The geometry of the Lubin-Tate space of deformations of a formal group is studied via an étale, rigid analytic map from the deformation space to projective space. This leads to a simple description of the equivariant canonical bundle of the deformation space which, in turn, yields a formula for the dualizing complex in stable homotopy theory.
We formulate a theory of punctured affine formal schemes, suitable for certain problems within algebraic topology. As an application, we show that the Morava K-theoretic localizations of Morava E-theory corepresent a version of the Lubin-Tate moduli problem in this framework.
We compute recursive approximations of the action of the height $h \geq 2$ Morava stabilizer group on the associated Lubin-Tate deformation ring. We then specialize to the case $h=3$ and $p>2$ to calculate the action explicitly. These results are new for $h>2$ and agree with computations by Lader at height $h=2$.
We give a method for computing factorization homology of $\oper{E}_n$-algebra using as an input an algebraic version of higher Hochschild homology due to Pirashvili. We then show how to compute higher Hochschild homology and cohomology when the algebra is étale in a sense that we make precise. As an application, we compute higher Hochschild cohomology of the Lubin-Tate ring spectrum.
This expository paper introduces several ideas in chromatic homotopy theory around Morava's extraordinary E-theories. In particular, we construct various moduli problems closely related to Lubin-Tate deformation theory and study their symmetries. These symmetries are then used in conjunction with the Hopkins-Kuhn-Ravenel character theory to provide formulas for the power operations and the stabilizer group action on the E-cohomology of a finite group.
We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}\langle 2 \rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-periodic.
We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E_n$ with its $C_2$-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C_2$-fixed points.
We take a direct approach to computing the orbits for the action of the automorphism group $\mathbb{G}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{\mathbb{G}_2} \cong \mathbb{F}_p$. The result is new for $p=2$ and $p=3$. For primes $p\geq 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.
We construct a stable model structure on profinite symmetric spectra with a continuous action of an arbitrary profinite group. This provides a natural framework for a new construction of homotopy fixed point spectra and of homotopy fixed point spectral sequences for the action of the extended Morava stabilizer group on Lubin-Tate spectra. These continuous homotopy fixed points are canonically equivalent to the homotopy fixed points of Devinatz and Hopkins but have a drastically simplified construction.
We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$.
Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava stabilizer group $Γ$. We study the derived stack $\operatorname{Spf} E/Γ$. Descent-theoretic techniques allow us to deduce a theorem of Hopkins-Mahowald-Sadofsky on the $K(n)$-local Picard group, as well as a recent result of Barthel-Beaudry-Stojanoska on the Anderson duals of higher real $K$-theories.
Let E_n be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E^{nr}_n whose coefficients are built from the coefficients of E_n and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E^{nr}_n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E^{nr}_n with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.
We show that Lubin--Tate theories attached to algebraically closed fields are characterized among $T(n)$-local $\mathbb{E}_{\infty}$-rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$, the collection of $\mathbb{E}_\infty$-ring maps from $R$ to such Lubin-Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$ admits an $\mathbb{E}_\infty$-ring map to such a Lubin-Tate theory. As consequences, we construct $\mathbb{E}_{\inf…
When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (Z^{hH})^{hK/H}, where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G=G_n, the extended Morava stabilizer group, and Z=L_{K(n)}(E_n \wedge X), where L_{K(n)} is Bousfield localization with respect to Morava K-theory, E_n is the Lubin-Tate spectrum, and X is any spectrum with trivial G_n-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we …
We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$ is either a Lubin-Tate spectrum $E_n$ or the related 2-periodic Morava K-theory $K_n$, and $G$ is a finite group. When $R$ is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a $G$-Galois extension in the sense of John Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ these extensions are always faithful in the $K_n$ local category. However, for a cyclic $p$-group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \to F({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it…
We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusion of the Witt vectors into the Lubin-Tate ring induces a split injection on continuous stabilizer cohomology with torsion cokernel of bounded exponent, thereby proving Hopkins' chromatic splitting conjecture and the vanishing conjecture of Beaudry-Goerss-Henn rationally. The key ingredients are the equivalence between the Lubin-Tate tower and the Drinfeld tow…
In their work on the period map and the dualizing sheaf for Lubin-Tate space, Gross and the second author wrote down an equivalence between the Spanier-Whitehead and Brown-Comenetz duals of certain type $n$-complexes in the $K(n)$-local category at large primes. In the culture of the time, these results were accessible to educated readers, but this seems no longer to be the case; therefore, in this note we give the details. Because we are at large primes, the key result is algebraic: in the Picard group of Lubin-Tate space, two important invertible sheaves become isomorphic modulo $p$.
We propose a new method to compute the $C_{2^n}$-equivariant homotopy groups of the Eilenberg-Mac Lane spectrum $H\underline{\mathbb{Z}}$ as a $RO(C_{2^n})$-graded Green functor using the generalized Tate squares. As an example, we completely compute the $C_4$ case and investigate two $\mathscr{P}$-homotopy limit spectral sequences for the family $\mathscr{P}=\{e,C_2\}$.
Let G be a closed subgroup of G_n, the extended Morava stabilizer group. Let E_n be the Lubin-Tate spectrum, let X be an arbitrary spectrum with trivial G-action, and define E^(X) to be L_K(n)(E_n ^ X). We prove that E^(X) is a continuous G-spectrum with a G-homotopy fixed point spectrum, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is the homotopy groups of the G-homotopy fixed point spectrum of E^(X). We show that the homotopy fixed points of E^(X) come from the K(n)-localization of the homotopy fixed points of the spectrum (F_n…
We compute the mod(p) homotopy groups of the continuous homotopy fixed point spectrum E_2^{hH_2} for p>2, where E_n is the Landweber exact spectrum whose coefficient ring is the ring of functions on the Lubin-Tate moduli space of lifts of the height n Honda formal group law over F_{p^n}, and H_n is the subgroup WF^x_{p^n} wreath product Gal(F_{p^n}/F_p) of the extended Morava stabilizer group G_n. We examine some consequences of this related to Brown-Comenetz duality and to finiteness properties of homotopy groups of K(n)_*-local spectra. We also indicate a plan for computing pi_*(E_n^{hH_n} s…
We provide a short proof that the dimensions of the mod $p$ homology groups of the unordered configuration space $B_k(T)$ of $k$ points in a torus are the same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology has no $p$-power torsion. The same argument works for the punctured genus $g$ surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory.
We give a short proof of a theorem of Kuhn that Tate constructions for finite group actions vanish in telescopically localized stable homotopy theory. In particular, we observe that Kuhn's theorem is equivalent to the statement that the transfer $BC_{p+} \to S^0$ admits a section after telescopic localization, which in turn follows from the Kahn-Priddy theorem.
We study a certain construction designed to bring together the following two topics: $i$) Dyer--Lashof-operations in negative Tate-cohomology, $ii$) the description of negative Tate-cohomology in terms of joins. It has the merit of making (some) sense in a more general context: where the loop-space of the space under study no longer has to be of compact homotopy-type. The exposition given here is a streamlined version of a previous version of mine, available here at the Arxiv.
We deform the Ravenel-Wilson computation of the Morava K-homology of Eilenberg-Mac Lane spaces to obtain a similar description of their completed Morava E-homology. This yields both a cohomological description and an interpretation on the level of formal schemes: the scheme associated to the E-cohomology of the space K(Z/p^infty, q) is the qth exterior power of the p-divisible group associated to the versal infinitesimally deformed formal group over Lubin-Tate space.
We construct an integral model for Lubin-Tate curves as moduli of finite subgroups of formal deformations over complete Noetherian local rings. They are p-adic completions of the modular curves X_0(p) at a mod-p supersingular point. Our model is semistable in the sense that the only singularities of its special fiber are normal crossings. Given this model, we obtain a uniform presentation for the Dyer-Lashof algebra of Morava E-theories at height 2 as local moduli of power operations in elliptic cohomology.
In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the univers…
We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{hC_2}$ is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n = 1$.
We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable v_n-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computations to the algebraic geometry of isogenies between Lubin-Tate formal groups.
From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operad C_2 (which is an S^1 operad) we obtain variations on Getzler's gravity operad, which we show governs the Chas-Sullivan string bracket.
On the l-adic homotopy type of configuration spaces
Universitat de Barcelona（巴塞罗那大学）；Centre de Recerca Matemàtica（数学研究中心）；Université Sorbonne Paris Nord（索邦巴黎北大学）；Laboratoire Analyse, Géométrie et Applications, CNRS (UMR 7539)（分析、几何与应用实验室，法国国家科学研究中心（联合研究单位7539））；Institut Universitaire de France（法兰西学院）
We review the shape theory of $\infty$-topoi, and relate it with the usual cohomology of locally constant sheaves. Additionally, a new localization of profinite spaces is defined which allows us to extend the étale realization functor of Isaksen. We apply these ideas to define an étale homotopy type functor $\mathrm{\acute{e}t}(\mathcal{X})$ for Berkovich's non-archimedean analytic spaces $\mathcal{X}$ over a complete non-archimedean field $K$ and prove some properties of the construction. We compare the étale homotopy types coming from Tate's rigid spaces, Huber's adic spaces, and rigid model…
We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^hF where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is…
We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of infinity-categories C. We classify such Goodwillie towers in terms of the derivatives of the identity functor of C. As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras with Tate diagonals. Our classification of Goodwillie …
We construct spectral sequences for computing the cohomology of automorphism groups of formal groups with complex multiplication by a $p$-adic number ring. We then compute the cohomology of the group of automorphisms of a height four formal group law which commute with complex multiplication by the ring of integers in the field $\mathbb{Q}_p(\sqrt{p})$, for primes $p>5$. This is a large subgroup of the height four strict Morava stabilizer group. The group cohomology of this group of automorphisms turns out to have cohomological dimension $8$ and total rank $80$. We then run the $K(4)$-local $E…
We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the c…
