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Singer’s algebraic transfer for all ranks s>5 and the negation of the dimension results for the graded spaces F2⊗A F2[x1,..
  • 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

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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 $\mathbb F_2$:
$$ QP_s:= \mathbb F_2 \otimes_{\mathscr A} P_s = P_s/\mathscr A^+\cdot P_s,$$
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 N. Sum [Vietnam J. Math. 49 (2021), 1079-1096]. Next, we study the dimension of $QP_s$ in degrees $s+5$ for $8\leq s\leq 9.$ This study corrects some results in Moetele and Mothebe’s paper [East-West Journal of Mathematics 18 (2016), 151-170]. 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 $s > 0,\, s\neq 10.$ 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 $5.$ For higher ranks, we explore 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.