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A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer
  • Đặng Võ Phúc
Đặng Võ Phúc
University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa

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Let $\mathcal A$ be the classical, singly-graded Steenrod algebra over the prime order field $\mathbb F_2$ and let $P^{\otimes h}: = \mathbb F_2[t_1, \ldots, t_h]$ denote the polynomial algebra on $h$ generators, each of degree $1$, viewed as a module over $\mathcal A.$ Write $GL_h$ for the usual general linear group of rank $h$ over $\mathbb F_2.$ As well known,  the (mod 2) cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A}^{h, h+*}(\mathbb F_2, \mathbb F_2)$ is still largely mysterious for all homological degrees $h \geq 6.$ The $h$-th algebraic transfer $$Tr_h^{\mathcal A}: (\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{*})_n\longrightarrow {\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2),$$ defined by William Singer \cite{Singer}, is a helpful tool to describe that Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains open for any $h\geq 5.$ There is currently no information on the conjecture for $h = 6$. In this paper, we verify Singer’s conjecture for all homological degrees $h\geq 1$ in the internal degrees $\leq 10 = 6(2^{0}-1) + 10\cdot 2^{0}$ and for $h = 6$ in the degree of the general form $n_s:=6(2^{s}-1) + 10\cdot 2^{s}.$ This result is important, since it tells us that the non-zero elements $h_2^{2}g_1 = h_4Ph_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_1}(\mathbb F_2, \mathbb F_2)$, and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2)$ are not in the image of the sixth algebraic transfer. This Note is a continuation of our previous one \cite{Phuc11}, which will refer to as Part I. Our approach is based on explicitly solving the hit problem for the Steenrod algebra in the case $h = 6$ and degree $n_s.$ This extends a result in Mothebe, Kaelo and Ramatebele \cite{MKR}. At the same time, we also use this obtained result to establish the dimension result for the space of the unhit elements, $\mathbb F_2\otimes_{\mathcal A}P^{\otimes 7}$ in a certain general degree.
Jan 2023Published in Journal of Algebra volume 613 on pages 1-31. 10.1016/j.jalgebra.2022.08.028