Lubin-Tate Spectrum Research Articles

Chromatic Cardinalities via Redshift

Abstract Using higher descent for chromatically localized algebraic $K$-theory, we show that the higher semiadditive cardinality of a $\pi $-finite $p$-space $A$ at the Lubin–Tate spectrum $E_{n}$ is equal to the higher semiadditive cardinality of the free loop space $LA$ at $E_{n-1}$. By induction, it is thus equal to the homotopy cardinality of the $n$-fold free loop space $L^{n} A$. We explain how this allows one to bypass the Ravenel–Wilson computation in the proof of the $\infty $-semi-additivity of the ${T(n)} $-local categories. “Cardinalis sinuatus – the Pyrrhuloxia” by Dick Culbert licensed under CC BY 2.0.

Higher semiadditive algebraic K-theory and redshift

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$- and $\mathrm {T}(n)$-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$, which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$, we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$-localized K-theory, showing that they coincide for any $p$-invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$.

Algebraic theories of power operations

AbstractWe develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding well‐behaved theories of power operations for ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstruction‐theoretic machinery, we obtain tools for computing with algebras over and over Lubin–Tate spectra. As an application, we demonstrate the existence of periodic complex orientations at heights .

Dualizing spheres for compact p-adic analytic groups and duality in chromatic homotopy

The primary goal of this paper is to study Spanier–Whitehead duality in the K(n)-local category. One of the key players in the K(n)-local category is the Lubin–Tate spectrum \(E_n\), whose homotopy groups classify deformations of a formal group law of height n, in the implicit characteristic p. It is known that \(E_n\) is self-dual up to a shift; however, that does not fully take into account the action of the automorphism group \({\mathbb {G}}_n\) of the formal group in question. In this paper we find that the \({\mathbb {G}}_n\)-equivariant dual of \(E_n\) is in fact \(E_n\) twisted by a sphere with a non-trivial (when \(n>1\)) action by \({\mathbb {G}}_n\). This sphere is a dualizing module for the group \({\mathbb {G}}_n\), and we construct and study such an object \(I_{{\mathcal {G}}}\) for any compact p-adic analytic group \({\mathcal {G}}\). If we restrict the action of \({\mathcal {G}}\) on \(I_{{\mathcal {G}}}\) to certain type of small subgroups, we identify \(I_{{\mathcal {G}}}\) with a specific representation sphere coming from the Lie algebra of \({\mathcal {G}}\). This is done by a classification of p-complete sphere spectra with an action by an elementary abelian p-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the K(n)-local Spanier–Whitehead duals of \(E_n^{hH}\) for select choices of p and n and finite subgroups H of \({\mathbb {G}}_n\).

Brauer groups and Galois cohomology of commutative ring spectra

In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on$\infty$-categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra$E$, we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of$\pi _0(E)$, recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum$KU$are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization$KU[1/2]$is$\mathbb {Z}/8\times \mathbb {Z}/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum$KO$which become Morita-trivial$KU$-algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’$KO$-algebra with the same coefficient ring as$\mathrm {End}_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of$KU$, previously studied by Mathew and Stojanoska.

Galois theory and Lubin-Tate cochains on classifying spaces

Abstract We consider brave new cochain extensions F(BG +,R) → 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 extension because it ramifies. As a consequence, it follows that the E n-theory Eilenberg-Moore spectral sequence for G and BGdoes not always converge to its expected target.

Every $K(n)$–local spectrum is the homotopy fixed points of its Morava module

Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to \pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).

Galois extensions of Lubin-Tate spectra

Let En be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E nr n whose coecients are built from the coecients of En 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 with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context.

Uniqueness of $E_\infty$ structures for connective covers

We refine our earlier work on the existence and uniqueness of E∞ structures on K- theoretic spectra to show that at each prime p, the connective Adams summand l has a unique structure as a commutative S-algebra. For the p-completion lp we show that the McClure- Staffeldt model for lp is equivalent as an E∞ ring spectrum to the connective cover of the periodic Adams summand Lp. We establish a Bousfield equivalence between the connective cover of the Lubin-Tate spectrum En and BPh ni .

Homotopy fixed points for [formula omitted] using the continuous action

Let K ( n ) be the nth Morava K-theory spectrum. Let E n be the Lubin–Tate spectrum, which plays a central role in understanding L K ( n ) ( S 0 ) , the K ( n ) -local sphere. For any spectrum X, define E ∨ ( X ) to be L K ( n ) ( E n ∧ X ) . Let G be a closed subgroup of the profinite group G n , the group of ring spectrum automorphisms of E n in the stable homotopy category. We show that E ∨ ( X ) is a continuous G-spectrum, with homotopy fixed point spectrum ( E ∨ ( X ) ) hG . Also, we construct a descent spectral sequence with abutment π * ( ( E ∨ ( X ) ) hG ) .

Invertible modules for commutative -algebras with residue fields

The aim of this note is to understand under which conditions invertible modules over a commutative Open image in new window-algebra in the sense of Elmendorf, Kriz, Mandell & May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative Open image in new window-algebra R has coherent localizations Open image in new window for every maximal ideal Open image in new window, then for every invertible R-module U, U*=π*U is an invertible graded R*-module. In some non-connective cases we can carry the result over under the additional assumption that the commutative Open image in new window-algebra has ‘residue fields’ for all maximal ideals Open image in new window if the global dimension of R* is small or if R is 2-periodic with underlying Noetherian complete local regular ring R0. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra.

Gamma homology, Lie representations and E ? multiplications

We prove that the stable homotopy of any Γ-module F is the homology of a bicomplex Ξ(F), in which the (q−1)st row is the two-sided bar construction ℬ(Lie* q ,Σ q ,F[q]). This gives a natural homotopical cotangent bicomplex for graded commutative algebras, in a form suitable for use in a new obstruction theory for classifying E ∞ ring structures on spectra. The E ∞ structure on certain Lubin-Tate spectra is a corollary.

Continuous homotopy fixed points for Lubin-Tate spectra

We provide a new and conceptually simplified construction of continuous homotopy fixed point spectra for Lubin-Tate spectra under the action of the extended Morava stabilizer group. Moreover, our new construction of a homotopy fixed point spectral sequence converging to the homotopy groups of the homotopy fixed points of Lubin-Tate spectra is isomorphic to an Adams spectral sequence converging to the homotopy groups of the spectra constructed by Devinatz and Hopkins. The new idea is built on the theory of profinite spectra with a continuous action by a profinite group.

Profinite $G$-spectra

We construct a stable model structure on profinite spectra with a continuous action of an arbitrary profinite group. The motivation is to provide a natural framework in a subsequent paper for a new and conceptually simplified construction of continuous homotopy fixed point spectra and of continuous homotopy fixed point spectral sequences for Lubin-Tate spectra under the action of the extended Morava stabilizer group.