Morava K-theory Research Articles

A chromatic vanishing result for TR

In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that T ( k ) T(k) -local TR vanishes on connective L n p , f L_n^{p,f} -acyclic E 1 \mathbb {E}_1 -rings for every 1 ≤ k ≤ n 1 \leq k \leq n and deduce consequences for connective Morava K-theory and the Thom spectra y ( n ) y(n) . The proof relies on the relationship between TR and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive ∞ \infty -categories which was recently established by Córdova Fedeli [Topological Hochschild homology of adic rings, Ph.D. thesis, University of Copenhagen, 2023].

Isotropic and numerical equivalence for Chow groups and Morava K-theories

In this paper we prove the conjecture claiming that, over a flexible field, isotropic Chow groups coincide with numerical Chow groups (with Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\Bbb {F}}_{p}$\\end{document}-coefficients). This shows that Isotropic Chow motives coincide with Numerical Chow motives. In particular, homs between such objects are finite groups and ⊗ has no zero-divisors. It provides a large supply of new points for the Balmer spectrum of the Voevodsky motivic category. We also prove the Morava K-theory version of the above result, which permits to construct plenty of new points for the Balmer spectrum of the Morel-Voevodsky A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{A}}^{1}$\\end{document}-stable homotopy category. This substantially improves our understanding of the mentioned spectra whose description is a major open problem.

Algebraic 𝐾-theory of the two-periodic first Morava 𝐾-theory

Using the root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of T ( 2 ) ∗ K ( k u ) T(2)_*\mathrm {K}(ku) for p > 3 p>3 . Through this, we also produce a new algebraic K K -theory computation; namely we obtain T ( 2 ) ∗ K ( k u / p ) T(2)_*\mathrm {K}(ku/p) , where k u / p ku/p is the 2 2 -periodic Morava K K -theory spectrum of height 1 1 .

MORAVA K-THEORY AND FILTRATIONS BY POWERS

Abstract We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin–Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabiliser group. As an application, we compute the zeroth limit at all primes and heights.

Tate blueshift and vanishing for Real oriented cohomology theories

We study transchromatic phenomena for the Tate construction of Real oriented cohomology theories. First, we show that after suitable completion, the Tate construction with respect to a trivial Z/2-action on height n Real Johnson–Wilson theory splits into a wedge of height n−1 Real Johnson–Wilson theories. This is the first example of Tate blueshift at all chromatic heights outside of the complex oriented setting. Second, we prove that the Tate construction with respect to a trivial finite group action on Real Morava K-theory vanishes, refining a classical Tate vanishing result of Greenlees–Sadofsky. In the course of proving these results, we develop some ideas in equivariant chromatic homotopy theory (e.g., completions of module spectra over Real cobordism, C2-equivariant chromatic Bousfield localizations) and apply the parametrized Tate construction.

A relation among Hopkins’ Picard groups of the localized categories with respect to finite wedges of the Morava 𝐾-theories

We work in the stable homotopy category of p p -local spectra for a fixed prime number p p . Let E E be a spectrum and L E \mathcal {L}_E denote the stable homotopy category of localized spectra with respect to E E in the sense of Bousfield. Then, M. Hopkins introduced a Picard group P i c ( L E ) Pic(\mathcal {L}_E) of the category L E \mathcal {L}_E . If the spectra E E and F F satisfy the relation ⟨ E ⟩ ≥ ⟨ F ⟩ \left \langle {E}\right \rangle \ge \left \langle F\right \rangle of the Bousfield classes, then we have a homomorphism ℓ : P i c ( L E ) → P i c ( L F ) \ell \colon Pic(\mathcal {L}_E)\to Pic(\mathcal {L}_F) . We consider the spectra K m n = E ( n ) ∧ M J m K_m^n=E(n)\wedge MJ_m for the n n -th Johnson-Wilson spectrum E ( n ) E(n) and a type m m generalized Moore spectrum M J m MJ_m for 0 ≤ m ≤ n 0\le m\le n . For E = K m n E=K_m^n , we have a subgroup P i c 0 ( L E ) Pic^0({\mathcal {L}_E}) of P i c ( L E ) Pic(\mathcal {L}_E) consisting of exotic elements. In this paper, we study the homomorphism ℓ : P i c 0 ( L E ( n ) ) → P i c 0 ( L K m n ) \ell \colon Pic^0({\mathcal {L}_{E(n)}})\to Pic^0({\mathcal {L}_{K_m^n}}) , and give conditions under which it is an isomorphism. This is a generalization of the result P i c 0 ( L 2 ) ≅ κ 2 Pic^0(\mathcal {L}_2)\cong \kappa _2 (P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk [J. Topol. 8 (2015), 267–294, Remark. 6.5]) for ( p , n , m ) = ( 3 , 2 , 2 ) (p,n,m)=(3,2,2) .

Twisted Morava K–theory and connective covers of Lie groups

Twisted Morava K-theory, along with computational techniques, including a universal coefficient theorem and an Atiyah-Hirzebruch spectral sequence, was introduced by Craig Westerland and the first author. We employ these techniques to compute twisted Morava K-theory of all connective covers of the stable orthogonal group and stable unitary group, and their classifying spaces, as well as spheres and Eilenberg-MacLane spaces. This extends to the twisted case some of the results of Ravenel and Wilson and of Kitchloo, Laures, and Wilson for Morava K-theory. This also generalizes to all chromatic levels computations by Khorami (and in part those of Douglas) at chromatic level one, i.e. for the case of twisted K-theory. We establish that for natural twists in all cases, there are only two possibilities: either that the twisted Morava homology vanishes, or that it is isomorphic to untwisted homology. We also provide a variant on the twist of Morava K-theory, with mod 2 cohomology in place of integral cohomology.

Monochromatic homotopy theory is asymptotically algebraic

In previous work, we used an ∞-categorical version of ultraproducts to show that, for a fixed height n, the symmetric monoidal ∞-categories of En,p-local spectra are asymptotically algebraic in the prime p. In this paper, we prove the analogous result for the symmetric monoidal ∞-categories of Kp(n)-local spectra, where Kp(n) is Morava K-theory at height n and the prime p. This requires ∞-categorical tools suitable for working with compactly generated symmetric monoidal ∞-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the En,p-local ∞-categories.

Algebraic slice spectral sequences

For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the \mathbb{R} -motivic effective slice spectral sequence is completely determined by the \rho -Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra.

The Omega spectrum for mod 2 KO-theory

The 8-periodic theory that comes from the KO-theory of the mod 2 Moore space is the same as the real first Morava K-theory obtained from the homotopy fixed points of the ℤ∕(2) action on the first Morava K-theory. The first Morava K-theory, K(1), is just mod 2 KU-theory. We compute the homology Hopf algebras for the spaces in this Omega spectrum.

Excellent rings in transchromatic homotopy theory

The purpose of this note is to verify that several basic rings appearing in transchromatic homotopy theory are Noetherian excellent normal domains and thus amenable to standard techniques from commutative algebra. In particular, we show that the coefficients of iterated localizations of Morava E-theory at the Morava K-theories are normal domains and also that the coefficients in the transchromatic character map for a fixed group form a normal domain.

Spectral sequences in smooth generalized cohomology

We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic study of torsion in differential cohomology. We apply this in detail to smooth Deligne cohomology, differential topological complex K-theory, and to a smooth extension of integral Morava K-theory that we introduce. In each case we explicitly identify the differentials in the corresponding spectral sequences, which exhibit an interesting and systematic interplay between (refinement of) classical cohomology operations, operations involving differential forms, and operations on cohomology with U(1) coefficients.

Chern Classes from Algebraic Morava K-Theories to Chow Groups

Chern Classes from Algebraic Morava K-Theories to Chow Groups

Ring of operations from Morava K-theories to Chow groups

Ring of operations from Morava K-theories to Chow groups

Some explicit expressions concerning BP

Abstract This paper provides some explicit expressions concerning the formal group laws of the following cohomology theories: BP, the Brown–Peterson cohomology, G(s), the cohomology theory obtained from BP by putting v i = 0 ${v_i=0}$ for all i ≥ 1 with i ≠ s ${i \ne s}$ , the Morava K-theory and the Abel cohomology. The coefficient rings of Weierstrass universal elliptic curve formal group law are computed in low dimensions.

The MoravaK–theory ofBO(q) andMO(q)

We give an easy proof that the Morava K-theories for BO(q) and MO(q) are in even degrees. Although this is a known result, it had followed from a difficult proof that BP^*(BO(q)) was Landweber flat. Landweber flatness follows from the even Morava K-theory. We go further and compute an explicit description of K(n)_*(BO(q)) and K(n)_*(MO(q)) and reconcile it with the purely algebraic construct from Landweber flatness.

Twisted Morava K-theory and E-theory

For an integral cohomology class H of degree n+2 on a space X, we define twisted Morava K-theory K(n)(X; H) at the prime 2, as well as an integral analogue. We explore properties of this twisted cohomology theory, study a twisted Atiyah-Hirzebruch spectral sequence, and give a universal coefficient theorem (in the spirit of Khorami). We extend the construction to define twisted Morava E-theory, and provide applications to string theory and M-theory.

EQUIVARIANT ANDERSON DUALITY AND MACKEY FUNCTOR DUALITY

AbstractWe show that the$\mathbb{Z}$/2-equivariantnth integral MoravaK-theory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrumKOas a corollary. The study of$\mathbb{Z}$/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries ofRO($\mathbb{Z}$/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.

The mod 2 Hopf ring for connective Morava K-theory

This paper examines the mod 2 homology of the spaces in the Omega-spectrum for connective Morava K-theory, i.e., the mod 2 Hopf ring for connective Morava K-theory. A natural set of generators for this Hopf ring arising from the homology and homotopy of the connective Morava K-theory spectrum is calculated and the non-trivial circle product relations among the generators arising from homology and homotopy are determined. This Hopf ring calculation is accomplished using Dieudonne ring theory and Adams spectral sequences for the connective Morava K-theory of Brown–Gitler spectra.

Generalized Yang–Baxter Operators for Dieudonné Modules

An enrichment of a category of Dieudonné modules is made by considering Yang–Baxter conditions, and these are used to obtain ring and coring operations on the corresponding Hopf algebras. Some examples of these induced structures are discussed, including those relating to the Morava K-theory of Eilenberg–MacLane spaces.