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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
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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

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