A computational reduction for many base cases in profinite telescopic algebraic K-theory
aa r X i v : . [ m a t h . A T ] J a n A COMPUTATIONAL REDUCTION FOR MANY BASE CASESIN PROFINITE TELESCOPIC ALGEBRAIC K -THEORY DANIEL G. DAVIS
Abstract.
For primes p ≥ K ( KU p )—the algebraic K -theory spectrum of( KU ) ∧ p , Morava K -theory K (1), and Smith-Toda complex V (1), Ausoni andRognes conjectured (alongside related conjectures) that L K (1) S i −−−→ ( KU ) ∧ p induces a map K ( L K (1) S ) ∧ v − V (1) → K ( KU p ) h Z × p ∧ v − V (1) that isan equivalence. Since the definition of this map is not well understood, weconsider K ( L K (1) S ) ∧ v − V (1) → ( K ( KU p ) ∧ v − V (1)) h Z × p , which is in-duced by i and also should be an equivalence. We show that for any closed G < Z × p , π ∗ (( K ( KU p ) ∧ v − V (1)) hG ) is a direct sum of two pieces givenby (co)invariants and a coinduced module, for K ( KU p ) ∗ ( V (1))[ v − ]. When G = Z × p , the direct sum is, conjecturally, K ( L K (1) S ) ∗ ( V (1))[ v − ] and, byusing K ( L p ) ∗ ( V (1))[ v − ], where L p = (( KU ) ∧ p ) h Z / (( p − Z ) , the summandssimplify. The Ausoni-Rognes conjecture suggests that in( − ) h Z × p ∧ v − V (1) ≃ ( K ( KU p ) ∧ v − V (1)) h Z × p ,K ( KU p ) fills in the blank; we show that for any G , the blank can be filled by( K ( KU p )) dis O , a discrete Z × p -spectrum built out of K ( KU p ). Introduction
Motivation for our work.
Let p be any prime, with Z p the p -adic integers,let K (1) denote the first Morava K -theory spectrum, and let L K (1) ( S ) be theBousfield localization of the sphere spectrum. Also, let KU p be p -complete complex K -theory, so that π ∗ ( KU p ) = Z p [ u ± ] , where π ( KU p ) = Z p and | u | = 2, and let Z × p denote the group of units in Z p . By[16, 17], Z × p – as the group of p -adic Adams operations – acts on the commutative S -algebra KU p by maps of commutative S -algebras. Given a commutative S -algebra A , the algebraic K -theory spectrum of A , K ( A ), is a commutative S -algebra, so that K ( KU p ) is a commutative S -algebra, and by the functoriality of K ( − ), Z × p acts on K ( KU p ) by maps of commutative S -algebras.For the rest of this paper, we let p ≥
5. Let V (1) be the type 2 Smith-Todacomplex S / ( p, v ). Then there is a v -self-map v : Σ d V (1) → V (1), where d issome positive integer (see [20, Theorem 9]), and hence, v induces a sequence V (1) → Σ − d V (1) → Σ − d V (1) → · · · of maps of spectra, and we set v − V (1) = colim j ≥ Σ − jd V (1) , he mapping telescope associated to v . In [4, paragraph containing (0.1)], [5, Con-jecture 4.2], and [3, page 46; Remark 10.8], Christian Ausoni and John Rognesconjectured that the K (1)-local unit map i : L K (1) ( S ) → KU p induces a weak equivalence(1.1) K ( L K (1) ( S )) ∧ v − V (1) → K ( KU p ) h Z × p ∧ v − V (1) , where K ( KU p ) h Z × p = ( K ( KU p )) h Z × p is a continuous homotopy fixed point spectrum that is formed with respect to acontinuous action of the profinite group Z × p on K ( KU p ). Remark 1.2.
The above conjecture is a collection of n = 1 instances of a moregeneral conjecture made by Ausoni and Rognes for every positive integer and everyprime (for more information, see the references mentioned above).One difficulty with making progress on this conjecture is that there is no pub-lished construction of K ( KU p ) h Z × p and, according to [13, Remark 1.5], the onlymodels for it, currently, are a “candidate definition” that uses condensed spectra(in the sense of Clausen-Scholze) in the setting of ∞ -categories (the author learnedof this construction from Jacob Lurie) and, possibly, a pyknotic version of this con-struction (in the framework of [7]). Thus, due to the lack of a robust model for themap in (1.1), the conjecture is difficult to approach computationally.If G is any profinite group and X is a discrete G -spectrum (as in [8]; the cruxof this concept is that for every k , l ≥
0, the set of l -simplices of the pointedsimplicial set X k is a discrete G -set), then there is a continuous homotopy fixedpoint spectrum X hG [8, Section 3.1] (and we use this notation for the rest of thispaper). Thus, to address the above difficulty, the author showed in [13, Section 1.2]that K ( KU p ) ∧ v − V (1), with v − V (1) equipped with the trivial Z × p -action, canbe realized as a discrete Z × p -spectrum – written as C dis p in [13], and hence, one canform ( K ( KU p ) ∧ v − V (1)) h Z × p := ( C dis p ) h Z × p and, by [13, Theorem 1.8], the map i induces a canonical map i ′ : K ( L K (1) ( S )) ∧ v − V (1) → ( K ( KU p ) ∧ v − V (1)) h Z × p . Remark 1.3.
According to [13, Remark 1.5], the relationship between the targetof i ′ and K ∧ v − V (1), where K denotes the aforementioned candidate model for K ( KU p ) h Z × p , is unclear.Now we make some observations to understand the relationship between the map i ′ and the conjectural equivalence in (1.1). If X is a discrete Z × p -spectrum and Y is a finite spectrum with trivial Z × p -action, then X ∧ Y is a discrete Z × p -spectrumand, by [14, Remark 7.16],(1.4) ( X ∧ Y ) h Z × p ≃ X h Z × p ∧ Y. More generally, if { X i } i ∈ I is a diagram of discrete Z × p -spectra indexed by a cofilteredcategory I , then the equivalence(holim i X i ) ∧ Y ≃ holim i ( X i ∧ Y ) mplies that it is natural to make the definition((holim i X i ) ∧ Y ) h Z × p := (holim i ( X i ∧ Y )) h Z × p = holim i ( X i ∧ Y ) h Z × p , where the last step applies [8, Section 4.4], and thus, we have(1.5) ((holim i X i ) ∧ Y ) h Z × p ≃ (holim i X i ) h Z × p ∧ Y, becauseholim i ( X i ∧ Y ) h Z × p ≃ holim i (( X i ) h Z × p ∧ Y ) ≃ (holim i ( X i ) h Z × p ) ∧ Y = (holim i X i ) h Z × p ∧ Y. Also, by [13], for each j ≥ K ( KU p ) ∧ Σ − jd V (1) can be realized as a discrete Z × p -spectrum, and hence, there is ( K ( KU p ) ∧ Σ − jd V (1)) h Z × p . Then since each Σ − jd V (1)is a finite spectrum with trivial Z × p -action, the pattern in (1.4) and (1.5) suggeststhat there should be an equivalence(1.6) K ( KU p ) h Z × p ∧ Σ − jd V (1) ? ≃ ( K ( KU p ) ∧ Σ − jd V (1)) h Z × p . Here and elsewhere, we place a “?” over a relation to indicate that it is not knownto be true, but it is desired and expected to some degree.Now notice that there is the isomorphism(1.7) K ( KU p ) h Z × p ∧ v − V (1) ∼ = colim j ≥ ( K ( KU p ) h Z × p ∧ Σ − jd V (1))and, by [13, Theorem 1.7], there is an equivalence(1.8) ( K ( KU p ) ∧ v − V (1)) h Z × p ≃ colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) h Z × p . Thus, (1.6)–(1.8) imply that there should be an equivalence(1.9) K ( KU p ) h Z × p ∧ v − V (1) ? ≃ ( K ( KU p ) ∧ v − V (1)) h Z × p , and this observation suggests that if (1.9) holds and i ′ is a weak equivalence, thenone should be able to prove that the map in (1.1) is a weak equivalence, and therebyverify the conjecture of Ausoni and Rognes. This potentially fruitful strategy forproving this conjecture involves computing π ∗ (cid:0) ( K ( KU p ) ∧ v − V (1)) h Z × p (cid:1) , and thus, in this paper, we make progress on this computation by showing thatit is a direct sum of two pieces given by invariants and coinvariants involving the Z × p -action on π ∗ ( K ( KU p ) ∧ v − V (1)). Additionally, with L p := ( KU p ) h Z / (( p − Z ) (as in [4]), the p -complete Adams summand and a commutative S -algebra, where Z / (( p − Z ) is the usual subgroup of Z × p , we show that the direct sum can beexpressed as invariants and coinvariants of the Z p -action on π ∗ ( K ( L p ) ∧ v − V (1)).Given a profinite group G and a discrete G -spectrum X , if H is any closedsubgroup of G , then H is a profinite group, X is a discrete H -spectrum (by re-striction of the G -action), and hence, there is the continuous homotopy fixed pointspectrum X hH . Our work for the above computation is in line with this multiplic-ity of possibilities: our result is not just for the Z × p -homotopy fixed points, but is or the homotopy fixed points of any closed subgroup (though the aforementionedpresentation involving L p is only for the case G = H = Z × p ).1.2. The main results.
In (1.9) above, we said that there should be an equivalence( − ) h Z × p ∧ v − V (1) ? ≃ ( K ( KU p ) ∧ v − V (1)) h Z × p , where the blank “ − ” can be filled in with K ( KU p ). One of our intermediate stepsin obtaining the results mentioned above is to give a way to fill in this blank witha discrete Z × p -spectrum that is related to K ( KU p ). Theorem 1.10.
Let p ≥ and let G be a closed subgroup of Z × p . There is adiscrete Z × p -spectrum ( K ( KU p )) dis O with the property that for each j ≥ , there isan equivalence (cid:0) ( K ( KU p )) dis O (cid:1) hG ∧ Σ − jd V (1) ≃ ( K ( KU p ) ∧ Σ − jd V (1)) hG , and (cid:0) ( K ( KU p )) dis O (cid:1) hG ∧ v − V (1) ≃ ( K ( KU p ) ∧ v − V (1)) hG . The spectrum ( K ( KU p )) dis O is defined in Definition 3.2, with O specified at thebeginning of Section 4, the first equivalence in Theorem 1.10 is Theorem 4.2, andthe last equivalence follows immediately from the first one and the general versionof (1.8) that is stated in (2.2).After the following prefatory remarks, we state our result that for any closedsubgroup G in Z × p , π ∗ (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) can be reduced to a direct sum.Recall that Z p is the pro- p completion of Z and Z can be regarded as a subsetof Z p in a way that makes the inclusion Z ֒ → Z p a ring homomorphism. We define C p − := Z / (( p − Z )to be the cyclic group of order p − Z × p ∼ = Z p × C p − (since p ≥ M is a Z [ Z p ]-module (so that Z p acts on M ), then M is naturallya Z [ Z ]-module, and M Z denotes the coinvariants. If K is a closed subgroup of aprofinite group H and A is a discrete K -module, we let Coind HK ( A ) denote thecoinduced discrete H -module of continuous K -equivariant functions H → A .Let P ( v ) = F p [ v ] denote the polynomial algebra over F p generated by theperiodic element v ∈ π p − ( V (1)). Also, P ( v ± ) = F p [ v , v − ] is the algebra ofLaurent polynomials on v . To help manage the typography in the upcoming text,for a closed subgroup G of Z × p , we let KV ( p, G ) := (cid:0) ( K ( KU p )) dis O (cid:1) hG ∧ v − V (1)and E ( p, G ) ∗ := H c ( G, π ∗ ( K ( KU p ) ∧ v − V (1))) , a graded continuous cohomology group with coefficients in the stated discrete G -module. Theorem 1.11.
Let p ≥ and let G be any closed subgroup of Z × p . There is anisomorphism π ∗ (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) ∼ = π ∗ ( KV ( p, G )) , here the right-hand side is the middle term in a short exact sequence → E ( p, G ) ∗ +1 → π ∗ ( KV ( p, G )) → (cid:0) π ∗ ( K ( KU p ) ∧ v − V (1)) (cid:1) G → of P ( v ± ) -modules. In particular, in each degree t , where t ∈ Z , this sequence is asplit exact sequence of F p -modules and there is an isomorphism π t (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) ∼ = (cid:0)(cid:0) Coind Z × p G ( π t +1 ( K ( KU p ) ∧ v − V (1))) (cid:1) C p − (cid:1) Z ⊕ (cid:0) π t ( K ( KU p ) ∧ v − V (1)) (cid:1) G of abelian groups, where in the direct sum, the left summand is isomorphic to E ( p, G ) t +1 . The proof of this result is broken up into six steps: • in Section 2, we use various homotopy fixed point spectral sequences topresent π ∗ (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) as the middle term in a colimit ofshort exact sequences; • Section 3 makes some recollections of several constructions that are neededto go further; • for each j ≥
0, ( K ( KU p ) ∧ Σ − jd V (1)) hG is the continuous homotopy fixedpoints of, not literally, K ( KU p ) ∧ Σ − jd V (1), but a discrete Z × p -spectrumequivalent to this Z × p -spectrum, and in Section 4, we study the role of V (1)in the construction of this discrete Z × p -spectrum and its associated ho-motopy fixed point spectral sequence (and thereby prove Theorem 1.10, ofwhich the first isomorphism in Theorem 1.11 is an immediate consequence); • Section 5 shows that each of the just-mentioned spectral sequences is iso-morphic to a spectral sequence in the category of P ( v )-modules; • in Section 6, we obtain the desired short exact sequence of P ( v ± )-modulesand the chief desideratum is shown to be a direct sum with its secondsummand as specified in Theorem 1.11; and • we obtain the isomorphism between E ( p, G ) t +1 and the expression involving Z -coinvariants of C p − -invariants (in every integral degree t ) in Section 7. Remark 1.12.
In Lemma 7.1, we show that in Theorem 1.11, for each integer t , E ( p, G ) t +1 ∼ = Z p ⊗ Z p [[ Z p ]] (cid:0)(cid:0) Coind Z × p G ( π t +1 ( K ( KU p ) ∧ v − V (1))) (cid:1) C p − (cid:1) , where Z p is regarded as a Z p [[ Z p ]]-module by giving Z p the trivial Z p -group action.We give this result in case this form of E ( p, G ) t +1 is easier to compute than the Z -coinvariants of Theorem 1.11. We point out that “ ⊗ Z p [[ Z p ]] ” above denotes the usualtensor product (for the category of abstract Z p [[ Z p ]]-modules) and not a completedtensor product (formed in some category of topological Z p [[ Z p ]]-modules).Now we focus on the case G = Z × p : our result in this case – Corollary 1.13 below– consists of three isomorphisms, and the first one is an immediate consequenceof Theorem 1.11. As alluded to earlier, the last two isomorphisms involve K ( L p ),and so we note that π ∗ ( L p ) = Z p [ v ± ] and L p ≃ E (1) p , the p -completed firstJohnson-Wilson spectrum. Also, we make explicit the following, which was implic-itly referred to earlier: after taking C p − -homotopy fixed points to form L p , thereis a residual action by Z p on L p through morphisms of commutative S -algebras,and hence, K ( L p ) carries a Z p -action. The telescope v − V (1) is given the trivial Z p -action, and K ( L p ) ∧ v − V (1) is equipped with the diagonal Z p -action. orollary 1.13. Let p ≥ . There are isomorphisms π ∗ (cid:0) ( K ( KU p ) ∧ v − V (1)) h Z × p (cid:1) ∼ = (cid:0)(cid:0) π ∗ +1 ( K ( KU p ) ∧ v − V (1)) (cid:1) C p − (cid:1) Z ⊕ (cid:0) π ∗ ( K ( KU p ) ∧ v − V (1)) (cid:1) Z × p ∼ = (cid:0) π ∗ +1 ( K ( L p ) ∧ v − V (1)) (cid:1) Z ⊕ (cid:0) K ( L p ) ∗ ( V (1))[ v − ] (cid:1) Z p ∼ = (cid:0) Z p ⊗ Z p [[ Z p ]] (cid:0) π ∗ +1 ( K ( L p ) ∧ v − V (1)) (cid:1)(cid:1) ⊕ (cid:0) K ( L p ) ∗ ( V (1))[ v − ] (cid:1) Z p . Remark 1.14.
As discussed in more detail in Section 1.3, the Ausoni-Rognesconjecture suggests that for p ≥
5, the direct sum in Corollary 1.13 – expressed inthree different, but isomorphic, ways – is a conjectural description of π ∗ ( K ( L K (1) ( S )) ∧ v − V (1)) ∼ = K ( L K (1) ( S )) ∗ ( V (1))[ v − ] , and it seems that it would be helpful to have a more explicit form of this directsum. We note that [1, page 4; Theorem 1.5] describes a strategy for computing π ∗ ( K ( L K (1) ( S )) ∧ V (1)) and gives a result that begins making progress on thisstrategy.The second isomorphism of Corollary 1.13 comes from the first one and the factthat there is a Z p -equivariant isomorphism( π ∗ ( K ( KU p ) ∧ v − V (1))) C p − ∼ = K ( L p ) ∗ ( V (1))[ v − ] , which is deduced in Section 8 from the fact that K ( L p ) and K ( KU p ) hC p − areequivalent after p -completion (for this equivalence, see [25, the sentence above Re-mark 4.4]; a proof is in [3, pages 11-12]). The third isomorphism in the corollary isan application of Remark 1.12.1.3. Considerations for the future, terminology, and notation.
In our dis-cussion of (1.9), we saw that proving that i ′ is a weak equivalence would be asubstantial step towards verifying the Ausoni-Rognes conjecture (more precisely,the instances described earlier of this general conjecture), and by Corollary 1.13,this step can be done by showing that i ′ induces an isomorphism K ( L K (1) ( S )) ∗ ( V (1))[ v − ] ? ∼ = (cid:0)(cid:0) π ∗ +1 ( K ( KU p ) ∧ v − V (1)) (cid:1) C p − (cid:1) Z ⊕ (cid:0) π ∗ ( K ( KU p ) ∧ v − V (1)) (cid:1) Z × p . In [2, Theorem 8.3], under the assumption of two hypotheses, there is a descriptionof the graded abelian group π ∗ ( K ( KU p ) ∧ V (1)) as a certain type of module (see[ibid.] for the details), and progress in verifying this description was made by [10,page 2; Theorem 4.5].Also by Corollary 1.13, we see that another and perhaps easier way to take theaforementioned step is to prove that i ′ induces an isomorphism K ( L K (1) ( S )) ∗ ( V (1))[ v − ] ? ∼ = (cid:0) π ∗ +1 ( K ( L p ) ∧ v − V (1)) (cid:1) Z ⊕ (cid:0) K ( L p ) ∗ ( V (1))[ v − ] (cid:1) Z p . We make a comment related to computing more explicitly the right-hand side ofthis conjectural isomorphism. By [9] (as conjectured in [4, page 5]), there is alocalization cofiber sequence K ( Z p ) → K ( ℓ p ) → K ( L p ) → Σ K ( Z p ) , here ℓ p is the p -complete connective Adams summand, with π ∗ ( ℓ p ) = Z p [ v ].Thus, there is the cofiber sequence K ( Z p ) ∧ V (1) → K ( ℓ p ) ∧ V (1) → K ( L p ) ∧ V (1) → Σ K ( Z p ) ∧ V (1) , and as stated in [25, page 1267], from explicit computations of K ( Z p ) ∗ ( V (1))(known by [11]; see also [2, page 664]) and K ( ℓ p ) ∗ ( V (1)) [4, Theorem 9.1], thelong exact sequence for this cofiber sequence yields calculations of K ( L p ) ∗ ( V (1)),and some information about this is in [25, Example 5.3].The author did not push the computation of the last-mentioned “right-hand side”further and one reason is a lack of knowledge about the Z p -action on K ( L p ) ∗ ( V (1)).In this vein, we note that [4, Remark 1.4] mentions a gap in understanding of howa certain Adams operation on K ( ℓ p ) acts on a particular class in K p − ( ℓ p ) (werefer the reader to [ibid.] for the details).In this paper, we always work in the category Sp Σ of symmetric spectra ofsimplicial sets, so that “spectrum” always means symmetric spectrum (except fora few places in this introduction, where the context makes the meaning clear). Welet ( − ) f : Sp Σ → Sp Σ , Z Z f denote a fibrant replacement functor, so that given the spectrum Z , there is anatural map Z → Z f that is a trivial cofibration, with Z f fibrant. If K is any groupand X is a K -spectrum, then X f is also a K -spectrum and the trivial cofibration X → X f is K -equivariant.Given a spectrum Z and an integer t , π t ( Z ) denotes [ S t , Z ], the set of morphisms S t → Z in the homotopy category of symmetric spectra, where here, S t denotes afixed cofibrant and fibrant model for the t -th suspension of the sphere spectrum.Outside of this introduction, “holim” denotes the homotopy limit for Sp Σ , as definedin [19, Definition 18.1.8]. If Z • is a cosimplicial spectrum that is objectwise fibrant,then by “the homotopy spectral sequence for holim ∆ Z • ,” we mean the conditionallyconvergent spectral sequence E s,t = H s [ π t ( Z ∗ )] = ⇒ π t − s (holim ∆ Z • ) , where π t ( Z ∗ ) is the usual cochain complex associated to the cosimplicial abeliangroup π t ( Z • ).2. Step i: a reduction to a colimit of short exact sequences
Let G be any closed subgroup of Z × p . If M is a discrete G -module, then we let H ∗ c ( G, M ) denote the continuous cohomology groups of G with coefficients in M .By [13, Theorem 1.7], there is a strongly convergent homotopy spectral sequence { E ∗ , ∗ r } r ≥ = { E ∗ , ∗ r } that has the form E s,t = H sc ( G, π t ( K ( KU p ) ∧ V (1))[ v − ]) = ⇒ π t − s (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) , with E s,t = 0, for all s ≥ t ∈ Z . Since the E -page has only two nontrivialcolumns, there is a short exact sequence(2.1) 0 → E ,t +12 → π t (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) → E ,t → , for each t ∈ Z .By [13, Theorem 1.7], there is an equivalence of spectra(2.2) ( K ( KU p ) ∧ v − V (1)) hG ≃ colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) hG . his result, coupled with the fact that H ∗ c ( G, − ) commutes with colimits of discrete G -modules indexed by directed posets, implies that for every t ∈ Z , the threenontrivial terms in (2.1) satisfy the following: E ,t +12 ∼ = colim j ≥ H c ( G, π t +1 ( K ( KU p ) ∧ Σ − jd V (1))) ,π t (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) ∼ = colim j ≥ π t (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) hG (cid:1) ,E ,t ∼ = colim j ≥ (cid:0) π t ( K ( KU p ) ∧ Σ − jd V (1)) (cid:1) G . Also, for each j ≥
0, by [13, Remark 1.20, Theorem 7.6, (8.3)], there is a stronglyconvergent homotopy spectral sequence { j E ∗ , ∗ r } having the form j E s,t = H sc ( G, π t ( K ( KU p ) ∧ Σ − jd V (1))) = ⇒ π t − s (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) hG (cid:1) , with j E s,t = 0, for all s ≥ t ∈ Z , so that there is a short exact sequence(2.3) 0 → j E ,t +12 → π t (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) hG (cid:1) → j E ,t → , where t ∈ Z .The above facts allow us to conclude that spectral sequence { E ∗ , ∗ r } is the colimitover { j ≥ } of the spectral sequences { j E ∗ , ∗ r } , and hence, the short exact sequencein (2.1) is the colimit over { j ≥ } of the short exact sequences in (2.3). Moreexplicitly, there is a commutative diagram0 / / E , ∗ +12 / / π ∗ (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) / / E , ∗ / / / / colim j ≥ j E , ∗ +12 / / ∼ = O O colim j ≥ π ∗ (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) hG (cid:1) / / ∼ = O O colim j ≥ j E , ∗ / / ∼ = O O Step ii: a recollection of various constructions with spectra
To go further, we need to better understand spectral sequence { j E ∗ , ∗ r } , for each j ≥
0, and to do this, we need to recall several constructions. In this section, H isan arbitrary profinite group.Given a spectrum Z , let Sets( H, Z ) be the H -spectrum whose k th pointed sim-plicial set Sets( H, Z ) k has l -simplices Sets( H, Z ) k,l equal to the H -set Sets( H, Z k,l )of all functions H → Z k,l , for each k, l ≥
0, where the H -action on Sets( H, Z k,l ) isdefined by(3.1) ( h · f )( h ′ ) = f ( h ′ h ) , f ∈ Sets(
H, Z k,l ) , h, h ′ ∈ H. As explained in [13, Section 2], given any H -spectrum X , there is a cosimplicial H -spectrum Sets( H • +1 , X ), where for each n ≥
0, the spectrum of n -cosimplicesof Sets( H • +1 , X ) is obtained by applying Sets( H, − ) iteratively n + 1 times to X . Definition 3.2.
Let X be an H -spectrum and let O = { N λ } λ ∈ Λ be an inversesystem of open normal subgroups of H ordered by inclusion, over a directed posetΛ. Following [13, Definition 4.4], X dis O := colim λ ∈ Λ holim ∆ Sets( H • +1 , X f ) N λ , here the colimit is formed in spectra (this definition is slightly more general thanthat of “ X dis N ” in [ibid.]: N satisfies several hypotheses that we do not require from O ). Each spectrum holim ∆ Sets( H • +1 , X f ) N λ is an H/N λ -spectrum, and hence, adiscrete H -spectrum, via the canonical projection H → H/N λ , so that X dis O is adiscrete H -spectrum. Also, X dis O is a fibrant spectrum (this follows from [13, stepstaken between (4.12) and (4.13)] and the fact that a homotopy limit of fibrantspectra is again fibrant).Let O be as in Definition 3.2. By [13, Lemma 4.7, proof of Theorem 4.9], forany H -spectrum X , there is a zigzag X ≃ −→ i X holim ∆ Sets( H • +1 , X f ) φ X ←−− X dis O of H -equivariant maps, where i X is a weak equivalence of spectra and φ X is inducedby the inclusions Sets( H • +1 , X f ) N λ → Sets( H • +1 , X f ).Now suppose that X is a discrete H -spectrum. As in [8, Sections 2.4, 3.2], there isa cosimplicial spectrum Γ • H X , where for each n ≥
0, the spectrum of n -cosimplicesof Γ • H X satisfies the isomorphism(Γ • H X ) n ∼ = colim U ⊳ o H n Y H n /U X, where H n is the n -fold cartesian product of copies of H ( H is the trivial group { e } ) and the colimit is over all the open normal subgroups of H n . By [8, Theorem3.2.1] and [14, page 330, Remark 7.5], if H = G , a closed subgroup of Z × p , then(3.3) X hG ≃ holim ∆ Γ • G X fib , where X fib is any discrete G -spectrum that is fibrant as a spectrum and is equippedwith a G -equivariant map X ≃ −→ X fib that is a weak equivalence of spectra.4. Step iii: the role of V (1) in the spectral sequences { j E ∗ , ∗ r } Now we focus on understanding the part played by V (1) in spectral sequence { j E ∗ , ∗ r } , where j ≥ G is any closed subgroup of Z × p ). Let O = { p m Z p } m ≥ , where each p m Z p is the open normal subgroup of Z × p that corresponds to( p m Z p ) × { e } ⊳ o Z p × C p − . In the introduction, we noted that K ( KU p ) ∧ v − V (1) is realized by the discrete Z × p -spectrum C dis p , which we can now define: C dis p := colim j ≥ (cid:0) (( K ( KU p ) ∧ Σ − jd V (1)) f ) dis O (cid:1) . By [13, Remark 1.20, (8.1)], spectral sequence { E ∗ , ∗ r } is the homotopy spectralsequence for holim ∆ Γ • G C dis p . In Section 2, we noted that there is the isomorphism { E ∗ , ∗ r } ∼ = colim j ≥ { j E ∗ , ∗ r } of spectral sequences; for each j , { j E ∗ , ∗ r } is the homotopy spectral sequence forholim ∆ Γ • G (cid:0) (( K ( KU p ) ∧ Σ − jd V (1)) f ) dis O (cid:1) . ix any j ≥
0. To increase readability and when the additional intuition carriedby the original notation is not needed, we will sometimes use the abbreviation K j := K ( KU p ) ∧ Σ − jd V (1) . Since the fibrant replacement morphism K j → ( K j ) f is a weak equivalence ofspectra that is Z × p -equivariant, the induced map ( K j ) dis O → (( K j ) f ) dis O is a weakequivalence that is Z × p -equivariant, by [13, Remark 1.20, paragraph after (8.4)]. If X is a discrete G -spectrum, then for each n ≥
0, the spectrum of n -cosimplicesof Γ • G X is obtained by applying iteratively n times to X a functor that preservesweak equivalences of spectra, by [8, Lemma 2.4.1]. Thus, the induced morphismΓ • G ( K j ) dis O ≃ −→ Γ • G (cid:0) (( K j ) f ) dis O (cid:1) is an objectwise weak equivalence of cosimplicial spectra, so spectral sequence { j E ∗ , ∗ r } is isomorphic to the homotopy spectral sequence forholim ∆ Γ • G ( K j ) dis O . Hence, we shift our focus to this latter spectral sequence.For each n ≥
0, the spectrum of n -cosimplices of Γ • G ( K j ) dis O satisfies (cid:0) Γ • G ( K j ) dis O (cid:1) n ∼ = colim U ⊳ o G n Y G n /U colim m ≥ holim ∆ Sets(( Z × p ) • +1 , ( K j ) f ) p m Z p . Now choose any m ≥
0. Again at the level of n -cosimplices, we have (cid:0) Sets(( Z × p ) • +1 , ( K j ) f ) p m Z p (cid:1) n ∼ = Y Z × p / ( p m Z p ) Y ( Z × p ) n ( K ( KU p ) ∧ Σ − jd V (1)) f ≃ (cid:0) Sets(( Z × p ) • +1 , ( K ( KU p )) f ) p m Z p (cid:1) n ∧ Σ − jd V (1) , where the isomorphism is as in [13, proof of Lemma 2.1] and the second stepapplies the fact that smashing with a finite spectrum commutes with any product.If Z • : ∆ → Sp Σ is a cosimplicial spectrum and Z ′ is any spectrum, then there isthe functor ( − ) ∧ Z ′ : Sp Σ → Sp Σ , Y Y ∧ Z ′ , and we let Z • ∧ Z ′ denote the cosimplicial spectrum (( − ) ∧ Z ′ ) ◦ Z • . Then we haveholim ∆ Sets(( Z × p ) • +1 , ( K j ) f ) p m Z p ≃ holim ∆ (Sets(( Z × p ) • +1 , ( K ( KU p )) f ) p m Z p ∧ Σ − jd V (1)) f ≃ (cid:0) holim ∆ Sets(( Z × p ) • +1 , ( K ( KU p )) f ) p m Z p (cid:1) ∧ Σ − jd V (1) , where the last step is because Σ − jd V (1) is a finite spectrum.Our last conclusion implies that for each n ≥
0, we have (cid:0) Γ • G ( K j ) dis O (cid:1) n ≃ colim U ⊳ o G n Y G n /U colim m ≥ (cid:0)(cid:0) holim ∆ Sets(( Z × p ) • +1 , ( K ( KU p )) f ) p m Z p (cid:1) ∧ Σ − jd V (1) (cid:1) f ≃ (cid:0) colim U ⊳ o G n Y G n /U colim m ≥ holim ∆ Sets(( Z × p ) • +1 , ( K ( KU p )) f ) p m Z p (cid:1) ∧ Σ − jd V (1) ∼ = (cid:0) Γ • G ( K ( KU p )) dis O (cid:1) n ∧ Σ − jd V (1) , here the second step uses that the smash product commutes with colimits andfinite products (which are weakly equivalent to finite coproducts). This showsthat there is a zigzag of objectwise weak equivalences between the following twocosimplicial spectra:(4.1) Γ • G (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis O (cid:1) ≃ (cid:0)(cid:0) Γ • G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f . If Z • is a cosimplicial spectrum that is objectwise fibrant, we let hss ( Z • ) denotethe associated homotopy spectral sequence. We have shown that there are isomor-phisms { j E ∗ , ∗ r } ∼ = hss (cid:0) Γ • G (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis O (cid:1)(cid:1) ∼ = hss (cid:0)(cid:0)(cid:0) Γ • G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f (cid:1) of spectral sequences; the first isomorphism was obtained earlier in this section andthe second one is by (4.1), which also yields the following result. Theorem 4.2.
Let p ≥ . If G is a closed subgroup of Z × p and j ≥ , then ( K ( KU p ) ∧ Σ − jd V (1)) hG ≃ (( K ( KU p )) dis O ) hG ∧ Σ − jd V (1) . Proof.
We have( K ( KU p ) ∧ Σ − jd V (1)) hG := (( K ( KU p ) ∧ Σ − jd V (1)) dis O ) hG ≃ holim ∆ Γ • G (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis O (cid:1) ≃ holim ∆ (cid:0)(cid:0) Γ • G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f ≃ (cid:0) holim ∆ Γ • G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) , where each step is justified by [13, end of Section 1.2], (3.3), (4.1), and the factthat Σ − jd V (1) is a finite spectrum, respectively, and the last expression above isequivalent to the right-hand side in the desired result (again, by (3.3)). (cid:3) Step iv: each spectral sequence is one of P ( v ) -modules In this section, j ≥ G is any closed subgroup of Z × p .Since p ≥ V (1) is a homotopy commutative and homotopy associative ringspectrum. Then by Theorem 4.2, π ∗ (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) hG (cid:1) ∼ = π ∗ ((( K ( KU p )) dis O ) hG ∧ Σ − jd V (1))is a right π ∗ ( V (1))-module, and hence, it is a P ( v )-module. This observationsuggests that spectral sequence HS j := hss (cid:0)(cid:0)(cid:0) Γ • G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f (cid:1) is one of P ( v )-modules, and now we show that this is the case.If Z • is a cosimplicial spectrum, let Q ∗ Z • be its cosimplicial replacement. Also,let C • := Γ • G ( K ( KU p )) dis O , so thatholim ∆ (cid:0)(cid:0) Γ • G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f = Tot (cid:0)Q ∗ ( C • ∧ Σ − jd V (1)) f (cid:1) . For each l ≥
0, let(5.1) F l → Tot l (cid:0)Q ∗ ( C • ∧ Σ − jd V (1)) f (cid:1) → Tot l − (cid:0)Q ∗ ( C • ∧ Σ − jd V (1)) f (cid:1) e a homotopy fiber sequence (when l = 0, the last term above is ∗ , the trivialspectrum) and to conserve space, let T l ( − ) := Tot l ( Q ∗ ( − )) and C • j := ( C • ∧ Σ − jd V (1)) f . Then HS j is the spectral sequence obtained from the exact couple formed from thelong exact sequences · · · → π t ( F l ) → π t ( T l ( C • j )) → π t ( T l − ( C • j )) → π t − ( F l ) → · · · associated to the above homotopy fiber sequences.As done earlier, we now exploit the fact that smashing with a finite spectrumcommutes with products and homotopy limits. Notice that for each n ≥ (cid:0)Q ∗ ( C • ∧ Σ − jd V (1)) f (cid:1) n = Q { [ j →···→ [ jn ] } ( C j n ∧ Σ − jd V (1)) f ≃ (cid:0)Q { [ j →···→ [ jn ] } C j n (cid:1) ∧ Σ − jd V (1)= ( Q ∗ C • ) n ∧ Σ − jd V (1) , where the middle two products are indexed over all length n compositions in thecategory ∆, so that Q ∗ ( C • ∧ Σ − jd V (1)) f ≃ (cid:0) ( Q ∗ C • ) ∧ Σ − jd V (1) (cid:1) f , which depicts a zigzag of objectwise weak equivalences between cosimplicial spectra.Then for each l ≥
0, with ∆ ( l ) equal to the full subcategory of ∆ consisting ofobjects of cardinality less than l + 2, and – given a cosimplicial spectrum Z • –using holim ∆ ( l ) Z • to denote holim ∆ ( l ) (cid:0) ∆ ( l ) ֒ → ∆ Z • −−→ Sp Σ (cid:1) , we haveTot l (cid:0)Q ∗ ( C • ∧ Σ − jd V (1)) f (cid:1) ≃ holim ∆ ( l ) Q ∗ ( C • ∧ Σ − jd V (1)) f ≃ holim ∆ ( l ) (cid:0) ( Q ∗ C • ) ∧ Σ − jd V (1) (cid:1) f ≃ (cid:0) holim ∆ ( l ) Q ∗ C • (cid:1) ∧ Σ − jd V (1) ≃ Tot l ( Q ∗ C • ) ∧ Σ − jd V (1) , where the first and last steps are by [15, Proposition 3.10]. Thus, in the stablehomotopy category, for l ≥
0, we can regard the homotopy fiber sequence in (5.1)as having the form(5.2) F l → Tot l ( Q ∗ C • ) ∧ Σ − jd V (1) → Tot l − ( Q ∗ C • ) ∧ Σ − jd V (1) . Since the stable model structure on Sp Σ is proper [21, Theorem 5.5.2], by [19,Remark 19.1.6, Propositions 13.4.4 and 19.5.3], we can regard a homotopy fiber asa homotopy limit. For each l ≥
0, let b F l γ l −→ Tot l ( Q ∗ C • ) α l −→ Tot l − ( Q ∗ C • ) β l −→ Σ b F l be a homotopy fiber sequence (our names for the maps follow [27, (5.29)]): by anapplication of ( − ) ∧ Σ − jd V (1), we obtain the homotopy fiber sequence b F l ∧ Σ − jd V (1) γ l ∧ −−−→ Tot l ( Q ∗ C • ) ∧ Σ − jd V (1) α l ∧ −−−→ Tot l − ( Q ∗ C • ) ∧ Σ − jd V (1) . By comparing this fiber sequence with (5.2), another application of commuting ahomotopy limit with smashing with a finite spectrum yields F l ≃ b F l ∧ Σ − jd V (1) , l ≥ . t follows that HS j is the spectral sequence obtained from the exact couple formedfrom the long exact sequences · · · → π ∗ ( b F l ∧ Σ − jd V (1)) ( γ l ∧ ∗ −−−−−→ π ∗ ( T l ( C • ) ∧ Σ − jd V (1)) −··· ( α l ∧ ∗ −−−−−→ π ∗ ( T l − ( C • ) ∧ Σ − jd V (1)) ( β l ∧ ∗ −−−−−→ π ∗− ( b F l ∧ Σ − jd V (1)) → · · · (the top row ends with a morphism that is continued in the bottom row), where l ≥
0. As recalled earlier, V (1) is a homotopy commutative and homotopy asso-ciative ring spectrum, so that this long exact sequence is in the category of P ( v )-modules. Thus, the associated exact couple and, consequently, spectral sequence HS j live in the category of P ( v )-modules.6. Step v: the P ( v ) -module spectral sequences give a direct sum As usual, G is any closed subgroup of Z × p , and Mod P ( v ) is the category of P ( v )-modules. We recall from Section 2 that there is the isomorphism π ∗ (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) ∼ = colim j ≥ π ∗ (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) hG (cid:1) , where the right-hand side is the middle term in the colimitcolim j ≥ (cid:0) → j E , ∗ +12 → π ∗ (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) hG (cid:1) → j E , ∗ → (cid:1) of short exact sequences. For each j ≥ HS j is a spectral sequence in Mod P ( v ) and since it is isomorphic to spectral sequence { j E ∗ , ∗ r } , the associated short exactsequence (displayed above, inside the parentheses) is in Mod P ( v ) . It will be helpfulto write out this short exact sequence explicitly: omitting the trivial terms on theends and letting K denote K ( KU p ), this sequence of P ( v )-modules has the form H c ( G, π ∗ +1 ( K ∧ Σ − jd V (1))) → π ∗ (( K dis O ) hG ∧ Σ − jd V (1)) → ( π ∗ ( K ∧ Σ − jd V (1))) G , where the middle term resulted from applying Theorem 4.2.If Z is any spectrum, then the diagram { π ∗ ( Z ∧ Σ − jd V (1)) } j ≥ is in Mod P ( v ) ,so that the isomorphism π ∗ ( Z ∧ v − V (1)) ∼ = colim j ≥ π ∗ ( Z ∧ Σ − jd V (1))is in the category of P ( v ± )-modules (for example, see [6, Corollary 1.2]). Thedirect system of spectra { Σ − jd V (1) } j ≥ induces a direct system (cid:8)(cid:0)(cid:0) Γ • G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f (cid:9) j ≥ of cosimplicial spectra, and hence, a direct system { HS j } j ≥ of homotopy spectralsequences. Thus, there is the direct system (cid:8) π ∗ (cid:0)(cid:0)(cid:0) Γ ∗ G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f (cid:1)(cid:9) j ≥ of associated cochain complexes in Mod P ( v ) , the cohomology of which induces thedirect system (cid:8) H sc ( G, π ∗ ( K ( KU p ) ∧ Σ − jd V (1))) (cid:9) j ≥ in Mod P ( v ) , for s = 0 ,
1. Therefore, the diagram (cid:8) → j E , ∗ +12 → π ∗ ((( K ( KU p )) dis O ) hG ∧ Σ − jd V (1)) → j E , ∗ → (cid:9) j ≥ f short exact sequences is in Mod P ( v ) , so that the exact sequence0 → colim j ≥ j E , ∗ +12 → colim j ≥ π ∗ ((( K ( KU p )) dis O ) hG ∧ Σ − jd V (1)) → colim j ≥ j E , ∗ → P ( v ± )-modules, where the isomorphismscolim j ≥ j E s, ∗ ∼ = H s h colim j ≥ π ∗ (cid:0)(cid:0)(cid:0) Γ ∗ G ( K ( KU p )) dis O (cid:1) ∧ Σ − jd V (1) (cid:1) f (cid:1)i , s = 0 , , show that the two outer nontrivial terms in the exact sequence are indeed modulesover P ( v ± ). In particular, in every degree t , the sequence is one of F p -modulesand is split exact, giving π t (cid:0) ( K ( KU p ) ∧ v − V (1)) hG (cid:1) ∼ = (cid:0) colim j ≥ H c ( G, π t +1 ( K ( KU p ) ∧ Σ − jd V (1))) (cid:1) ⊕ (cid:0) π t ( K ( KU p ) ∧ v − V (1)) (cid:1) G , an isomorphism of F p -modules.7. Step vi: simplifying H c ( G, π ∗ ( K ( KU p ) ∧ V (1))[ v − ])Now we work on reducing the first summand in the direct sum obtained at theend of the previous section to a more familiar object. Fix j ≥ t ∈ Z , andrecall that π t ( K j ) = π t ( K ( KU p ) ∧ Σ − jd V (1))is a finite abelian group (this fact is explained in [13, Section 1.2]; the author didnot play a role in the hard work behind the explanation, which was done by others,as noted by the references in [ibid.]) and, as a unitary F p -module, it is a p -torsiongroup (that is, pm = 0, for every element m ). Notice that H c ( G, π t ( K ( KU p ) ∧ Σ − jd V (1))) ∼ = H c ( Z × p , Coind Z × p G ( π t ( K j ))) ∼ = colim N ⊳ o Z × p H c ( Z × p , C N ( t,j ) ) , where the first isomorphism is by Shapiro’s Lemma, the second one is by [24,Proposition 6.10.4, (a)] – with C N ( t,j ) := Coind Z × p /NGN/N (( π t ( K j )) N ∩ G ) , and each C N ( t,j ) is a Z × p /N -module (by definition), which makes C N ( t,j ) a discrete Z × p -module via the projection Z × p → Z × p /N .Let N be fixed. As a set, C N ( t,j ) is finite and, for every element f in this abeliangroup, pf = 0. This last fact – together with p − p being relatively prime– implies that the cohomology H ∗ ( C p − , C N ( t,j ) ) for the C p − -module C N ( t,j ) (byrestriction of the Z × p -action) vanishes in positive degrees, so that in the Lyndon-Hochschild-Serre spectral sequence E p,q = H pc (cid:0) Z p , H q ( C p − , C N ( t,j ) ) (cid:1) = ⇒ H p + qc ( Z × p , C N ( t,j ) ) , we have E p,q = ( H pc (cid:0) Z p , (cid:0) C N ( t,j ) (cid:1) C p − (cid:1) , q = 0;0 , q > . This gives H c ( Z × p , C N ( t,j ) ) ∼ = H c (cid:0) Z p , (cid:0) C N ( t,j ) (cid:1) C p − (cid:1) ∼ = H (cid:0) Z , (cid:0) C N ( t,j ) (cid:1) C p − (cid:1) ∼ = (cid:0)(cid:0) C N ( t,j ) (cid:1) C p − (cid:1) Z , here the third expression above is a non-continuous cohomology group and thesecond isomorphism is because (cid:0) C N ( t,j ) (cid:1) C p − is finite and p -torsion (for example, see[22, Example 4.6, Lemma 4.7]).Now we put the pieces together as j varies. Given a group K , let Z [ K ]-Modbe the category of K -modules, and let Ab denote the category of abelian groups.Also, given mathematical expressions A and B , notation of the form A K/e/L ∼ = B or A e/K/L ∼ = B means that (a) in Ab, A ∼ = B ; (b) in expression A , any colimits are in Z [ K ]-Modor Ab (signified by “ e ” in “ e/K/L ”), respectively, but these colimits can be formedin Ab or Z [ K ]-Mod, respectively, since the forgetful functor Z [ K ]-Mod → Ab is aleft adjoint; (c) part (b) explains the commuting of any colimits with the evidentfunctor and this commuting underlies the isomorphism A ∼ = B ; and (d) L denotesa group, and in B , any colimits are in Z [ L ]-Mod, by which we mean Ab, when L is “ e .” (To avoid any confusion, we note that if K = Z , then Z [ K ]-Mod means Z [ Z ]-Mod.) We havecolim j ≥ H c ( G, π t ( K ( KU p ) ∧ Σ − jd V (1))) ∼ = colim N ⊳ o Z × p colim j ≥ (cid:0)(cid:0) C N ( t,j ) (cid:1) C p − (cid:1) Z e/e/ Z ∼ = (cid:16) colim N ⊳ o Z × p colim j ≥ (cid:0) C N ( t,j ) (cid:1) C p − (cid:17) Z Z /e/e ∼ = (cid:16)(cid:16) colim N ⊳ o Z × p colim j ≥ C N ( t,j ) (cid:17) C p − (cid:17) Z . Again, let N ⊳ o Z × p be fixed and, as is standard, given A ∈ Z [ GN/N ]-Mod, letInd Z × p /NGN/N ( A ) = Z [ Z × p /N ] ⊗ Z [ GN/N ] A, and set P j := ( π t ( K j )) N ∩ G . Then there are isomorphisms C N ( t,j ) ∼ = Hom Z [ GN/N ]-Mod ( Z [ Z × p /N ] , P j ) ∼ = Ind Z × p /NGN/N ( P j )of Z [ Z × p /N ]-modules, since Z × p /N is finite (for example, see [24, proof of Proposition6.10.4]) and because ( Z × p /N ) / ( GN/N ) ∼ = Z × p /GN is finite [12, Proposition 5.9],respectively. Hence, there are the following isomorphisms of Z [ Z × p /N ]-modules (inthe first use below of the “ e/K/L ∼ = ” notation, part (c) of its meaning does not apply):colim j ≥ C N ( t,j ) e/ ( Z × p /N ) / ( Z × p /N ) ∼ = colim j ≥ Ind Z × p /NGN/N ( P j ) e/ ( Z × p /N ) / ( GN/N ) ∼ = Ind Z × p /NGN/N (colim j ≥ P j ) ∼ = Coind Z × p /NGN/N (colim j ≥ P j ) ∼ = Coind Z × p /NGN/N (( π t ( K ( KU p ) ∧ v − V (1))) N ∩ G ) . These four isomorphisms are of Z [ Z × p ]-modules (via the projection Z × p → Z × p /N )and, by [24, Proposition 6.10.4, (a)], we conclude thatcolim j ≥ H c ( G, π t ( K ( KU p ) ∧ Σ − jd V (1))) ∼ = (cid:0)(cid:0) Coind Z × p G ( π t ( K ( KU p ) ∧ v − V (1))) (cid:1) C p − (cid:1) Z , completing the proof of Theorem 1.11.In case it is easier to compute colim j ≥ H c ( G, π t ( K ( KU p ) ∧ Σ − jd V (1))) by notrestricting the Z p -action to the Z -action, as done on the right-hand side in the lastisomorphism above, we take another look at each H c (cid:0) Z p , (cid:0) C N ( t,j ) (cid:1) C p − (cid:1) to obtain the ollowing result, which is the content of Remark 1.12. Here (as in the remark), Z p is regarded as having the trivial Z p -group action. Lemma 7.1.
When p ≥ , G is any closed subgroup of Z × p , and t ∈ Z , there is anisomorphism H c ( G, π t ( K ( KU p ) ∧ v − V (1))) ∼ = Z p ⊗ Z p [[ Z p ]] (cid:0)(cid:0) Coind Z × p G ( π t ( K ( KU p ) ∧ v − V (1))) (cid:1) C p − (cid:1) . Proof.
Notice that every coefficient group (cid:0) C N ( t,j ) (cid:1) C p − is a finite discrete p -torsion Z p [[ Z p ]]-module. It is standard that there is a projective resolution0 → Z p [[ Z p ]] τ −→ Z p [[ Z p ]] → Z p → c Z p [[ Z p ]] ( − , (cid:0) C N ( t,j ) (cid:1) C p − ) of continuous modulehomomorphisms to this resolution: we obtain that H c (cid:0) Z p , (cid:0) C N ( t,j ) (cid:1) C p − (cid:1) ∼ = ( C N ( t,j ) ) Cp − (cid:30) (im( τ ∗ : ( C N ( t,j ) ) Cp − → ( C N ( t,j ) ) Cp − )) ∼ = H c (cid:0) Z p , (cid:0) C N ( t,j ) (cid:1) C p − (cid:1) = Z p b ⊗ Z p [[ Z p ]] (cid:0) C N ( t,j ) (cid:1) C p − ∼ = Z p ⊗ Z p [[ Z p ]] (cid:0) C N ( t,j ) (cid:1) C p − , where the right-hand side of the second isomorphism is a continuous homologygroup (see [26, Section 3.3]) and the last step is because (cid:0) C N ( t,j ) (cid:1) C p − is finite (andthus, a finitely generated object in the category of profinite Z p [[ Z p ]]-modules; see[24, Proposition 5.5.3]). The isomorphism in the second step is not quite immediate,and it can be justified in a sleek way: since Z is an orientable discrete Poincar´eduality group of dimension one and pro- p good (in the sense of [29, Section 3.1]; by[22, Example 4.6, Lemma 4.7]), the pro- p completion Z p is an orientable (profinite)Poincar´e duality group at p of dimension one, by [29, Proposition 3.2] (here, for“orientable (profinite) Poincar´e duality group at p ,” we use the definitions in [26,Section 4.4, page 394] and by [28, Remark 2.2], these are equivalent to those usedin [29]), and the desired isomorphism follows.Then the result follows by the manipulations that preceded this lemma. Tounderstand the abstract Z p [[ Z p ]]-module structure of the C p − -fixed points of thecoinduced module in the statement of the lemma (and of the various pieces involvedin the manipulations), it is helpful to note that if H is an arbitrary profinite group,then a p -torsion discrete H -module is canonically a discrete, and hence abstract, Z p [[ H ]]-module. (cid:3) A further reduction in the case when G = Z × p Let V (0) be the mod p Moore spectrum M ( p ), and more generally, for eachinteger i ≥
1, let M ( p i ) be the mod p i Moore spectrum. By restriction of the Z × p -action, C p − acts on K ( KU p ), so that there is the homotopy fixed point spectrum K ( KU p ) hC p − = ( K ( KU p )) hC p − , nd by [25, page 1267] (see [3, pages 11-12] for a proof), the canonical mapholim i ≥ ( K ( L p ) ∧ M ( p i )) ≃ −→ holim i ≥ ( K ( KU p ) hC p − ∧ M ( p i ))is a weak equivalence. It follows that the morphism L V (0) ( K ( L p )) ≃ −→ L V (0) ( K ( KU p ) hC p − )(between Bousfield localizations with respect to V (0)) is a weak equivalence, sothat the natural map K ( L p ) → K ( KU p ) hC p − is a V (0)-equivalence. The familiarcofiber sequence Σ p − V (0) v −→ V (0) i −→ V (1)induces the commutative diagram K ( L p ) ∧ Σ p − V (0) / / (cid:15) (cid:15) K ( L p ) ∧ V (0) / / (cid:15) (cid:15) K ( L p ) ∧ V (1) (cid:15) (cid:15) K ( KU p ) hC p − ∧ Σ p − V (0) / / K ( KU p ) hC p − ∧ V (0) / / K ( KU p ) hC p − ∧ V (1)in which the rows are cofiber sequences. Since the leftmost and middle vertical mapsare weak equivalences, the rightmost vertical map is a weak equivalence. Thus, foreach j ≥
0, the map K ( L p ) ∧ Σ − jd V (1) ≃ −→ K ( KU p ) hC p − ∧ Σ − jd V (1)is a weak equivalence. We apply this conclusion in the following way.There are the homotopy fixed point spectral sequences ♯ E s,t = H s ( C p − , π t ( K ( KU p ) ∧ v − V (1))) = ⇒ π t − s (( K ( KU p ) ∧ v − V (1)) hC p − )and j ♯ E s,t = H s ( C p − , π t ( K ( KU p ) ∧ Σ − jd V (1))) ⇒ π t − s (( K ( KU p ) ∧ Σ − jd V (1)) hC p − ) , for each j ≥
0. Since each π t ( K ( KU p ) ∧ Σ − jd V (1)) is p -torsion, both ♯ E s,t ∼ = colim j ≥ j ♯ E s,t and each j ′ ♯ E s,t vanish for s > t ∈ Z , and j ′ ≥
0. As a consequence, π ∗ (( K ( KU p ) ∧ v − V (1)) hC p − ) ∼ = ( π ∗ ( K ( KU p ) ∧ v − V (1))) C p − , ( K ( KU p ) ∧ v − V (1)) hC p − ≃ colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) hC p − , and π ∗ (( K ( KU p ) ∧ Σ − jd V (1)) hC p − ) ∼ = ( π ∗ ( K ( KU p ) ∧ Σ − jd V (1))) C p − , j ≥ { j | j ≥ } ); we state it here because of its intrinsic nterest), we have the isomorphisms( π ∗ ( K ( KU p ) ∧ v − V (1))) C p − ∼ = colim j ≥ π ∗ (( K ( KU p ) ∧ Σ − jd V (1)) hC p − ) ∼ = colim j ≥ π ∗ ( K ( KU p ) hC p − ∧ Σ − jd V (1)) ∼ = π ∗ ( K ( L p ) ∧ v − V (1)) ∼ = K ( L p ) ∗ ( V (1))[ v − ] . Each of the spectra K ( L p ) and K ( KU p ) hC p − have a natural action by Z p andthe map K ( L p ) → K ( KU p ) hC p − is Z p -equivariant; thus, each of the above fourisomorphisms is Z p -equivariant. References [1] Gabe Angelini-Knoll. On topological Hochschild homology of the K (1)-local sphere. 30 pp.,arXiv:1612.00548v3; January 8, 2021; accepted for publication in Journal of Topology .[2] Christian Ausoni. On the algebraic K -theory of the complex K -theory spectrum. Invent.Math. , 180(3):611–668, 2010.[3] Christian Ausoni and John Rognes. Algebraic K -theory of the fraction field of topological K -theory. 54 pp., arXiv:0911.4781; November 25, 2009.[4] Christian Ausoni and John Rognes. Algebraic K -theory of topological K -theory. Acta Math. ,188(1):1–39, 2002.[5] Christian Ausoni and John Rognes. The chromatic red-shift in algebraic K -theory. In Guido’sBook of Conjectures, Monographie de L’Enseignement Math´ematique , volume 40, pages 13–15, 2008.[6] Christian Ausoni and John Rognes. Algebraic K -theory of the first Morava K -theory. J. Eur.Math. Soc. (JEMS) , 14(4):1041–1079, 2012.[7] Clark Barwick and Peter Haine. Pyknotic objects, I. Basic notions. 39 pages,arXiv:1904.09966v2; April 30, 2019.[8] Mark Behrens and Daniel G. Davis. The homotopy fixed point spectra of profinite Galoisextensions.
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