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April 2021 A note on the Peterson hit problem for the Steenrod algebra
Nguyen Khac Tin
Proc. Japan Acad. Ser. A Math. Sci. 97(4): 25-28 (April 2021). DOI: 10.3792/pjaa.97.005

Abstract

Let $P_{k}=\oplus_{n\geqslant 0} (P_{k})_{n} \cong \mathbf{F}_{2}[x_{1},x_{2},\ldots ,x_{k}]$ be the graded polynomial algebra over the prime field of two elements $\mathbf{F}_{2}$, in $k$ generators $x_{1}, x_{2}, \ldots , x_{k}$, each of degree 1. Being the mod-2 cohomology of the classifying space $B(\mathbf{Z}/2)^{k}$, the algebra $P_{k}$ is a module over the mod-2 Steenrod algebra $\mathcal{A}$.

In this Note, we explicitly compute the hit problem of some generic degrees $r(2^{s}-1)+2^{s}m$ in $P_{k}$, where $r=k-1=4, m \in \{8; 10; 11 \}$ and $s$ an arbitrary non-negative integer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees and in the cases $k=5$ and 6.

Citation

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Nguyen Khac Tin. "A note on the Peterson hit problem for the Steenrod algebra." Proc. Japan Acad. Ser. A Math. Sci. 97 (4) 25 - 28, April 2021. https://doi.org/10.3792/pjaa.97.005

Information

Published: April 2021
First available in Project Euclid: 13 April 2021

Digital Object Identifier: 10.3792/pjaa.97.005

Subjects:
Primary: 55S05, 55S10, 55T15

Rights: Copyright © 2021 The Japan Academy

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Vol.97 • No. 4 • April 2021
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