A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

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

“…It is well-known that by Peterson [32], Kameko [22], Sum [53], Mothebe [27] and the author [42] 2 and a result in our previous work [42], we claim that dim (QP 5 ) 2 5−1 −5 = 60 and dim (QP 6 ) 2 6−1 −6 = 4056. Then, by Theorem 1.2 and the results in [32], [22], [53], [42], we notice that…”

“…Then, (QP r ) 2 r−1 −r is an F 2 -vector subspace of (QP r ) 2 r−1 −r . Basing on the data and the previous results by Peterson [32], Kameko [22], Sum [53], Mothebe [27] and the present writer [42], we propose the following.…”

“…al. [27,28,29,30], Walker and Wood [60], the present author [34,35,36,37,39,40,42,43,44], Sum [51,52,56,57] and others), but it was a well known unresolved problem for s 5. The problem was systematically studied for s 4 (see [4,22,32,53]).…”

“…Noting that C(0) and D 2 are the indecomposable elements in the sixth cohomology groups Ext 6,56 A (F 2 , F 2 ) and Ext 6,64 A (F 2 , F 2 ), respectively. We also have shown in [42] that D 2 is not in the image of (ϕ 6 ) * . In [16], Chơn and Hà proved that the elements k t ∈ Ext 7,7+d t+2 A (F 2 , F 2 ), t 1 are in the image of (ϕ 7 ) * .…”

“…Thus, we have seen that Singer's transfer is highly non-trivial. Since then, it has been extensively studied by many algebraists like Bruner, Hà and Hưng [9], Chơn and Hà [14,15,16], Hà [17], Hưng [18], Nam [31], the present author [34,36,37,38,39,40,41,42,43,44], Quỳnh [46], Sum [52,55,56,57] and others. Remarkably, in higher rank, Singer states that the transfer is not a monomorphism in bidegree (5,14) and gives the following prediction.…”