On Singerʹs invariant-theoretic description of the lambda algebra: A mod p analogue Original Research Article

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-moduleM 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 E1 term of the Adams spectral sequence for spheres. Notably, we naturalize a little bit Singerʹs way to define Γ+M. In fact, we replace Singerʹs 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 Singerʹs 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.