A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer
Let \({ P}^{\otimes h}: = {\mathbb F}_{2}[t_{1},t_{2},\ldots,t_{h}]\) denote the graded polynomial algebra over the prime field of two elements, \({\mathbb F}_2,\) in \(h\) variables \(t_i\) each of degree \(1.\) \({ P}^{\otimes h}\) is isomorphic to the mod \(2\) cohomology algebra of the product of \(h\) copies of the infinite dimensional real projective space \({\mathbb R}P^{\infty},\) and, as such, has a natural module structure over the mod \(2\) Steenrod algebra \(\mathcal{A}.\) Let \(GL_h\) denote the general linear group of rank \(h\) over \({\mathbb F}_2.\) In [Math. Z. 202, No. 4, 493--523 (1989; Zbl 0687.55014)] \textit{W. M. Singer} defined the algebraic transfer, which is a homomorphism \[ Tr_h^{\mathcal A} : ({\mathbb F}_2 \otimes_{GL_h} \mathrm{Ann}_{\overline{\mathcal A}}[{ P}^{\otimes h}]^*)_n \rightarrow \mathrm{Ext}_{\mathcal A}^{h,h+n}({\mathbb F}_2, {\mathbb F}_2) \] and he conjectured that it is a monomorphism for all \(h \geq 1\) and \(n \geq 0.\) The conjecture is known to be true for \(h = 1, 2, 3 \) and \(4\) but remains open for \(h \geq 5.\) The paper is concerned with the Peterson hit problem of finding a minimal set of generators for \({ P}^{\otimes h}\) as a module over \(\mathcal{A}\) or, equivalently, of finding an additive vector space basis for \({\mathbb F}_2 \otimes_{\mathcal A} { P}^{\otimes h}\) in each degree \(n \geq 0.\) The author considers the case \(h=6\) and explicitly computes the dimension of \({\mathbb F}_2 \otimes_{\mathcal A} { P}^{\otimes h}\) in degree \(n_s = 6(2^s-1) + 10 \cdot 2^s \) for all integers \(s \geq 0.\) As a consequence, he proves Singer's conjecture for the case \(h = 6\) and degree \(n_s.\)
scientific article; zbMATH DE number 7603200
scientific article