Related papers: Comparison of Morava E-theories
We compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups.
We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(\mathbb{G}_2, E_t)$, at $p=2$, for $0\leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same…
The Morava stabilizer groups play a dominating role in chromatic stable ho-motopy theory. In fact, for suitable spectra X, for example all finite spectra, thechromatic homotopy type of X at chromatic level n \textgreater{} 0 and a given…
We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with action of Morava stabilizer group G_n can be recovered from the system of some height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite…
Let $k$ be a perfect field of characteristic $p$. Associated to any (1-dimensional, commutative) formal group law of finite height $n$ over $k$ there is a complex oriented cohomology theory represented by a spectrum denoted $E(n)$ and…
Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava stabilizer group $\Gamma$. We study the derived stack $\operatorname{Spf} E/\Gamma$. Descent-theoretic techniques allow us to deduce a theorem of…
This is an announcement of some new computational methods in stable homotopy theory, in particular, methods for using the cohomology of small-height Morava stabilizer groups to compute the cohomology of large-height Morava stabilizer…
We calculate the cohomology of the extended Morava stabilizer group of height $n$, with trivial mod $p$ coefficients, for all heights $n$ and all primes $p>>n$. The result is an exterior algebra on $n$ generators. A brief sketch of the…
There is a natural action of a kind of Hecke algebra $\mathcal{H}_n$ on the $n$th Morava $E$-theory of spaces. We construct Hecke operators in an amalgamated cohomology theory of the $n$th and the $(n+1)$st Morava $E$-theories. These…
We construct a ``logarithmic'' cohomology operation on Morava E-theory, which is a homomorphism defined on the multiplicative group of invertible elements in the ring E^0(K) of a space K. We obtain a formula for this map in terms of the…
Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_*(e_n;\mathbb{F}_p)$ for any prime $p$ and $n \leq 4$ up to possible multiplicative extensions. In order to…
Working at the prime $2$ and chromatic height $2$, we construct a finite resolution of the homotopy fixed points of Morava $E$-theory with respect to the subgroup $\mathbb{G}_2^1$ of the Morava stabilizer group. This is an upgrade of the…
Let n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment pi_*(L_{K(n)}(X)) and E_2-term equal to the…
We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an…
For cohomology theories closely related to Morava E-theory, we provide an algebro-geometric interpretation of the cohomology of groups that arise as centralizers of tuples of commuting elements inside of symmetric groups. The interpretation…
We prove that the Morava-$K$-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a $p$-local finite Postnikov system with vanishing $(n+1)$st homotopy group.
In this note, we show how a continuous action of the Morava stabilizer group $\mathbb G_n$ on the Lubin-Tate spectrum $E_n$, satisfying the conclusion $E_n^{h\mathbb G_n}\simeq L_{K(n)} S$ of the Devinatz-Hopkins Theorem, may be obtained by…
We compute the MU-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n+1})$, of all $\mathbb{E}_1$-MU-algebra forms of connective Morava K-theory k(n). As qualitative consequences, we deduce the Lichtenbaum--Quillen conjecture, telescope…
We calculate an explicit closed formula for the action of the height 2 full Morava stabilizer group on the coefficient ring of height 2 Morava E-theory. In particular, this yields an explicit, surprisingly simple closed formula for the…
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 Ravenel-Wilson-Yagita and Kashiwabara from spaces to spectra and deduce that the…