therefore k p n q ˚ p ˆ Z p q is simple v n -torsion.Now we know that the multiplication-by- p j map S Ñ S on the sphere spec-trum is nulhomotopic after applying the function spectrum functor F p´ , H F p q . So,if we smash p j : S ÝÑ S with H Z , the resulting map p j : H Z ÝÑ H Z is nulho-motopic after applying F p´ , H F p q , by closedness of SHC as a monoidal category.Consequently the induced map H ˚ p H Z ; F p q Ñ H ˚ p H Z ; F p q is zero, and the longexact sequence induced in mod p cohomology by the fiber sequence H Z p j ÝÑ H Z Ñ H Z { p j Z splits into short exact sequences:(8) 0 (cid:15) (cid:15) H ˚ p H Z ; F p q o o – (cid:15) (cid:15) H ˚ p H Z { p j Z ; F p q o o “ (cid:15) (cid:15) H ˚ p Σ H Z ; F p q o o – (cid:15) (cid:15) o o (cid:15) (cid:15) A {{ E p q o o H ˚ p H Z { p j Z ; F p q o o Σ A {{ E p q o o o o Note that k p n q ^ H ˆ Z p is bounded below and has Ext- p -complete homotopy groups so it is H F p -nilpotently complete and the Adams spectral sequence indeed converges to k p n q ˚ H ˆ Z p . Since A {{ E p q is Q n -acyclic for positive n the long exact sequence induced in Q n -Margolis homology by the bottom row of (8) gives us that H ˚ p H Z { p j Z ; F p q isalso Q n -acyclic. Proposition A.7, from the appendix on Margolis homology, thengives us Ext sE p Q n q ` H ˚ p H Z { p j Z ; F p q , F p ˘ – H ` H ˚ p H Z { p j Z ; F p q ; Q n ˘ ˚ –
0. So the E -page of the Adams spectral sequence E s,t – Ext s,tA ` H ˚ p k p n q ^ H Z { p j Z ; F p q , F p ˘ – Ext s,tE p Q n q ` H ˚ p H Z { p j Z ; F p q , F p ˘ ñ π t ´ s `` k p n q ^ H Z { p j Z ˘˘ is concentrated on the s “ v n -multiplication increases Adams degreein this spectral sequence, π ˚ ` k p n q ^ H Z { p j Z ˘ must be simple v n -torsion. (cid:3) Lemma 3.6.
Let n be a positive integer, let p be a prime, and let t X i u be a sequenceof bounded spectra, each with maximal nontrivial homotopy group in degree M ´ ,such that c is a finite-type graded ˆ Z p -module for each i . Then lim i k p n q ˚ p X i q is v n -power-torsion, and consequently the inverse limit Adams spectral sequence lim i Ext ˚ , ˚ E p Q n q ˚ p F p , H ˚ p X i qq ñ lim i k p n q ˚ X i has v n -power-torsion E -page.Proof. We will compare two different exact couples in order to produce a spectralsequence(1) whose E -page is simple v n -torsion,(2) that strongly converges tolim i k p n q ˚ p X i q , (3) and that has a vanishing line that implies that there cannot be any in-finite v n -towers produced by passing from the E -page to the abutmentlim i k p n q ˚ p X i q .Applying k p n q ˚ to the Postnikov tower of X i yields the spectral sequence E s,t – k p n q s p Σ t Hπ t X i q ñ k p n q s X i (9) d r : E rs,t Ñ E rs ´ ,t ` r , with E -page isomorphic to H ˚ p k p n q ; π ˚ p X i qq . The spectral sequence is functorial,and its E -page is the associated graded of the filtration on k p n q ˚ p X i q in whichan element x P k p n q ˚ p X i q has filtration ě j if and only if the projection mapfrom X i to its j th Postnikov truncation X ď ji sends x to zero in k p n q -homology.In particular, this is a decreasing filtration, so the map k p n q ˚ p X i ` q Ñ k p n q ˚ p X i q may raise filtration, but cannot decrease filtration. OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 13
Beginning with the E -page, [32, Thm. 4.1] establishes that this spectral se-quence is isomorphic to the usual Atiyah-Hirzebruch spectral sequence H ˚ p k p n q ; π ˚ p X i qq ñ k p n q ˚ X i constructed by applying X i -homology to a CW-decomposition of k p n q . But spectralsequence (9) has an important property which the usual Atiyah-Hirzebruch spe-tral sequence (AHSS), arising from a CW-decomposition of k p n q , doesn’t obviouslyhave: the E -page of (9) consists of simple v n -torsion, by Lemma 3.5. Consequentlyall later pages in (9) are, as k p n q ˚ -modules, simple v n -torsion. Consequently, foreach r ě
2, the E r -page in the homological Atiyah-Hirzebruch spectral sequence H ˚ p k p n q ; π ˚ p X i qq ñ k p n q ˚ X i consists of simple v n -torsion, by the following ar-gument: for each pair i, t , the graded k p n q ˚ -module k p n q ˚ p Σ t Hπ t X i q consists of simple v n -torsion in grading degrees ě t , by Lemma 3.5.That observation is our first reason to consider spectral sequence (9) built usingthe Postnikov system of X i . For both spectral sequences, we will show that thehypothesis that π ˚ X i is a finite type graded ˆ Z p -module is sufficient to produce alimit of exact couples, however it is more clear from the AHSS that this spectralsequence strongly converges to the desired abutment.We now give an explicit construction of the AHSS by choosing a finite-typeCW-decomposition of k p n q , and letting(10) k p n q p q Ñ k p n q p q Ñ k p n q p q Ñ . . . be the associated skeletal filtration. Applying X i -homology to (10) yields the un-rolled (in the sense of [9]) exact couple(11) . . . / / p X i q ˚ p k p n q p q q (cid:15) (cid:15) / / p X i q ˚ p k p n q p q q (cid:15) (cid:15) / / . . . p X i q ˚ ` k p n q p q { k p n q p q ˘ g g ❖❖❖❖❖❖❖❖❖❖❖❖❖ p X i q ˚ ` k p n q p q { k p n q p q ˘ j j ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ which is natural in the variable i . Here we are writing k p n q p j q { k p n q p j ´ q for thehomotopy cofiber of the inclusion map k p n q p j ´ q Ñ k p n q p j q .Since π ˚ p X i q is a finite-type graded ˆ Z p -module, so is each of the graded abeliangroups in (11). So R lim i of each of those groups vanishes, so we get an unrolledexact couple(12) . . . / / lim i p X i q ˚ p k p n q p q q (cid:15) (cid:15) / / lim i p X i q ˚ p k p n q p q q (cid:15) (cid:15) / / . . . lim i p X i q ˚ ` k p n q p q { k p n q p q ˘ h h ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ lim i p X i q ˚ ` k p n q p q { k p n q p q ˘ j j ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ It is a classical theorem of Maunder [26] that the cohomological
Atiyah-Hirzebruch spectralsequence H ˚ p X ; π ˚ p E qq ñ r Σ ˚ X, E s , arising from a CW-decomposition of X , is isomorphic(beginning with the E -pages) to the spectral sequence H ˚ p X ; π ˚ p E qq ñ r Σ ˚ X, E s arising fromthe Postnikov system of E . As far as we know, the analogous results for the homological Atiyah-Hirzebruch spectral sequence did not appear in the literature until Tene’s paper. I.e., a CW-decomposition with finitely many cells in each dimension. This is possible becausethe homology of k p n q is finite-type. The spectral sequence of (11) is the classical homological Atiyah-Hirzebruchspectral sequence of the bounded spectrum X i , so it is strongly convergent to thecolimit, by section 12 of [9]. In particular, the limit lim j p X i q ˚ ` k p n q p j q ˘ vanishes.So we have lim j lim i p X i q ˚ ´ k p n q p j q ¯ – lim i lim j p X i q ˚ ´ k p n q p j q ¯ – , i.e., the spectral sequence of (12) is also strongly convergent to the colimit, andthat colimit is colim j lim i p X i q ˚ ´ k p n q p j q ¯ . Finally, since (10) is a finite-type
CW-decomposition, each skeleton k p n q p j q is afinite spectrum, so we have a Milnor sequence0 Ñ R lim i Σ k p n q p j q˚ p X i q Ñ k p n q p j q˚ p holim i X i q Ñ lim i k p n q j ˚ p X i q Ñ R lim i term vanishes due to our finiteness hypotheses. Consequently wehave isomorphismscolim j lim i p X i q ˚ ´ k p n q p j q ¯ – colim j π ˚ ` k p n q j ^ holim i X i ˘ – π ˚ ´ hocolim j ´ k p n q p j q ^ holim i X i ¯¯ – π ˚ p k p n q ^ p holim i X i qq– k p n q ˚ p holim i X i q , i.e., the spectral sequence of the exact couple (12) converges strongly to k p n q ˚ p holim i X i q . We want to understand the action of v n -multiplication on k p n q ˚ p holim i X i q ,but the exact couples (11) and (12) aren’t exact couples of k p n q ˚ -modules, andin particular, v n -multiplication isn’t defined on these exact couples! However, byTene’s homological version of Maunder’s theorem, the spectral sequence of (11) isisomorphic (starting with the E -page) to the spectral sequence (9), whose exactcouple is an exact couple of k p n q ˚ -modules, and whose functoriality gives us v n -multiplication as an endomorphism of the whole exact couple and consequently ofits spectral sequence. Let E i denote the unrolled exact couple(13) . . . / / k p n q ˚ p X ď i q x x qqqqqqqqqqqqqq / / k p n q ˚ p X ď i q u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ / / . . .k p n q ˚ ` Σ Hπ X i ˘ O O k p n q ˚ ` Σ Hπ X i ˘ O O Our finite-type assumptions on π ˚ p X i q give us that each of the groups in E i isfinite, and hence the relevant R lim i groups all vanish, so that lim i E i is an exactcouple. The spectral sequence of the exact couple lim i E i is isomorphic to that of(12), hence converges to k p n q ˚ p holim i X i q . We showed already that the E -pageof the spectral sequence (9) associated to E i is simple v n -torsion, and since a limitof simple v n -torsion k p n q ˚ -modules remains simple v n -torsion, we have that the E -page of the spectral sequence of lim i E i is simple v n -torsion. So its later pages,and its E -page (whose associated graded is k p n q ˚ p holim i X i q ) must also be simple v n -torsion. OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 15
We have therefore produced a spectral sequence whose E -page is simple v n -torsion and which strongly converges to lim i k p n q ˚ X i so it suffices to produce avanishing line. This vanishing line is clear from the description of the E -page aslim i H ˚ p k p n q , π ˚ p X i qq and the assumption that π j X i – j ě M for each i . (cid:3) Finally, we come to the main technical tool obtained in this paper.
Theorem 3.7.
Let n, N, M be integers such that n ą and N ě M , and let ¨ ¨ ¨ Ñ Y Ñ Y Ñ Y be a sequence of morphisms of bounded-below H F p -nilpotently complete spectra.Make the following assumptions: ‚ the graded Z p -modules π ˚ Y i are finite type ‚ there is an isomorphism lim i H p H ˚ p Y i ; F p q ; Q n q – , and ‚ each spectrum Y i satisfies condition H p M q .Then holim i Y i is K p n q -acyclic.Proof. We prove this in several steps:Step 1: We consider the homotopy fiber sequence K p n q ^ holim i Y ě Ni Ñ K p n q ^ holim i Y i Ñ K p n q ^ holim i Y ă Ni in which the right-hand term is contractible since bounded-above spectra are K p n q -acyclic for n ą
0. So holim i Y i is K p n q -acyclic if and only if holim i Y ě Ni is K p n q -acyclic. So we only need to show that holim i Y ě Ni is K p n q -acyclic. We thereforeneed to show that k p n q ˚ p holim i Y ě Ni q is v n -power-torsion. Here we note that sincethe spectra Y ě Ni are uniformly bounded below there is a weak equivalence k p n q ^ p holim i Y ě Ni q » holim i ` k p n q ^ Y ě Ni ˘ by Corollary 3.3. It therefore suffices to show that π ˚ p holim i k p n q ˚ Y ě Ni q is v n -power-torsion. Since π ˚ Y i is a finite type ˆ Z p -module and consequently π ˚ Y ě Ni is afinite type Z p -module, we know that H ˚ p Y i q and H ˚ p Y ě Ni q are finite type graded F p -modules. Since H ˚ p Y ě Ni ; F p q is finite type and k p n q ^ Y ě Ni is bounded belowand H F p -nilpotent, we can apply [23, Prop. 2.2] to produce a strongly convergentinverse-limit Adams spectral sequence of the formlim i Ext ˚ , ˚ gr Mod p E p Q n qq p H ˚ p Y ě Ni ; F p q , F p q ñ π ˚ p holim i k p n q ^ Y ě Ni q . (14)We therefore need to prove:(1) that the E -page of (14) is v n -power-torsion, and(2) there cannot exist an infinite sequence of nontrivial extensions in the E -page of (14) which could yield a non- v n -power-torsion element in π ˚ p holim i k p n q ^ Y ě Ni q . Step 2: The second task, to show that we cannot have an infinite sequence ofnontrivial v n -extensions, is easy as we will now show explicitly. The E -page in question is isomorphic tolim i Ext ˚ , ˚ gr Comod p A ˚ q p F p , H ˚ p k p n q ; F p q b F p H ˚ p Y ě Ni ; F p qq– lim i Ext ˚ , ˚ gr Comod p A ˚ q ` F p , p A ˚ l E p Q n q ˚ F p q b F p H ˚ p Y ě Ni ; F p q ˘ (15) – lim i Ext ˚ , ˚ gr Comod p E p Q n q ˚ q ` F p , H ˚ p Y ě Ni ; F p q ˘ (16) – lim i Ext ˚ , ˚ grMod p E p Q n qq ` H ˚ p Y ě Ni ; F p q , F p ˘ (17)with isomorphism (15) by the well-known description of H ˚ p k p n q ; F p q given by[7], isomorphism (16) by a standard change-of-rings isomorphism as in CorollaryA1.3.13 of [30], and isomorphism (17) by the assumption that H ˚ p Y ě Ni ; F p q is finitetype and Y ě Ni is bounded below.Since every graded E p Q n q -module is a coproduct of cyclic E p Q n q -modules (aspecial case of the Cohen-Kaplansky theorem, from [13]), we knowExt ˚ , ˚ grMod p E p Q n qq ` H ˚ p Y ě Ni ; F p q , F p ˘ is a direct product of copies ofExt ˚ , ˚ gr Mod p E p Q n qq p Σ m F p , F p q and(18) Ext ˚ , ˚ gr Mod p E p Q n qq p Σ m E p Q n q , F p q (19)for various integers m and m . The Ext-module (18) is isomorphic to F p r v n s , whilethe Ext-module (19) is isomorphic to F p r v n s{ v n , concentrated on the cohomologicaldegree 0 line.Since Y ě Ni is N -connective, H ˚ p k p n q ^ Y ě Ni ; F p q vanishes below grading degree N . So, while we may have a copy of F p r v n s generated in bidegree p , N q in the E -page of the Adams spectral sequence for Y ě Ni , the E -page of that spectralsequence vanishes above the line formed by such a potential v n -tower, i.e., E s,t – t ă p p n ´ q s ` N . Of course the E -page must then vanish above the sameline. So, given an element x P E s,t , although we may have that v n x “ E but v n x ‰ k p n q ˚ p Y ě Ni q due to a filtration jump in the E -page, a fixed choice ofelement x cannot support an infinite sequence of such filtration jumps, since afterfinitely many such filtration jumps the resulting element in E would necessarilybe above the vanishing line, and hence would be zero. Write E ˚ , ˚ r p Y ě Ni q for the E r -page of the Adams spectral sequence for Y ě Ni and simply E ˚ , ˚ r for the E r -pageof (14). Then there are isomorphismslim i E ˚ , ˚ r p Y ě Ni q – E ˚ , ˚ r , by the construction of the inverse limit Adams spectral sequence (see remark aboveProposition 2.2 in [23]). Since there is a uniform vanishing line for each Adamsspectral sequence for Y ě Ni , we have the same vanishing line in (22) and the sameargument implies that there cannot be any infinite v n -towers after resolving multi-plicative extensions.Step 3: Therefore, it suffices to prove that the E -page of (14) is v n -power torsion.By Theorem 2.6, the sequence0 Ñ H ˚ p Y i ; F p q Ñ H ˚ p Y ă Ni ; F p q Ñ H ˚ p Σ Y ě Ni ; F p q Ñ OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 17 is exact for each i . Since H ˚ p Y i ; F p q and H ˚ p Y ă Ni ; F p q , and consequently H ˚ p Y ě Ni ; F p q ,are finite type, and since H p H ˚ p X ; F p q ; Q n q is a subquotient of H ˚ p X q for any spec-trum X , we know that the we know that H p H ˚ p Y i ; F p q ; Q n q , H p H ˚ p Y ă Ni ; F p q ; Q n q ,and H p H ˚ p Y ě Ni ; F p q ; Q n q are each finite type. Since R lim vanishes on sequencesof finite-dimensional vector spaces, we now have R lim H p H ˚ p Y i ; F p q ; Q n q – R lim H p H ˚ p Y ě Ni ; F p q ; Q n q– R lim i H p H ˚ p Y ă Ni ; F p q ; Q n q – . We apply Margolis homology to the sequence (20) to get a long exact sequencefor each i , and consequently an inverse sequence of long exact sequences. In general,applying lim to an inverse sequence of long exact sequences yields a chain complexwith no guarantee of exactness anywhere at all. But in our case, we have just shownthat R lim i H p´ ; Q n q vanishes on each term in (20), so applying lim i does producea long exact sequence. In this long exact sequence, we know that every third termis some suspension of lim i H p H ˚ p Y i ; F p q ; Q n q . Since this vanishes by assumption, we get an isomorphismlim i H p H ˚ p Σ ´ Y ă Ni ; F p q ; Q n q Ñ lim i H p H ˚ p Y ě Ni ; F p q ; Q n q . (21)induced by the right-hand map in (20).We note that Y ă Ni is a bounded spectrum and k p n q ^ Y ă Ni is therefore boundedbelow and H F p -nilpotent complete so we may use functoriality of [23, Prop. 2.2]to produce a map of inverse limit Adams spectral sequences fromlim i Ext ˚ , ˚ gr Mod p E p Q n qq p H ˚ p Σ ´ Y ă Ni ; F p q , F p q ñ π ˚ p holim i k p n q ^ p Σ ´ Y ă Ni qq (22)to the spectral sequence (14).We then observe that this map of spectral sequences induces the compositeisomorphism lim i Ext s ` ,t gr Comod p E p Q n q ˚ q ` H ˚ p Σ ´ Y ă Ni ; F p q , F p ˘ – lim i ´ Σ ´ s p p p n ´ qq H ` H ˚ p Σ ´ Y ă Ni ; F p q ; Q n ˘¯ ´p t ` p n ´ q (23) – lim i ´ Σ ´ s p p p n ´ qq H ` H ˚ p Y ě Ni ; F p q ; Q n ˘¯ ´p t ` p n ´ q (24) – lim i Ext s ` ,t gr Mod p E p Q n qq ` H ˚ p Y ě Ni ; F p q , F p ˘ (25)with isomorphisms (23) and (25) due to Proposition A.4 from the appendix onMargolis homology, and with isomorphism (24) given by (21). To see that thisisomorphism is the same as the the map of E pages induced by the mapΣ ´ Y ă Ni Ñ Y ě Ni , as we have claimed, we must simply observe that the isomorphism (23) is a nat-ural isomorphism. Also, to understand the grading shift of Margolis homology inisomorphism (23) and isomorphism (25) we point the reader to our grading con-vention in Conventions A.2, though this grading shift plays no significant role inthe argument. Note that we are not claiming that these two spectral sequenceshave isomorphic E pages, but rather that their E pages are isomorphic above the s “ By Lemma 3.4 the E -page of the H F p -Adams spectral sequence for k p n q^ Y ă Ni is v n -power-torsion for each i . To draw the same conclusion for the spectral sequence(14), we need more information. In fact, spectral sequence (22) has a vanishing lineof slope 1 {| v n | that crosses the x -axis at N , as we argued earlier. Consequently, the E -page of the inverse-limit Adams spectral sequence (14) vanishes above the s “ E -page of (14) is a limit of v n -power-torsion E -pages, theonly possible problem would be if a sequence of v n -power-torsion elements became v n -power-torsion free in the limit, but the vanishing region and Lemma 3.6 makesthis impossible. Therefore, the E -page of (14) and consequently the E -page of(22) is v n -power-torsion. (cid:3) A higher chromatic height analogue of Mitchell’s theorem
In [28], Mitchell proved that K p m q ˚ p K p Z qq – m ě K p m q ˚ p K p R qq – H Z -algebra R . Throughout this section we will write BP x n y for the truncated Brown-Peterson spectrum with coefficients π ˚ p BP x n yq – Z p p q r v , . . . , v n s . We also use the convention that BP x´ y is H F p . Since BP x y “ H Z p p q , we may consider the following higher chromatic height analogue of Mitchell’sresult. Question 4.1.
Suppose n is some integer, n P r´ , . If R is a BP x n y -algebraspectrum, then does K p m q ˚ p K p BP x n yqq vanish for all m ě n ` K p BP x n yq sothat if there is a “red-shift” in algebraic K-theory of a BP x n y -algebra spectra, thenthis shift is a shift of at most one. The main goal of this section is to answer thisquestion for all p n, p q such that BP x n y can be modeled by an E ring spectrum,which is known to be n “ ´ , , n “ p “ ,
3. The cases n “ ´ n “ H F p and H Z p are E ring spectra. There is an E ring spectrum model for BP x y by McClure-Staffeldt [27] and there is an E ring spectrum model for BP x y at the prime p “ p “ Conventions 4.2.
When n “ ´ , , n “ p “ p “
3, we write BP x n y for an E ring spectrum model for the truncatedBrown-Peterson spectrum, which exist by the discussion above. Remark 4.3.
One can ask whether our positive answer to Question 4.1, Theorem4.10, in this section can be extended to those values of p n, p q for which BP x n y is not known to admit an E ring structure—or even those values of p n, p q for which BP x n y is known not to admit an E ring structure, when n ě p “ p ą E multiplication on BP x n y : if the results of Bruner-Rognes [12] can be extended to the setting of E ring spectra leading to the samecomputations of H c ˚ p T C ´ p BP x n yq ; F p q and H c ˚ p T P p BP x n yq ; F p q as appear there,then our main theorem in this section, Theorem 4.10, will apply for any p n, p q suchthat BP x n y has a model as an E ring spectrum. The spectrum BP was shown toadmit an E ring spectrum model by Basterra-Mandell [8]. We have heard othermathematicians claim that the spectra BP x n y also admit E multiplications, but OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 19 as far as we have been able to tell, this is folklore and does not appear in theliterature. There is currently work in progress of Hahn-Wilson on showing that BP x n y is indeed E as a BP -algebra.If BP x n y admits an E multiplication, and if the calculations of [12] can be madeto work using only the E Dyer-Lashof-Kudo-Araki operations rather than theclassical ( E ) Dyer-Lashof-Kudo-Araki operations, then our Theorem 4.10 wouldgive a positive answer to Question 4.1 for all primes p and heights n .We first briefly recall the setup for topological periodic cyclic homology andtopological negative cyclic homology. Let R be an E ring spectrum and write T for the circle group. Recall that the topological Hochschild homology of R , denoted T HH p R q has a canonical T -action. We define topological negative cyclic homology as the homotopy fixed-point spectrum T C ´ p R q : “ T HH p R q h T and topological periodic cyclic homology as the Tate spectrum T P p R q : “ T HH p R q t T . There is a homological T -homotopy fixed point spectral sequence E ˚ , ˚ p R q “ H ˚ p T , H ˚ p T HH p R q ; F p qq ñ H c ˚ p T C ´ p R q ; F p q (26)with abutment H c ˚ p T C ´ p R q ; F p q : “ lim k Ñ´8 H ˚ p T C ´ p R qr k s ; F p q where T C ´ p R qr k s : “ F p E T p k q , T HH p R qq T and E T p k q is the k -skeleton of E T . There is also a homological T -Tate spectralsequence ˆ E ˚ , ˚ p R q “ ˆ H ˚ p T , H ˚ p T HH p R q ; F p qq ñ H c ˚ p T P p R q ; F p q (27)with abutment H c ˚ p T P p R q ; F p q : “ lim k Ñ´8 H ˚ p T P p R qr k s ; F p q where T P p R qr k s : “ ´Ą E T { Ą E T k ^ F p E T , T HH p R qq ¯ T (28)is the Greenlees filtration [16] where Ą E T k is the cofiber of the map E T p k q` Ñ S for k ě Ą E T ´ k ´ if k ă E T and on Ą E T only in a range of dimensions,there is also a spectral sequence E ˚ , ˚ p R q “ H ˚ě´ k p T , H ˚ p T HH p R q ; F p qq ñ H ˚ p T C ´ p R qr k s ; F p q (29)whose input is P k ` p t q b H ˚ p T HH p R q ; F p q with | t | “ ´ E ˚ , ˚ p R q “ ˆ H ˚ě´ k p T , H ˚ p T HH p R q ; F p qq ñ H ˚ p T P p R qr´ k s ; F p q (30)whose input is P p t ´ qt t k u b H ˚ p T HH p R q ; F p q for k ě More generally, for a homology theory E ˚ we write E c ˚ p T P p R qq : “ lim k Ñ´8 E ˚ p T P p R qr k sq and E c ˚ p T C ´ p R qq : “ lim k Ñ´8 E ˚ p T C ´ p R qr k sq . We now recall that by Angeltveit-Rognes [4, Thm. 5.12], there is an isomorphism H ˚ p T HH p BP x n yq ; F p q – H ˚ p BP x n y ; F p q b E p σ ¯ ξ , σ ¯ ξ , . . . , σ ¯ ξ n ` q b P p σ ¯ τ n ` q of A ˚ -comodules with A ˚ -coaction ν n : H ˚ p T HH p BP x n yq ; F p q Ñ A ˚ b H ˚ p T HH p BP x n yq ; F p q given by the restriction of the coproduct of A ˚ to H ˚ p BP x n y ; F p q Ă A ˚ on elementsin H ˚ p BP x n y ; F p q , and by the formula ν n p σx q “ p b σ qp ν n p x qq in [4, Eq. 511] for elements of the form σ ¯ ξ i for 1 ď i ď n and σ ¯ τ n ` , and then forthe remaining elements by the formula ν n p xy q “ ν n p x q ν n p y q . Bruner-Rognes [12] then compute H c ˚ p T C ´ p BP x n yq ; F p q and H c ˚ p T P p BP x n yq ; F p q ,as we recall below. We will focus on odd primes for simplicity, but essentially thesame results will hold at the prime 2. In [12], they compute E ˚ , ˚8 p BP x n yq – P p t q b P p ¯ ξ pk | ď k ď n ` q b P p ¯ ξ k ` | k ě n ` qb E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q ‘ T where T consists of classes x in filtration s “ tx “ τ k ` “ ¯ τ k ` ´ ¯ τ k p σ ¯ τ k q p ´ for k ě m . One can easily deduce the computationˆ E ˚ , ˚8 p BP x n yq – P p t, t ´ q b P p ¯ ξ pk | ď k ď n ` q b P p ¯ ξ k ` | k ě n ` qb E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q for topological periodic cyclic homology from their computation. There are nopossible additive extensions because the abutment is an F p -vector space.It will also be useful to record the computations for each spectrum in the filtra-tion:ˆ E ˚ , ˚8 p BP x n yqr´ k s “r P p t ´ qt t k ´ u b P p ¯ ξ pk | ď k ď n ` q b P p ¯ ξ k ` | k ě n ` qb E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n qs ‘ V p n, k q where V p n, k q : “ rXt ker d s,t | s “ ´ k, t ě us (31)and we write d s,t for the differential in the spectral sequence computing H c ˚ p T P p BP x n yq ; F p q rather than the truncated one. We now compute the “continuous Margolis ho-mology,” with respect to the Greenlees filtration, of the homology groups of thetopological periodic homology of BP x n y ; i.e., the limit of the Margolis homologyof H ˚ p T P p BP x n yqr k s ; F p q as k goes to negative infinity. OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 21
Proposition 4.4.
There are isomorphisms lim k Ñ´8 H p H ˚ p T P p BP x n yqr k s ; F p q , Q m q – for all m ě n ` .Proof. First, we note that Q m acts trivially on any P p t ´ q module contained in P p t ˘ q for all m ě t is in an even degree. We also claim that the tensorproduct P p t ´ qt t k ´ u b P p ¯ ξ k ` | k ě n ` q b E p ¯ τ k ` | k ě n ` qb P p ¯ ξ pk | ď k ď n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q b V p n, k q in the abutment of the truncated homological T -Tate spectral sequence is still atensor product of Q m -modules, in other words, we will argue that there are nopossible hidden E p Q n q -module extensions.To see this we note that there is no room for hidden Q m -module extension inthe truncated homological T -Tate spectral sequence for the sphere spectrum P p t qt t k ´ u ñ H ˚ p T P p S qr´ k sq which maps to the truncated homological T -Tate spectral sequence for BP x n y .The truncated T -Tate spectral sequence for the sphere spectrum collapses andhas abutment P p t ´ qt t k ´ u with trivial Q m -action. The map to the abutment H ˚ p T P p BP x n yqr k s ; F p q is an injection and it is a map of Q n -modules so the el-ements t k must have trivial Q m -action. Here we use the fact that the abutmentis a graded F p -module so there are no hidden additive extensions involving t j .Therefore, all elements of the form xt j in the abutment must have Q m -action Q m p xt j q “ Q m p x q t j ` xQ m p t j q “ Q m p x q t j . So the Q m -action preserves the filtra-tion in the truncated homological T -Tate spectral sequence. Since hidden comoduleextensions would shift filtration, we know that the abutment is a tensor product of E p Q m q -modules and it is isomorphic to the E -page as E p Q m q -modules.Now that we have proven the claim, we can apply the K¨unneth isomorphismfor Margolis homology [24, Prop. 18.1.2(a)] and the fact that Margolis homologycommutes with coproducts of E p Q m q -modules to produce the isomorphism H p H ˚ p T P p BP x n yqr k s ; F p q , Q m q – ` P p t ´ qt t k ´ u b H p M , Q m q b H p M , Q m q ˘ ‘ H p V p n, k q , Q m qt t k u where M : “ P p ¯ ξ k ` | k ě n ` q b E p ¯ τ k ` | k ě n ` q ,M : “ P p ¯ ξ pk | ď k ď n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q , and V p n, k q is defined as in (31), above. We also recall that Q m p ¯ τ k ` q “ ¯ ξ p m ` k ´ m for k ě m `
1. This holds by the coaction ν p ¯ τ k ` q “ ¯ τ m b ¯ ξ p m ` k ´ m ` ¯ ν p ¯ τ k ` q where the ¯ ν p ¯ τ k ` q consists of terms such that ¯ τ m b x does not appear for anynontrivial x . It is easy to observe that the same will be true of τ k ` for k ě n ` Q k ` -Margolis homology of M b M as the tensorproduct of the chain complex p c k ` q : “ ` F p t u Ð F p t τ k ` u ˘ with the chain complex whose Margolis homology is H p M b M , Q k ` q where M – M b E p ¯ τ k ` q . Since the chain complex p c k ` q is acyclic, we observe thatthe Margolis homology of P p t ´ qt t k ´ u b M b M is trivial for k ě n `
1. We alsoobserve that the map H ˚ p H ˚ p T P p BP x n yqr k s ; F p q , Q m q Ñ H ˚ p H ˚ p T P p BP x n yqr k ` s ; F p q , Q m q maps the summand H ˚ p V p n, k q , Q m q to zero and maps zero to H ˚ p V p n, k ` q , Q m q by examination of the map of truncated Tate spectral sequences and consequentlythe map is the zero map. Therefore, t H p H ˚ p T P p BP x n yqr k s ; F p q , Q m qu is pro-equivalent to the constant pro-object on the zero object and in particularlim k Ñ´8 H p H ˚ p T P p BP x n yqr k s ; F p q , Q m q “ . (cid:3) Lemma 4.5.
Let R be a connective p -local E ring spectrum such that π p R q – Z p p q and such that π ˚ p R q is finite type as a graded Z p p q -module. Then the homotopygroups of the enveloping algebra π ˚ p R ^ R op q are also a finite type graded Z p p q -module.Proof. We first show that π ˚ p R ^ S p p q R op q is finite type as a Z p p q -module. We applythe K¨unneth spectral sequenceTor π ˚ p S p p q q˚ , ˚ p π ˚ p R q , π ˚ p R op qq ñ π ˚ p R ^ ˆ S p p q R op q . Since π ˚ p R q is a finite type Z p p q -module and π ˚ p S q is a finite type Z p p q -module andthe unit map S p p q Ñ R is an isomorphism on π , we may see by an easy resolutionargument that each bidegree in the spectral sequence is a finite type Z p p q -module.Since this is a first quadrant spectral sequence graded with the Serre convention,we know that only finitely many bidegrees contribute to each homotopy degree inthe abutment.Note that R ^ S p p q R op is defined as the equalizer R ^ S p p q ^ R op p ψ R q p p q ^ / / ^p ψ L q p p q / / R ^ R op (32)where p ψ R q p p q and p ψ L q p p q are the usual right and left action of S p p q on the p -localspectrum R respectively. We may then simply observe that Bousfield localizationat the Moore spectrum S Z p p q is smashing so the equalizer (32) is equivalent to theequalizer S p p q ^ p R ^ S ^ R op q ^ ψ R ^ / / ^ ^ ψ L / / S p p q ^ p R ^ R op q , which in turn is equivalent to p R ^ R op q p p q where we write p´q p p q for Bousfieldlocalization at S Z p p q . Since R ^ R op is already p -local, there is an equivalence to R ^ R op » p R ^ R op q p p q and consequently, π ˚ p R ^ R op q – π ˚ p R ^ S p p q R op q as graded Z p p q -modules. (cid:3) OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 23
We now show that the filtration t ˆ T P p BP x n yqr i s p u satisfies the remaining hy-potheses of Theorem 3.7. Lemma 4.6.
The graded Z p p q -modules π ˚ p T P p BP x n yqr i sq are finite type for all i and all pairs p n, p q such that BP x n y is an E ring spectrum. Consequently, π ˚ p ˆ T P p BP x n yqr i s p q is a finite type graded Z p -modules for all i under the sameconditions on the pairs p n, p q .Proof. Since the truncated Tate spectral sequenceˆ H ˚ě´ k p T ; π ˚ p T HH p BP x n yqqq ñ π ˚ p T P p BP x n yqr´ k sq has a vanishing line of vertical slope with the Serre convention, which implies onlyfinitely many bidegrees contribute to each homotopy degree in the abutment, itsuffices to show that π ˚ p T HH p BP x n yqq is a finite type graded Z p p q -module for all n . To show this, we apply [2, Lem. 5.2.5], which states that if R is a connective E ring spectrum whose homotopy groups π ˚ p R b R q are finite type π p R q -modulesthen, in particular, T HH ˚ p R q is finite type. Note that π ˚ p BP x n yq – Z p p q r v , . . . v n s is a connective commutative graded π p BP x n yq – Z p p q -module and by assumption BP x n y is E , so to apply [2, Lem. 5.2.5] it suffices to show that π ˚ p BP x n y ^ BP x n yq is finite type, but this follows by Lemma 4.5. (cid:3) Remark 4.7.
The only place where the E ring spectrum structure is used in theresult above is in the hypothesis of [2, Lem. 5.2.5]. However, [2, Lem. 5.2.5] canbe easily generalized to E ring spectra when we are only applying it to topologicalHochschild homology, and not to the “higher THH” constructions obtained bytensoring with simplicial sets other than the standard simplicial circle, which wereunder consideration in the paper [2] and which motivated the assumption of an E ring structure in that paper. Lemma 4.8.
The spectra ˆ p T P p BP x n yqr´ k sq p satisfy condition H p M q for all k ě and a fixed M depending on n , but not on k .Proof. Note that H ˚ p ˆ p T P p BP x n yqr´ k sq p q – H ˚ p T P p BP x n yqr´ k sqq . It suffices tocheck that the sub- A ˚ -comoduleHom A p F p , H ˚ p T P p BP x n yr k sq ; F p qq Ă H ˚ p T P p BP x n yqr k s ; F p q of A ˚ -comodule primitives is bounded above. We first choose a cofinal sequencethat always contains elements in homotopy that are not t -divisible to make theargument easier. Recall thatˆ E ˚ , ˚8 p BP x n yqr´ k s “ p P p t ´ qt t k ´ u b P p ¯ ξ pk | ď k ď n ` q b P p ¯ ξ k ` | k ě n ` qb E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n qq‘p H ˚ p T HH p BP x n yq ; F p q{ im p d ´ k ` , ˚ qq where we assume k ą A ˚ -comodule extensions. We claim thatafter resolving hidden comodule extensions there cannot be additional comoduleprimitives. Suppose x P H ˚ p T P p BP x n yqr´ k s ; F p q is not a comodule primitive, and has coaction ν n p x q “ ÿ x p q b x p q , adapting Sweedler’s notation used for coproducts in coalgebras to this context.Then after resolving hidden A ˚ -comodule extensions, we will have ν n p x q “ ÿ x p q b x p q ` ÿ y p q b y p q , and the terms in the sum ř y p q b y p q must be in higher filtration in the spectralsequence’s E -page, and therefore cannot cancel with terms in the sum ř x p q b x p q .Thus, we cannot have an element that is not a comodule primitive in the E -page become a comodule primitive after resolving hidden comodule extensions. Ittherefore suffices to consider the comodule primitives in ˆ E ˚ , ˚8 p BP x n yqr k s , which bythe argument above, are the same as the comodule primitives in the abutment.Now, we know that the only A ˚ -comodule primitive in P p ¯ ξ pk | ď k ď n ` q b P p ¯ ξ k ` | k ě n ` q Ă A ˚ is the element 1 because it is a sub-Hopf algebra of the dual Steenrod algebra andits coaction is the restriction of the coalgebra structure on A ˚ . Also, E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q is clearly bounded above by the same m independent of k . Also, t j for j ‰ t j such that 2 j ą k , which is in degree ´ k ` ψ : ˆ H ˚ p S , F p q Ñ ˆ H ˚ p T , F p q b A ˚ of the dual Steenrod algebra on ˆ H ˚ p T , F q p is ψ p t q “ b t ` ¯ ξ b t ` ¯ ξ b t ` . . . when j ą ψ p t q ¨ ψ p t ´ q “ b j ă
0. The coaction ψ k : ˆ H ˚ě´ k p T ; F p q Ñ ˆ H ˚ě´ k p T ; F p q b A ˚ can then be computed as ψ k p t q “ ψ p t q mod t k ` . So the only way a power t k could be a comodule primitive is when k “ j ą k ` t j x for x P p P p t ´ qt t k ´ u b P p ¯ ξ pk | ď k ď n ` q b P p ¯ ξ k ` | k ě n ` qb E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n qq will always have a summand ¯ ξ j b t j x when k is sufficiently large so t j x cannot be a comodule primitive unless 2 j ą k .Therefore the only comodule primitives are elements in E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q , possibly plus some correcting terms, or elements in E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n qt t j u OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 25 such that 2 j ą k plus some correcting terms. We claim that these are both boundedabove by the maximal degree of E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q . This follows since t j is in a negative degree and if y P E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q is not a comodule primitive then it will be of the form y ÞÑ b y ` ÿ y p q b y p q , but then y p q b t j y p q will be nonzero, since t j is not a zero divisor. Therefore, t j y will not be a comodule primitive unless y is also a comodule primitive, which provesthe claim.We still need to show that the comodule primitives in H ˚ p T HH p BP x n yq ; F p q{ im p d ´ k ` , ˚ q , are bounded above, which is not immediately obvious because the comodule prim-itives in H ˚ p T HH p BP x n yqq are not bounded above. However, the elements p σ p ¯ τ n qq k for k ě p d q andall remaining potential comodule primitives in H ˚ p T HH p BP x n yq ; F p q are boundedabove by the same bound as the highest degree element in the A ˚ -comodule E p ¯ τ k ` | k ě n ` q b E p ¯ ξ p ´ k σ ¯ ξ k | ď k ď n q . Therefore, it remains to check that sums of homogeneous elements in the abut-ment coming from different filtrations in the truncated T -tate spectral sequence arenot comodule primitives. We divide into two cases. First, consider t j x where x is anot comodule primitive. If x is not a comodule primitive, then the coaction on x is1 b x ` ÿ i ě x p q i b x p q i and there cannot be a term in the coaction on t j ` k y that cancels out the term x p q b x p q t j in the coaction on t j x . Second, consider the case t j x where x is acomodule primitive. The only comodule primitives are products of σ ¯ ξ i or possibly τ k ` plus some additional classes. In each case, there will be a term of the form¯ ξ j b t j x when j ą ξ j b j ą (cid:3) Remark 4.9.
Note that the comodule primitives are not necessarily bounded abovein H ˚ p T HH p BP x n yq ; F p q , but they are bounded above in H ˚ p T P p BP x n yqr k s ; F p q nonetheless. Theorem 4.10.
When BP x n y is an E ring spectrum, there are isomorphisms K p m q ˚ p T P p BP x n yqq – and K p m q ˚ p T C ´ p BP x n yqq – for m ě n ` for all n ě . Proof.
The statement K p m q ˚ p T P p BP x n yqq – K p m q ˚ p T C ´ p BP x n yqq – m ě n ` T HH p BP x n yq h T Ñ T HH p BP x n yq h T Ñ T HH p BP x n yq t T and the claim that K p m q ˚ p Σ T HH p BP x n yq h T q – m ě n `
2, which we now prove. Since K p m q ˚ has a K¨unneth isomorphism,there is a B¨okstedt spectral sequence HH K p m q ˚ ˚ p K p m q ˚ p BP x n yqq ñ K p m q ˚ T HH p BP x n yq . When m ě n ` K p m q ˚ p BP x n yq –
0, so clearly the spectral sequenceconverges and is trivial so that K p m q ˚ p T HH p BP x n yqq – m ě
2. We then simply use the fact that smashing with K p m q commutes withhomotopy colimits to show that K p m q ˚ p Σ T HH p BP x n yq h T q – m ě n ` (cid:3) Corollary 4.11.
When BP x n y is an E ring spectrum, there are isomorphisms K p m q ˚ p K p BP x n yqq – for m ě n ` ě .Proof. The result for topological cyclic homology follows by Theorem 4.10 togetherwith the long exact sequence in Morava K-theory associated to fiber sequence
T C p BP x n yq p Ñ T C ´ p BP x n yq p Ñ T P p BP x n yq p of [29, Cor. 1.5]. For algebraic K-theory, the result follows by [14, Thm. 7.3.1.8 ],which produces a fiber sequence K p BP x n yq p Ñ T C p BP x n yq p Ñ Σ ´ H Z p and the associated long exact sequence in Morava K-theory, since K p m q ˚ p Σ ´ H Z p q “ m ě . Note that our result does not give a new proof in the case n “ ´ F p . (cid:3) Appendix A. Review of Margolis homology.
This appendix, which does not logically rely on anything earlier in the paper,consists of material that is well-known to users of Margolis homology. We thinkit is useful to provide these results, which in a few cases we do not know writtenreferences for (but are nevertheless not difficult, and certainly not new).
OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 27
A.1.
Basics of Margolis homology.
Given a graded ring R , we write gr Mod p R q for the category of graded R -modules and grading-preserving R -module homomor-phisms. Definition A.1.
Let k be a field and let E p Q q be the exterior k -algebra on a singlehomogeneous generator Q in an odd grading degree | Q | . By Q -Margolis homology we mean the functor H p´ ; Q q : gr Mod p E p Q qq Ñ gr Mod p E p Q qq given on a graded E p Q q -module M by the quotient H p M ; Q q “ ´ ker p M Q ÝÑ Σ ´| Q | M q ¯ { ´ im p Σ | Q | M Q ÝÑ M q ¯ . Of course we shouldn’t use the word “homology” in the phrase “Margolis ho-mology” unless H p´ ; Q n q is a functor which turns short exact sequences of E p Q n q -modules to long exact sequences, but happily it does have this property: given ashort exact sequence of graded E p Q q -modules0 Ñ M Ñ M Ñ M Ñ . . . / / Σ | Q | H p M ; Q q BECDGF (cid:15) (cid:15) H p M ; Q q / / H p M ; Q q / / H p M ; Q q BECDGF (cid:15) (cid:15) Σ ´| Q | H p M ; Q q / / . . . . Here is a quick note on gradings; it is extremely elementary, but not taking amoment to “fix notations” on this point tends to lead to sign errors in the gradings.
Conventions A.2.
Given a graded ring R and graded R -modules M and N , wewrite hom R p M, N q for the degree-preserving R -linear morphisms M Ñ N , andwe write hom R p M, N q for the graded abelian group whose degree n summandis hom R p Σ n M, N q . We write Ext s,tR p M, N q for the s th right-derived functor ofhom R p´ , Σ t N q on the opposite category of graded R -modules applied to M , i.e.(up to isomorphism), the s th right-derived functor of hom R p Σ ´ t M, ´q on the cat-egory of graded R -modules applied to N . We write Ext s, ˚ R p M, N q for the gradedabelian group whose degree t summand is Ext s,tR p M, N q , and we refer to this grad-ing as the internal or topological grading, to distinguish it from the cohomological degree given by s .In particular, the k -linear dual of a graded k -vector space has the signs of thegradings reversed, i.e., p Σ n V q ˚ “ hom k p Σ n V, k q – Σ ´ n p V ˚ q . Now given a spectrum X , the action of Q n on H ˚ p X ; F p q is the one inducedin homotopy by the map of function spectra F p X, H F p q Ñ F p X, Σ p n ´ H F p q in-duced by the composite (6). Somewhat less famous than the action of Steenrod Some references, e.g. [11], use the opposite grading on hom R —hence the need to give Con-vention A.2 explicitly. The argument for our choice of gradings is that it is the one that yieldsthe hom-tensor adjunction hom R p L, hom R p M, N qq – hom R p L b R M, N q . operations on mod p cohomology, we have also the dual action of Steenrod op-erations on mod p homology: the action of Q n on H ˚ p X ; F p q is the one inducedin homotopy by the map of spectra X ^ H F p Ñ X ^ Σ p n ´ H F p induced bythe composite (6). These operations are F p -linearly dual under the isomorphism H i p X ; F p q – hom F p p H i p X ; F p q , F p q , which holds for all spectra X ; see Proposi-tion III.13.5 of [1] or Theorem IV.4.5 of [15]. Lemma A.3.
Let k, E p Q q , | Q | be as in Definition A.1. Let M be a graded E p Q q -module. Then we have an isomorphism of graded E p Q q -modules hom E p Q q p M, E p Q qq – hom k p M, Σ | Q | k q natural in the choice of M .Proof. Let pr : E p Q q Ñ Σ | Q | k denote the morphism of k -vector spaces given bypr p a ` bQ q “ b for all a, b P k , and let g p M q denote the morphism of graded k -vector spaces given by g : hom E p Q q p M, E p Q qq Ñ hom k p M, Σ | Q | k qp g p f qqp a ` bQ q “ pr p f p a ` bQ qq“ aβ ` bα, where f p q “ α ` βQ. We claim that g p M q is actually a morphism of E p Q q -modules, where the E p Q q -action on hom k p M, Σ | Q | k q is by precomposition, i.e., p Qf qp m q “ f p Qm q for f P hom k p M, Σ | Q | k q . Clearly g p M q is k -linear, so we only need to show that g p M q commutes with multiplication by Q , which is easily verified: p g p M qp Qf qq p a ` bQ q “ aα “ p g p M qp f qq p Q p a ` bQ qq , where f p q “ α ` βQ . The function g p M q is natural in M , i.e., g is a naturaltransformation g : hom E p Q q p´ , E p Q qq Ñ hom k p´ , Σ | Q | k q of the functorshom E p Q q p´ , E p Q qq , hom k p´ , Σ | Q | k q : gr Mod p E p Q qq Ñ gr Mod p E p Q qq op . (33)The domain and codomain of g are each coproduct-preserving functors (because ofthe op in (33)), so if we can show that the two claims(1) every object of gr Mod p E p Q qq is a coproduct of indecomposable objects,and(2) g is an isomorphism when evaluated on each indecomposable object ingr Mod p E p Q qq ,are true, then we will know that g is a natural isomorphism.The first claim is true, since every E p Q q -module decomposes as a direct sum ofcyclic E p Q q -modules, e.g. by the Cohen-Kaplansky theorem, [13]. In particular,every object of gr Mod p E p Q qq is isomorphic to a coproduct of suspensions of E p Q q and suspensions of k .We have that p g p Σ n E p Q qqq p f qp a ` bQ q “ αb “ ´ aβ , where f p q “ α ` βQ , so f P ker g p Σ n E p Q qq if and only if α “ “ β , i.e., if and only if f “
0. So g p Σ n E p Q qq is injective. Since g p Σ n E p Q qq is an injective homomorphismbetween finite-dimensional k -vector spaces, it is bijective. The same argumentapplies with Σ n k in place of Σ n E p Q q , so the second claim is also true. So g is anatural isomorphism. (cid:3) OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 29
Proposition A.4.
Let k, E p Q q , | Q | be as in Definition A.1. Then we have isomor-phisms of graded E p Q q -modules: H p M ˚ ; Q q – H p M ; Q q ˚ , (34) Ext s, ´˚ E p Q q p k, M q – Σ ´ s | Q | H p M ; Q q if s ą , (35) Ext s, ´˚ E p Q q p M, k q – Σ ´p s ` q| Q | H ´ hom E p Q q p M, E p Q qq ; Q ¯ if s ą , (36) – Σ ´ s | Q | H p M ; Q q ˚ . (37) natural in the choice of graded E p Q q -module M . (The notation M ˚ is for the graded k -linear dual of M , i.e., M ˚ “ hom k p M, k q .)Proof. We handle each of the isomorphisms (34) through (37) in turn:The Q -Margolis homology of M is the homology of the chain complex ¨ ¨ ¨ Q ÝÑ Σ | Q | M Q ÝÑ M Q ÝÑ Σ ´| Q | M Q ÝÑ . . . (38)and so, since the k -linear dual of the multiplication-by- Q map on a E p Q q -module isthe multiplication-by- Q map on the k -linear dual of that module, the cohomologyof the k -linear dual of the chain complex (38) is H p M ˚ ; Q q . So the classical uni-versal coefficient sequence for chain complexes (e.g. as in 3.6.5 of [33]) yields theisomorphism (34).Applying hom E p Q q p´ , M q to the projective graded E p Q q -module resolution of k Ð E p Q q Q ÐÝ Σ | Q | E p Q q Q ÐÝ Σ | Q | E p Q q Q ÐÝ . . . (39)yields the cochain complex0 Ñ M Q ÝÑ Σ ´| Q | M Q ÝÑ Σ ´ | Q | M Q ÝÑ . . . whose homology is Σ ´ s | Q | H p M ; Q q in each cohomological degree s ą
0. This givesus isomorphism (35). (See Convention A.2 for the sign change in grading degrees.)We take advantage of the fact that E p Q q is self-injective, so that0 Ñ Σ ´| Q | E p Q q Q ÝÑ Σ ´ | Q | E p Q q Q ÝÑ Σ ´ | Q | E p Q q Q ÝÑ . . . (40)is an injective graded E p Q q -module resolution of k . Applying hom E p Q q p M, ´q to(40) yields the cochain complex0 Ñ hom E p Q q p M, Σ ´| Q | E p Q qq Q ÝÑ hom E p Q q p M, Σ ´ | Q | E p Q qq Q ÝÑ . . . , (41)hence isomorphism (36).Isomorphism (37) then follows from the chain of isomorphismsΣ ´p s ` q| Q | H ´ hom E p Q q p M, E p Q qq ; Q ¯ – Σ ´p s ` q| Q | H ´ hom k p M, Σ | Q | k q ; Q ¯ – Σ ´p s ` q| Q | H ´ Σ | Q | M ˚ ; Q ¯ – Σ ´ s | Q | H p M ; Q q ˚ , due to Lemma A.3. (cid:3) To be absolutely clear: it is not a typo that (40) has Σ ´| Q | E p Q q , not E p Q q , in cohomologicaldegree 0. This is because the kernel of multiplication by Q on E p Q q is Σ | Q | k , so we need todesuspend to get an injective resolution of k and not Σ | Q | k . A.2.
The relationships between Margolis homology and Morava K -theory. Recall that, for each prime number p and each positive integer n , we have the ringspectrum K p n q , the p -primary height n Morava K -theory spectrum, whose ring ofhomotopy groups is given by π ˚ p K p n qq – F p r v ˘ n s with | v n | “ p p n ´ q . Theconnective cover of K p n q , written k p n q and called p -primary connective height n Morava K -theory , is also a ring spectrum, and of course has its ring of coefficientsgiven by π ˚ p k p n qq – F p r v n s . The ring spectra K p n q and k p n q each depend on achoice of prime number p , but the prime number p is traditionally suppressed fromthe notations for K p n q and k p n q . Proposition A.5.
The mod p cohomology of the connective p -primary height n Morava K -theory is given, as a graded module over the Steenrod algebra A , by H ˚ p k p n q ; F p q – A b E p Q n q F p , where Q n is the n th Milnor primitive in A . Meanwhile, H ˚ p K p n q ; F p q – . Proof.
See [7]. (cid:3)
Theorem A.6, like everything else in this appendix, is basically well-known; forexample, that X finite and H p H ˚ p X ; F p q ; Q n q is trivial implies that K p n q ^ X iscontractible appears as Corollary 4.9 in [18]. We do not know any written reference,however, which is stated or worked out in the level of generality of Theorem A.6. Theorem A.6.
Let p be a prime number and let n be a positive integer. Let X be aspectrum with finite-type mod p homology groups, and such that H p H ˚ p X ; F p q ; Q n q vanishes. ‚ Then the homotopy colimit of the sequence ¨ ¨ ¨ v n ÝÑ ´ k p n q ^ Σ p n ´ X ¯ ˆ H F p v n ÝÑ p k p n q ^ X q ˆ H F p v n ÝÑ . . . (42) is contractible. ‚ If X is furthermore assumed to be bounded below, then K p n q ^ X is alsocontractible, and the mapping spectrum F p X, K p n qq is also contractible.Proof. Consider the spectral sequence E s,t – Ext s,t gr Comod p A ˚ q p F p , H ˚ p k p n q ^ X ; F p qq , (43) ñ π t ´ s ´ p k p n q ^ X q ˆ H F p ¯ d r : E s,tr Ñ E s ` r,t ` r ´ r . We have the change-of-rings isomorphismExt s,t gr Comod p A ˚ q p F p , H ˚ p k p n q ^ X ; F p qq– Ext s,t gr Comod p A ˚ q ` F p , E p Q n q ˚ l A ˚ F p b F p H ˚ p X ; F p q ˘ – Ext s,t gr Comod p E p Q n q ˚ q p F p , H ˚ p X ; F p qq– Ext s,t gr Mod p E p Q n qq p H ˚ p X ; F p q , F p q so isomorphism (37) in Proposition A.4 gives us that, if H p H ˚ p X ; F p q ; Q q vanishes,then the E -page of spectral sequence (43) collapses on to the s “ ¨ ¨ ¨ v n ÝÑ π ˚ ´ p k p n q ^ Σ p n ´ X q ˆ H F p ¯ v n ÝÑ π ˚ ´ p k p n q ^ X q ˆ H F p ¯ v n ÝÑ . . . (44) OMMUTING UNBOUNDED HOMOTOPY LIMITS WITH MORAVA K-THEORY 31 is zero. Compactness of the sphere spectrum gives us that (44) computes thehomotopy groups of the homotopy colimit of (42).If X is assumed to be bounded below, then the nilpotent completion map k p n q ^ X Ñ p k p n q ^ X q ˆ H F p is a weak equivalence, since k p n q is a ring spectrumand so π ˚ p k p n q ^ X q is a π p k p n qq -module, i.e., an F p -module, hence π ˚ p k p n q ^ X q is p -adically complete, hence k p n q ^ X is H F p -nilpotently complete by Theorem6.6 of [10]. So the homotopy colimit of (44) is weakly equivalent to the telescopeof v n on k p n q ^ X , i.e., K p n q ^ X , since smashing with X commutes with ho-motopy colimits. Finally, since K p n q is a field spectrum, we have K p n q ˚ p X q – hom K p n q ˚ p K p n q ˚ p X q , K p n q ˚ q (a nice general way to prove this duality isomor-phism is by using the universal coefficient theorem IV.4.5 of [15], but the resultwas certainly known earlier). (cid:3) The boundedness hypothesis in the statement of Theorem A.6 cannot be doneaway with: if X “ K p n q , for example, then neither K p n q ^ X nor F p K p n q , K p n qq are contractible. The sequence (42), of course, still has contractible homotopycolimit in that case since each term in that sequence is contractible.Proposition A.7 is a simple cohomological duality. For clarity, we drop thegradings: Proposition A.7.
Let k, E p Q q , | Q | be as in Definition A.1. For each E p Q q -module M , we have an isomorphism of E p Q q -modules Ext sE p Q q p M, k q –
Ext s, ˚ E p Q q p k, M ˚ q (45) for all integers s . If s ą , then each side of (45) is furthermore isomorphic to Ext s, ˚ E p Q q p k, M q ˚ . Proof.
The s “ k is a closed monoidal product on the category of E p Q q -modules.Consequently,for the rest of this proof we assume s ą
0, and consequently the hypotheses ofProposition A.4 are fulfilled. Stringing together isomorphisms from PropositionA.4: Ext sE p Q q p M, k q – H p M ; Q q ˚ (46) – H p M ˚ ; Q q– Ext sE p Q q p k, M ˚ q . The right-hand side of (46) is also isomorphic to Ext sE p Q q p k, M q ˚ , by isomorphism(35). (cid:3) References [1] J. F. Adams.
Stable homotopy and generalised homology . Chicago Lectures in Mathematics.University of Chicago Press, Chicago, IL, 1995. Reprint of the 1974 original.[2] Gabe Angelini-Knoll and Andrew Salch. A May-type spectral sequence for higher topologicalHochschild homology.
Algebr. Geom. Topol. , 18(5):2593–2660, 2018.[3] Gabriel Angelini-Knoll and J. D. Quigley. Chromatic complexity of the algebraic k-theory of y p n q . Preprint, 2019.[4] Vigleik Angeltveit and John Rognes. Hopf algebra structure on topological Hochschild ho-mology. Algebr. Geom. Topol. , 5:1223–1290, 2005.[5] Christian Ausoni and John Rognes. Algebraic K -theory of topological K -theory. Acta Math. ,188(1):1–39, 2002. [6] Christian Ausoni and John Rognes. The chromatic red-shift in algebraic K-theory.
Enseigne-ment Math´ematique , 54(2):13–15, 2008.[7] Nils Andreas Baas and Ib Madsen. On the realization of certain modules over the Steenrodalgebra.
Math. Scand. , 31:220–224, 1972.[8] Maria Basterra and Michael A. Mandell. The multiplication on BP.
J. Topol. , 6(2):285–310,2013.[9] J. Michael Boardman. Conditionally convergent spectral sequences. In
Homotopy invariantalgebraic structures (Baltimore, MD, 1998) , volume 239 of
Contemp. Math. , pages 49–84.Amer. Math. Soc., Providence, RI, 1999.[10] A. K. Bousfield. The localization of spectra with respect to homology.
Topology , 18(4):257–281, 1979.[11] R. Bruner. An adams spectral sequence primer, 2009. preprint available at.[12] Robert R. Bruner and John Rognes. Differentials in the homological homotopy fixed pointspectral sequence.
Algebr. Geom. Topol. , 5:653–690, 2005.[13] I. S. Cohen and I. Kaplansky. Rings for which every module is a direct sum of cyclic modules.
Math. Z. , 54:97–101, 1951.[14] Bjørn Ian Dundas, Thomas G. Goodwillie, and Randy McCarthy.
The local structure ofalgebraic K-theory , volume 18 of
Algebra and Applications . Springer-Verlag London, Ltd.,London, 2013.[15] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May.
Rings, modules, and algebrasin stable homotopy theory , volume 47 of
Mathematical Surveys and Monographs . AmericanMathematical Society, Providence, RI, 1997. With an appendix by M. Cole.[16] J. P. C. Greenlees. Representing Tate cohomology of G -spaces. Proc. Edinburgh Math. Soc.(2) , 30(3):435–443, 1987.[17] Michael Hill and Tyler Lawson. Automorphic forms and cohomology theories on Shimuracurves of small discriminant.
Adv. Math. , 225(2):1013–1045, 2010.[18] Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy theory. II.
Ann.of Math. (2) , 148(1):1–49, 1998.[19] Mark Hovey and Neil P. Strickland. Morava K -theories and localisation. Mem. Amer. Math.Soc. , 139(666):viii+100, 1999.[20] Markus Land, Lennart Meier, and Georg Tamme. Vanishing results for chromatic localizationsof algebraic K -theory. arXiv e-prints , page arXiv:2001.10425, Jan 2020.[21] Tyler Lawson. Secondary power operations and the Brown-Peterson spectrum at the prime2. Ann. of Math. (2) , 188(2):513–576, 2018.[22] Tyler Lawson and Niko Naumann. Commutativity conditions for truncated Brown-Petersonspectra of height 2.
J. Topol. , 5(1):137–168, 2012.[23] Sverre Lunø e Nielsen and John Rognes. The topological Singer construction.
Doc. Math. ,17:861–909, 2012.[24] H. R. Margolis.
Spectra and the Steenrod algebra , volume 29 of
North-Holland MathematicalLibrary . North-Holland Publishing Co., Amsterdam, 1983.[25] Akhil Mathew, Niko Naumann, and Justin Noel. Nilpotence and descent in equivariant stablehomotopy theory.
Adv. Math. , 305:994–1084, 2017.[26] C. R. F. Maunder. The spectral sequence of an extraordinary cohomology theory.
Proc.Cambridge Philos. Soc. , 59:567–574, 1963.[27] J. E. McClure and R. E. Staffeldt. On the topological Hochschild homology of b u. I. Amer.J. Math. , 115(1):1–45, 1993.[28] S. A. Mitchell. The Morava K -theory of algebraic K -theory spectra. K -Theory , 3(6):607–626,1990.[29] Thomas Nikolaus and Peter Scholze. On topological cyclic homology. Acta Math. , 221(2):203–409, 2018.[30] Douglas C. Ravenel.
Complex cobordism and stable homotopy groups of spheres , volume 121of
Pure and Applied Mathematics . Academic Press Inc., Orlando, FL, 1986.[31] Andrew Senger. The Brown-Peterson spectrum is not E p p ` q at odd primes. arXiv e-prints ,page arXiv:1710.09822, October 2017.[32] Haggai Tene. A geometric description of the Atiyah-Hirzebruch spectral sequence for B -bordism. M¨unster J. Math. , 10(1):171–188, 2017.[33] Charles A. Weibel.
An introduction to homological algebra , volume 38 of