Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-19T05:59:36.178Z Has data issue: false hasContentIssue false
  • English
  • Français

A global structure theorem for the mod 2 Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo–Araki–May algebras

Published online by Cambridge University Press:  16 October 2000

DAVID J. PENGELLEY ,
FRANKLIN P. PETERSON  and
FRANK WILLIAMS
DAVID J. PENGELLEY
Affiliation:
New Mexico State University, Las Cruces, NM 88003, U.S.A.; e-mail: davidp@nmsu.edu
FRANKLIN P. PETERSON
Affiliation:
Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.; e-mail: fpp@math.mit.edu
FRANK WILLIAMS
Affiliation:
New Mexico State University, Las Cruces, NM 88003, U.S.A.; e-mail: frank@nmsu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The Dickson algebra Wn+1 of invariants in a polynomial algebra over [ ]2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ]n−2 on a fundamental class in degree 2n – a. This involves a new family of left ideals [Iscr ]a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ]n−2 in the Steenrod algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 2 , September 2000 , pp. 263 - 276
Copyright
2000 Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)