On Singer's invariant-theoretic description of the lambda algebra: A mod p analogue

Abstract

Let p be an odd prime. The purpose of the paper is to give a mod p analogue for the Singer invariant-theoretic description of the lambda algebra. In other words, we give an invariant-theoretic interpretation for the homology of the mod p Steenrod algebra A. More precisely, we associate to any left A-module M a chain complex Γ + M whose homology is isomorphic to Tor ∗ A( Z/p, M) . This chain complex is built from the Dickson-Mùi invariants of the general linear group GL(n, Z/p) for n > 0. Particularly, in the case M = Z/p , the complex Γ + = Γ + Z/p is dual to the lambda algebra, which is the E 1 term of the Adams spectral sequence for spheres. Notably, we naturalize a little bit Singer's way to define Γ + M. In fact, we replace Singer's Γ + M by its image under the so-called total power. Consequently, the action of A on the new Γ + M is diagonal, meanwhile that on Singer's Γ + M is not. Also, the differential of the new Γ + M becomes simpler. This naturalization is valid for p = 2 as well as for p an odd prime.