# Morava K-theories and localisation

@article{Hovey1999MoravaKA, title={Morava K-theories and localisation}, author={Mark Hovey and Neil P. Strickland}, journal={Memoirs of the American Mathematical Society}, year={1999}, volume={139}, pages={0-0} }

Introduction Basic definitions $E$ theory $K$-injective spectra Generalised Moore spectra Bousfield classes The $E(n)$-local category General properties of the $K(n)$-local category Smallness and duality Homology and cohomology functors Brown-Comenetz duality The natural topology Dualisable spectra $K$-nilpotent spectra Grading over the Picard group Examples Questions and conjectures Completion Small objects in other categories References Index.

## 243 Citations

On localization sequences in the algebraic K-theory of ring spectra

- Mathematics
- 2014

We identify the $K$-theoretic fiber of a localization of ring spectra in terms of the $K$-theory of the endomorphism algebra spectrum of a Koszul-type complex. Using this identification, we provide a…

Commuting unbounded homotopy limits with Morava K-theory

- Mathematics
- 2020

This paper provides conditions for Morava K-theory to commute with certain homotopy limits. These conditions extend previous work on this question by allowing for homotopy limits of sequences of…

Multicurves and Equivariant Cohomology

- Mathematics
- 2011

Let $A$ be a finite abelian group. The author sets up an algebraic framework for studying $A$-equivariant complex-orientable cohomology theories in terms of a suitable kind of equivariant formal…

Brown–Peterson cohomology from Morava $E$ -theory

- MathematicsCompositio Mathematica
- 2017

We prove that the $p$ -completed Brown–Peterson spectrum is a retract of a product of Morava $E$ -theory spectra. As a consequence, we generalize results of Kashiwabara and of Ravenel, Wilson and…

Vanishing results for chromatic localizations of algebraic $K$-theory

- Mathematics
- 2020

We show that algebraic $K$-theory preserves $n$-connective $L_{n}^{f}$-equivalences between connective ring spectra, generalizing a result of Waldhausen for rational algebraic $K$-theory to higher…

Operations and co-operations in Morava $E$-theory

- Mathematics
- 2004

Let E = En denote the Morava E-theory spectrum, and let i be the Morava stabilizer group of ring spectrum isomorphisms of E. We revisit the isomorphism …⁄LK(n)(E ^ E) » C(i;E⁄) of graded formal Hopf…

Affineness and chromatic homotopy theory

- Mathematics
- 2015

Given an algebraic stack X, one may compare the derived category of quasi‐coherent sheaves on X with the category of dg‐modules over the dg‐ring of functions on X. We study the analogous question in…

Completed power operations for Morava E-theory

- Mathematics
- 2015

We construct and study an algebraic theory which closely approximates the theory of power operations for Morava E-theory, extending previous work of Charles Rezk in a way that takes completions into…

Morava K-theory and Filtrations by Powers

- Mathematics
- 2021

We prove the convergence of the Adams spectral sequence based on Morava Ktheory and relate it to the filtration by powers of the maximal ideal in the Lubin–Tate ring through a Miller square. We use…

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Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the…

Complex Cobordism and Stable Homotopy Groups of Spheres

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An introduction to the homotopy groups of spheres Setting up the Adams spectral sequence The classical Adams spectral sequence $BP$-theory and the Adams-Novikov spectral sequence The chromatic…

Life after the Telescope Conjecture

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We discuss the chromatic filtration in stable homotopy theory and its connections with algebraic K-theory, specifically with some results of Thomason, Mitchell, Waldhausen and McClure-Staffeldt. We…

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THE MORAVA K-THEORIES OF EILENBERG-MACLANE SPACES AND THE CONNER-FLOYD CONJECTURE

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Introduction. Of the many generalized homology theories available, very few are computable in practice except for the simplest of spaces. Standard homology and K-theory are the only ones which can be…

Morava Hopf algebras and spaces K ( n ) equivalent to finite Postnikov systems

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We have three somewhat independent sets of results. Our first results are a mixed blessing. We show that Morava K-theories don’t see k-invariants for homotopy commutative H-spaces which are finite…

The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory

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The geometry of the Lubin-Tate space of deformations of a formal group is studied via an \'etale, rigid analytic map from the deformation space to projective space. This leads to a simple description…

Bousfield Localization Functors and Hopkins' Chromatic Splitting Conjecture

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This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be p-local for a prime p throughout this paper. Recall that if E is a…