A phenomenon of reciprocity in the universal Steenrod algebra

Abstract

In this paper we compute the cohomology algebra of certain subalgebras L r {L_r} and certain quotients K s {K_s} of the mod 2 \bmod \, 2 universal Steenrod algebra Q Q , the algebra of cohomology operations for H ∞ {H_\infty } -ring spectra (see [ M ] [\text {M}] ). We prove that \[ Ext L r ⁡ ( F 2 , F 2 ) ≅ K − k + 1 , Ext K s ⁡ ( F 2 , F 2 ) ≅ L − s + 1 \operatorname {Ext}_{{L_r}}({F_2},{F_2}) \cong {K_{ - k + 1}}, \qquad \operatorname {Ext}_{{K_s}}({F_2},{F_2}) \cong {L_{ - s + 1}} \] with r r , s s integers and r ≤ 1 r \leq 1 , s ≥ 0 s \geq 0 . We also observe that some of the algebras L r {L_r} , K s {K_s} are well known objects in stable homotopy theory and in fact our computation generalizes the fact that H ∗ ( A L ) ≅ Λ opp {H^{\ast } }({A_L}) \cong \Lambda ^{{\text {opp}}} and H ∗ ( Λ opp ) ≅ A L {H^{\ast } }({\Lambda ^{{\text {opp}}}}) \cong {A_L} (see, for instance, [ P ] [\text {P}] ). Here A L {A_L} is the Steenrod algebra for simplicial restricted Lie algebras and Λ \Lambda is the E 1 {E_1} -term of the Adams spectral sequence discovered in [ B-S ] [\text {B-S}] .