On the 𝓐-generators of the polynomial algebra as a module over the Steenrod algebra, I

Abstract

Let 𝓟 n := H *((ℝP ∞) n ) ≅ ℤ2[x 1, x 2, …, x n ] be the graded polynomial algebra over ℤ2, where ℤ2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 𝓟 n , viewed as a graded left module over the mod-2 Steenrod algebra, 𝓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2 r − 1) + 4.2 r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 𝓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗𝓐𝓟 n . The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗𝓐𝓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2 r − 1) + 4.2 r is also discussed at the end of this paper.