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Ambidexterity in K(N)-Local Stable Homotopy Theory

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Ambidexterity in K(N)-Local Stable Homotopy Theory Ambidexterity in K(n)-Local Stable Homotopy Theory Michael Hopkins and Jacob Lurie December 19, 2013 Contents 1 Multiplicative Aspects of Dieudonne Theory 4 1.1 Tensor Products of Hopf Algebras . 6 1.2 Witt Vectors . 12 1.3 Dieudonne Modules . 19 1.4 Disconnected Formal Groups . 28 2 The Morava K-Theory of Eilenberg-MacLane Spaces 34 2.1 Lubin-Tate Spectra . 35 2.2 Cohomology of p-Divisible Groups . 41 2.3 The Spectral Sequence of a Filtered Spectrum . 47 2.4 The Main Calculation . 50 3 Alternating Powers of p-Divisible Groups 63 3.1 Review of p-Divisible Groups . 64 3.2 Group Schemes of Alternating Maps . 68 3.3 The Case of a Field . 74 3.4 Lubin-Tate Cohomology of Eilenberg-MacLane Spaces . 82 3.5 Alternating Powers in General . 86 4 Ambidexterity 89 4.1 Beck-Chevalley Fibrations and Norm Maps . 91 4.2 Properties of the Norm . 96 4.3 Local Systems . 103 4.4 Examples . 107 5 Ambidexterity of K(n)-Local Stable Homotopy Theory 112 5.1 Ambidexterity and Duality . 113 5.2 The Main Theorem . 121 5.3 Cartier Duality . 126 5.4 The Global Sections Functor . 135 1 Introduction Let G be a finite group, and let M be a spectrum equipped with an action of G. We let M hG denote the homotopy fixed point spectrum for the action of G on M, and MhG the homotopy orbit spectrum for the hG action of G on X. These spectra are related by a canonical norm map Nm : MhG ! M . Our starting point is the following result of Hovey and Sadofsky (see [9]): Theorem 0.0.1. Let K(n) be a Morava K-theory, let M be a spectrum which is K(n)-local, and let G be a finite group acting on M. Then the norm map hG Nm : MhG ! M hG exhibits M as a K(n)-localization of MhG. In other words, the map Nm induces an isomorphism of hG K(n)-homology groups K(n)∗MhG ! K(n)∗M . Our goal in this paper is to place Theorem 0.0.1 into a larger context. The collection of all K(n)-local spectra can be organized into an 1-category, which we will denote by SpK(n). If M is a K(n)-local spectrum with an action of a finite group G, then M determines a diagram ρ : BG ! SpK(n). The homotopy fixed point spectrum M hG can be identified with a limit of the diagram ρ, and the localized homotopy orbit spectrum LK(n)MhG can be identified with a colimit of the diagram ρ. The main result of this paper is the following variant of Theorem 0.0.1: Theorem 0.0.2. Let X be a Kan complex. Assume that, for every vertex x 2 X, the sets πn(X; x) are finite for every integer n, and trivial for n 0. Let ρ : X ! SpK(n) be a diagram of K(n)-local spectra, indexed by X. Then there is a canonical homotopy equivalence Nm : lim(ρ) !∼ lim(ρ): X −! − Remark 0.0.3. In the special case where X is the classifying space of a finite group G, and ρ : X ! SpK(n) classifies an action of G on a K(n)-local spectrum M, the map Nm : lim(ρ) ! lim(ρ) we will construct X −! − hG in our proof of Theorem 0.0.2 agrees with the K(n)-localization of the usual norm map MhG ! M . Consequently, Theorem 0.0.2 can be regarded as a generalization of Theorem 0.0.1. Example 0.0.4. The simplest instance of Theorem 0.0.2 occurs when the Kan complex X is discrete. In this case, Theorem 0.0.2 asserts that for any finite collection of objects M1;:::;Mk 2 SpK(n), the product Q ` 1≤i≤k Mi and the coproduct 1≤i≤k Mi are canonically equivalent. Let us briefly outline our approach to Theorem 0.0.2. Our assumptions on X guarantee that there exists an integer n ≥ 0 such that the homotopy groups πm(X; x) vanish for m > n. We proceed by induction on n. The case n = 0 reduces to Example 0.0.4. The inductive step can be broken into two parts: (a) The construction of the norm map Nm : lim(ρ) ! lim(ρ). X −! − (b) The proof that NmX (ρ) is an equivalence. To carry out (a), we note that a map from lim(ρ) to lim(ρ) can be identified with a collection of maps −! − φx;y : ρ(x) ! ρ(y), depending functorially on the pair (x; y) 2 X × X. Note that every point e of the path ∆1 space Px;y = fxg ×X X ×X fyg determines an equivalence ρ(e): ρ(x) ! ρ(y) in the 1-category C. We will choose NmX to correspond to the family of maps φx;y given heuristically by the formula Z φx;y = ρ(e)dµ, e2Px;y where the integral is taken with respect to a \measure" µ which is defined using the inverse of the norm map NmPx;y (which exists by virtue of our inductive hypothesis). Making this idea precise will require some rather intricate categorical constructions, which we explain in detail in x4. 2 The core of the proof is in the verification of (b). Using formal properties of the norm maps NmX , we can reduce to proving that the map Nm : lim(ρ) ! lim(ρ) is an equivalence in the special case where X is X −! − an Eilenberg-MacLane space K(Z =p Z; m), and ρ is the constant functor taking the value K(n) 2 SpK(n). −∗ After passing to homotopy groups, NmX induces a map of graded abelian groups ι : K(n)∗X ! K(n) X, and we wish to show that ι is an isomorphism. ∗ Here we proceed by explicit calculation. The groups K(n)∗K(Z =p Z; m) and K(n) K(Z =p Z; m) were computed by Ravenel and Wilson in [18]. Their results are most neatly summarized using the language of 0 1 Dieudonne theory. Let κ = π0K(n), let G denote the formal group over κ given by Spf K(n) CP , and let M be the covariant Dieudonne module of G (so that M is a module over the Dieudonne ring W (κ)[F; V ], d which is free of n as a module over W (κ)). For each d ≥ 1, the exterior power ^W (κ)M inherits an action (d) n of W (κ)[F; V ], which is the covariant Dieudonne module of a smooth formal group G of height d and n−1 dimension d−1 . We have a canonical isomorphism ∗ K(n) K(Z =p Z; m) ' π∗K(n) ⊗κ A; where A is the ring of functions on p-torsion subgroup of G(m). We will give a proof of this result in x2 which is somewhat different from the proof given in [18]: it relies on multiplicative aspects of the theory of Dieudonne modules, which we review in x1. We can summarize the preceding discussion by saying that the group scheme Spec K(n)0K(Z =p Z; m) behaves, in some sense, like an mth exterior power of the group scheme Spec K(n)0K(Z =p Z; 1). In x4, we will make this idea more precise by introducing, for every integer m > 0 and every finite flat commutative (m) group scheme G over a commutative ring R, another group scheme AltG . We will see that the calculation of Ravenel and Wilson yields an isomorphism of commutative group schemes (m) Spec K(n)0K(Z =p Z; m) ' AltG[p]; where G[p] denotes the p-torsion subgroup of the formal group G (Theorem 2.4.10). Moreover, this isomor- phism can be lifted to characteristic zero: if E denotes the Lubin-Tate spectrum associated to the formal group G, and G denotes the universal deformation of G over the Lubin-Tate ring R = π0E, then we have a canonical isomorphism Spec E^K(Z =p Z; m) ' Alt(m) 0 G[p] of finite flat group schemes over R (Theorem 3.4.1). We will use this isomorphism to identify the bilinear ^ form β with (the reduction of) a certain multiple of the trace pairing on the algebra E0 K(Z =p Z; m). In x5, we will use this identification to prove the nondegeneracy of β, and thereby obtain a proof of Theorem 0.0.2. Remark 0.0.5. Most of the results proven in the first three sections of this paper have appeared elsewhere in print, though with a somewhat different exposition. In particular, the material of x1 was inspired by [4] (see also [2]). The calculations of x2 were originally carried out by in [18] (at least for odd primes; an extension to the prime 2 is indicated in [12]). The algebraic results of x3 concerning alternating powers of finite flat group schemes can be found in [8] (at least for odd primes), and the relationship with the generalized cohomology of Eilenberg-MacLane spaces is described in [2] (for Morava K-theory) and [17] (for Morava E-theory). Notation and Terminology In the later sections of this paper, we will freely use the language of 1-categories. We refer the reader to [13] for the foundations of this theory, and to [14] for an exposition of stable homotopy theory from the 1-categorical point of view.
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