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

Abstract

(Volume 129 (2000), 263–275)It is with very great regret that we record the death of one of the authors, Frank Peterson, shortly before the appearance of the paper. The Authors and Editor wish to express their great personal and professional respect for him.

Global structure of the mod two symmetric algebra,H∗(BO; 𝔽2), over the Steenrod algebra

“…To achieve this we show that the epimorphism G → S induces a monomorphism QG → QS , on the indecomposable quotients, by computing a basis for QG . For this we appeal to our earlier understanding [9], via the Kudo-Araki-May algebra K [10] (see Appendix II), of bases for the unstable cyclic A-modules arising in the analogous structure theorem for the Dickson algebras. With QG → QS an isomorphism, G → S must be an isomorphism also, since S is a free commutative algebra.…”

“…We continue our study [9] of invariant algebras as unstable algebras over the Steenrod algebra A by proving a structure theorem for the algebra S of symmetric invariants over the field F 2 . The algebra S is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles [8], and we identify the two.…”

“…M(p, 1) is defined in[9] as the quotient of the free unstable A-module on a class in degree 2 p − 1 modulo the action of Sq 2 i for i ≤ p − 2; in other words, in usual notation, M(p, 1) = F (2 p − 1) /AA p−2 .Algebraic & Geometric Topology, Volume 3(2003) …”

The odd-primary Kudo–Araki–May algebra of algebraic Steenrod operations and invariant theory

“…We have also used this structure in work to appear [16] to give a minimal presentation for the mod p cohomology of CP(∞) as a module over the Steenrod algebra, which in turn allows us to give a minimal presentation of the cohomology of the classifying space BU (ie the algebra of symmetric invariants) as an algebra over the Steenrod algebra. Corresponding results at the prime 2, some joint with Peterson, appear in [15,17,18]. In [17] we analyzed the analogous algebra of operations D i at the prime 2 and named it the Kudo-Araki-May algebra K. Where results in this paper are completely analogous to those in [17], we shall omit their proofs; proofs that are not analogous or immediate will be given in subsequent sections.…”

“…Theorem 1. 15 The diagonal maps ∆ in U and U respect the relations, and hence K and K inherit the structure of bialgebras.…”