Tyler Lawson

Commutativity conditions for truncated Brown-Peterson spectra of height 2

An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized truncated Brown-Peterson spectra of height 2 as E1-ring spectra. The criterion is checked for an example at the prime 2 derived from the universal elliptic curve equipped with a level 1(3) structure. 2000MSC: 55P42, 55P43, 55N22 and 14L05

Completed representation ring spectra of nilpotent groups

In this paper, we examine the “derived completion” of the representation ring of a pro-p group G ^ p with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg‐MacLane spectrum HZ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor RŒ c from groups to ring spectra, and show that the map RŒG ^ p c ! RŒGc becomes an equivalence after completion when G is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the p‐adic Heisenberg group. 55P60; 55P43, 19A22

Commutative ring objects in pro-categories and generalized Moore spectra

We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M J Hopkins that certain towers of generalized Moore spectra, closely related to the K.n/‐local sphere, are E1 ‐algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads. 55P43, 55U35; 18D20, 18D50, 18G55

Vanishing of some galois cohomology groups for elliptic curves

Let \(E/\mathbb {Q}\) be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of \(\mathbb {Q}\) obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group \(H^1\bigl ( G, E[p]\bigr )\) does not vanish, and investigate the analogous question for \(E[p^i]\) when \(i>1\). We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald–Wang problem for elliptic curves.

A descent spectral sequence for arbitrary K(n)-local spectra with explicit e 2-term

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 continuous cohomology of G_n, the extended Morava stabilizer group, with coefficients in a certain discrete G_n-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local E_n-Adams spectral sequence for pi_*(L_{K(n)}(X)), whose E_2-term is not known to always be equal to a continuous cohomology group.

Realizability of the adams-novikov spectral sequence for formal a-modules

We show that the formal A-module Adams-Novikov spectral sequence of Ravenel does not naturally arise from a filtration on a map of spectra by examining the case A = Z[i]. We also prove that when A is the ring of integers in a nontrivial extension of Qp, the map (L, W) → (L A , W A ) of Hopf algebroids, classifying formal groups and formal A-modules respectively, does not arise from compatible maps of E ∞ -ring spectra (MU, MU^MU)→ (R, S).

Strictly commutative complex orientation theory

For a multiplicative cohomology theory E, complex orientations are in bijective correspondence with multiplicative natural transformations to E from complex bordism cohomology MU. If E is represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientation

A note on _{∞} structures

MU \rightarrow E