Martin Frankland

Completed power operations for Morava E–theory

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 account. These algebraic structures are made explicit in the case of K-theory. Methodologically, we emphasize the utility of flat modules in this context, and prove a general version of Lazards flatness criterion for module spectra over associative ring spectra.

Behavior of Quillen (co)homology with respect to adjunctions

This paper aims to answer the following question: Given an adjunction between two categories, how is Quillen (co)homology in one category related to that in the other? We identify the induced comparison diagram, giving necessary and sufficient conditions for it to arise, and describe the various comparison maps. Examples are given. Along the way, we clarify some categorical assumptions underlying Quillen (co)homology: cocomplete categories with a set of small projective generators provide a convenient setup.

Higher Toda brackets and the Adams spectral sequence in triangulated categories

The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B. Shipley based on J. Cohens approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipleys, and show that they are self-dual. Our main result is that the Adams differential

Moduli spaces of 2-stage Postnikov systems

d_r

The realizability of operations on homotopy groups concentrated in two degrees

in any Adams spectral sequence can be expressed as an

Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy

(r+1)

Towards the dual motivic Steenrod algebra in positive characteristic

-fold Toda bracket and as an

A user's guide: Completed power operations for Morava

r^{\text{th}}

E

order cohomology operation. We also show how the result simplifies under a sparseness assumption, discuss several examples, and give an elementary proof of a result of Heller, which implies that the three-fold Toda brackets in principle determine the higher Toda brackets.

-theory

Abstract Using the obstruction theory of Blanc, Dwyer and Goerss, we compute the moduli space of realizations of 2-stage Π-algebras concentrated in dimensions 1 and n or in dimensions n and n + 1 . The main technical tools are Postnikov truncation and connected covers of Π-algebras, and their effect on Quillen cohomology.