On double Poisson structures on commutative algebras

Double Poisson structures (a la Van den Bergh) on commutative algebras are studied; the main result shows that there are no non-trivial such structures on polynomial algebras of Krull dimension greater than one. For a general commutative algebra A, this places significant restrictions on possible double Poisson structures. Exotic double Poisson structures are exhibited by the case of the polynomial algebra on a single generator, previously considered by Van den Bergh.Comment: 12 pages; very minor revisio

On the derived functors of destabilization at odd primes

An explicit chain complex is constructed to calculate the derived functors of destabilization at an odd prime, generalizing constructions of Zarati and of Hung and Sum. The methods are based on the ideas of Singer and Miller and also apply at the prime two. A structural result on the derived functors of destabilization is deduced.Comment: Significant revision (27 pages); accepted for publication in Acta Mathematica Vietnamica (AMV

Truncated projective spaces, Brown-Gitler spectra and indecomposable A(1)-modules

A structure theorem for bounded-below modules over the subalgebra A(1) of the mod 2 Steenrod algebra generated by Sq^1, Sq^2 is proved; this is applied to prove a classification theorem for a family of indecomposable A(1)-modules. The action of the A(1)-Picard group on this family is described, as is the behaviour of duality. The cohomology of dual Brown-Gitler spectra is identified within this family and the relation with members of the A(1)-Picard group is made explicit. Similarly, the cohomology of truncated projective spaces is considered within this classification; this leads to a conceptual understanding of various results within the literature. In particular, a unified approach to Ext-groups relevant to Adams spectral sequence calculations is obtained, englobing earlier results of Davis (for truncated projective spaces) and recent work of Pearson (for Brown-Gitler spectra).Comment: Revision - mathematical content essentially unchanged (39 pages

On the derived functors of destabilization and of iterated loop functors

These notes explain how to construct small functorial chain complexes which calculate the derived functors of destabilization (respectively iterated loop functors) in the theory of modules over the mod 2 Steenrod algebra; this shows how to unify results of Singer and of Lannes and Zarati.Comment: Minor revision; more detail provided on the Lannes-Zarati morphism and a new section giving perspectives. 32 pages. (v2 Minor revision. 25 pages.

Essential extensions, the nilpotent filtration and the Arone-Goodwillie tower

The spectral sequence associated to the Arone-Goodwillie tower for the n-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod 2 Steenrod algebra, together with properties of a generalized class of almost unstable modules which is introduced here. An essential ingredient of the proof is a non-vanishing result for certain extension groups in the category of unstable modules localized away from nilpotents.Comment: Minor revision. Sketch of case p odd added. This version 27 page

On the double transfer and the f-invariant

The purpose of this paper is to investigate an algebraic version of the double complex transfer, in particular the classes in the two-line of the Adams-Novikov spectral sequence which are the image of comodule primitives of the MU-homology of the product of two copies of infinite complex projective space via the algebraic double transfer. These classes are analysed by two related approaches; the first, p-locally for an odd prime, by using the morphism induced in MU-homology by the chromatic factorization of the double transfer map together with the f'-invariant of Behrens (for p>=5). The second approach uses the algebraic double transfer and the f-invariant of Laures.Comment: 25 pages (revised