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- Title
On the 퓐-generators of the polynomial algebra as a module over the Steenrod algebra, I.
- Authors
Tin, Nguyen Khac; Dung, Phan Phuong; Ly, Hoang Nguyen
- Abstract
Let 퓟n := H*((ℝP∞)n) ≅ ℤ2[x1, x2, ..., xn] 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 퓟n, viewed as a graded left module over the mod-2 Steenrod algebra, 퓐. 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(2r − 1) + 4.2r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 퓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗퓐퓟n. The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗퓐퓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2r − 1) + 4.2r is also discussed at the end of this paper.
- Subjects
MODULES (Algebra); ALGEBRA; POLYNOMIALS; VECTOR spaces
- Publication
Mathematica Slovaca, 2024, Vol 74, Issue 3, p763
- ISSN
0139-9918
- Publication type
Article
- DOI
10.1515/ms-2024-0058