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

Abstract

The Dickson algebra Wn+1 of invariants in a polynomial algebra over F2 is an unstable algebra over the mod 2 Steenrod algebra A, or equivalently, over the Kudo-Araki-May algebra K 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 analyze the interplay of actions over A and K to characterize unstable cyclic modules with trivial action by the subalgebra An−2 on a fundamental class in degree 2 n − a, thereby verifying unstable instances of a conjectured basis for A/AAn−2. This involves a new family of left ideals Ia in K, which play the role filled by the ideals AAn−2 in the Steenrod algebra.

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.…”

“…Let K n denote the direct sum of the A-module M(n, 0) on t 2 n with the free unstable A-module on the t 2 k , k ≥ n + 1. Here M(n, 0) is as defined in [9], namely the free unstable A-module on one generator t 2 n modulo the left A-submodule generated by…”

“…1 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…”

“…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.…”

“…Finally, we shall produce an A-algebra epimorphism from S = H * (BO; F 2 ) to each of the mod two Dickson algebras (Theorem 4.4), which we characterized in [9] as unstable A-algebras. In fact we shall show that the (n + 1)-st Dickson algebra has the role of capturing precisely the quotient of S = H * (BO; F 2 ) common to the cohomology of the n-th distinct connected cover BO φ(n) and to BO(2 n+1 − 1).…”