Singer’s algebraic transfer for all ranks s in some degrees and the negation of the dimension results for the graded spaces F2 ⊗ AF2 [x1,...,xs] in degrees s+ 5

Abstract

<p>Let Ps:=F2[x1,x2,...,xs] be the graded polynomial algebra over the prime field of two elements,F2, insvariables x1,x2,...,xs, each of degree one. This algebra is considered as a graded moduleover the mod-2 Steenrod algebra, A. The classical "hit problem", initiated by Frank Peterson[Abstracts Amer. Math. Soc. 833 (1987), 55-89], concerned with seeking a minimal set of A-module Ps. Equivalently, when F2 is an A-module concentrated in degree 0, one can write downexplicitly a monomial basis for theZ-graded vector space overF2:</p> <p> QPs:=F2⊗APs=Ps/A+·Ps,</p> <p>where A+ denotes the augmentation ideal of A.The problem is unresolved in general. In thispaper, we study the hit problem for Ps with s>5. More explicitly, we first compute explicitly thedimension of QPs for s= 5 in the generic degree 21·2t−5witht= 1. Note that the problem when t= 0 was solved by Nguyễn Sum [Vietnam J. Math. 49 (2021), 1079-1096]. Next, we shall reject aresult of Meshack Moetele and Mbakiso Fix Mothebe [East-West Journal of Mathematics 18 (2016),151-170] on the dimension of QPs in degreess+ 5 for 8< s<9. We also give an explicit formula forthe dimension of QPs in degree 14 for alls >0 and in degree 15 for all s >0, s,10. In addition, a conjecture on the dimension of QPs in degree 2s−1−s has been given. As applications, weinvestigate William Singer’s conjecture [Math. Z. 202 (1989), 493-523] on the algebraic transfer of rank 5 in degrees 21·2t−5 for all t>0 and of ranks s >0 in internal degreesd, 130}and{D1(t) = (Sq0)t(D1(0))∈Ext5,57·2tA(F2,F2)|t>0} belongs to theimage of the algebraic transfer of rank 5. We also study the behavior of the algebraic transfer of rank 7 in the generic degrees 23·2t−7 fort= 0and`·2t−7for`∈ {9,16}, t63. Our results then claim that the non-zero elements Pc0∈Ext7,23·20A(F2,F2), k0=k∈Ext7,9·22A(F2,F2)and h0} belongs to the image of the transfer.</p>

On modules over the mod 2 Steenrod algebra and hit problems

“…As applications, we establish the dimension result for the cohit F 2 -module Q ⊗6 in generic degree 5(2 s+4 −1)+n 1 .2 s+4 and examine Conjecture 1.1 for bidegrees (5, 5 + n s ) with s ≥ 0. Next, as a continuation of the previous works in [ Boa93,Sum15,Phu23b,Phu22a,Phu22b,Phu22c,Phu22d], we shall investigate Conjecture 1.1 in bidegree (m, m + n) for n = 12 and arbitrary m > 0. The cases of ranks m = 1, 2, 3 are relatively easy to compute (cf.…”

“…Although Z is the minimal spike, ω(X) = (5, 0, 0, 4, 0) > ω(Z) = (3, 3, 1, 1, 1). The reader is referred to [Phu22c] for further information about a basis of Q ⊗5 37 .…”

On modules over the mod 2 Steenrod algebra and hit problems

“…Despite the fact that Z is the minimal spike, it can be observed that ω(X) = (5, 0, 0, 4, 0) > ω(Z) = (3, 3, 1, 1, 1). For further details about a basis of Q ⊗5 37 , we refer the reader to [Phu22c].…”