Singer's algebraic transfer for all ranks $s$ in some degrees and the negation of the dimension results for the graded spaces $\mathbb F_2\otimes_{\mathscr A} \mathbb F_2[x_1, \ldots, x_s]$ in degrees $s+5$
Authors: Võ Phúc Đặng 
Published: July 2022

<p>Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded module over the  mod-2 Steenrod algebra, $\mathscr {A}$. The classical "hit problem", initiated by Frank Peterson [Abstracts Amer. Math. Soc. 833 (1987), 55-89], concerned with seeking a minimal set of $\mathscr A$-module $P_s.$ Equivalently, when $\mathbb F_2$ is an $\mathscr A$-module concentrated in degree 0, one can write down explicitly a monomial basis for the $\mathbb Z$-graded vector space over $\mathbb F_2$:</p>
<p>$$ QP_s:= \mathbb F_2 \otimes_{\mathscr A} P_s = P_s/\mathscr A^+\cdot P_s,$$</p>
<p>where $\mathscr A^{+}$ denotes the augmentation ideal of $\mathscr A.$ The problem is unresolved in general. In this paper, we study the hit problem for $P_s$ with $s\geq 5.$ More explicitly, we first compute explicitly the dimension of $QP_s$ for $s = 5$ in the generic degree $21\cdot 2^{t}-5$ with $t = 1.$ Note that the problem when $t = 0$ was solved by Nguyễn Sum [Vietnam J. Math. 49 (2021), 1079-1096]. Next, we shall reject a result of Meshack Moetele and Mbakiso Fix Mothebe [East-West Journal of Mathematics 18 (2016), 151-170] on the dimension of $QP_s$ in degrees $s+5$ for $8\leq s\leq 9.$ We also give an explicit formula for the dimension of $QP_s$ in degree $14$ for all $s > 0$ and in degree $15$ for all $s > 0,\, s\neq 10.$ In addition, a conjecture on the dimension of $QP_s$ in degree $2^{s-1}-s$ has been given. As applications, we investigate William Singer's conjecture [Math. Z. 202 (1989), 493-523] on the algebraic transfer of rank $5$ in degrees $21\cdot 2^{t}-5$ for all $t\geq 0$ and of ranks $s > 0$ in internal degrees $d,\, 13\leq d\leq 15.$ This is a completely new result of proving the Singer conjecture for all ranks $s$ in certain internal degrees. In particular, our results have shown that any element in the $Sq^{0}$-families $\{\chi_t=(Sq^{0})^{t}(\chi_0)\in {\rm Ext}_{\mathscr A}^{5, 21\cdot 2^{t+1}}(\mathbb F_2, \mathbb F_2)|\, t\geq 0\}$ and $\{D_1(t)=(Sq^{0})^{t}(D_1(0))\in {\rm Ext}_{\mathscr A}^{5, 57\cdot 2^{t}}(\mathbb F_2, \mathbb F_2)|\, t\geq 0\}$ belongs to the image of the algebraic transfer of rank $5.$  We also study the behavior of the algebraic transfer of rank 7 in the generic degrees $23\cdot 2^{t}-7$ for $t = 0$ and $\ell\cdot 2^{t}-7$ for $\ell\in \{9,\, 16\},\,  t\leq 3.$ Our results then claim that the non-zero elements $Pc_0\in {\rm Ext}_{\mathscr A}^{7, 23\cdot 2^{0}}(\mathbb F_2, \mathbb F_2),$\ $k_0 = k\in {\rm Ext}_{\mathscr A}^{7, 9\cdot 2^{2}}(\mathbb F_2, \mathbb F_2)$ and $h_6D_2\in {\rm Ext}_{\mathscr A}^{7, 2^{7}}(\mathbb F_2, \mathbb F_2)$ are not in the image of the transfer, and that every indecomposable element in the $Sq^{0}$-family $\{Q_2(t) = (Sq^{0})^{t}(Q_2(0))\in {\rm Ext}^{7, 2^{t+6}}_{\mathscr A}(\mathbb F_2, \mathbb F_2):\, t\geq 0\}$ belongs to the image of the transfer. </p>