2026/02/28 by Dang Vo Phuc
#math.AT
The motivic hit problem asks for a minimal set of module generators of H*,*(BVn;\mathbb F2) over the mod~2 motivic Steenrod algebra. Kameko proved that the motivic Peterson-type analogue of Wood's theorem fails by constructing monomials zk which are not hit even when the corresponding topological degree may satisfy β(d)>n. His proof passes to Nn=Mn/(τ) and analyzes, in degree d=k+2d1 with d1=(n-1)(2k-1), a distinguished summand whose basis consists of the monotone translates of zk. In this work, we isolate the local content of this summand before quotienting by hit elements. More precisely, we construct a linear projection ϑ:Nnd,*\longrightarrow V, where V is the M1--summand spanned by the images of the monomials σ(zk), and define a parity functional ε:V→\mathbb F2 by summing the coefficients of these basis vectors. We prove that the local image of the hit subspace is exactly the parity-zero hyperplane: ϑ(A^\sharp+(Nn)∩ Nnd,*)=ker(ε). Consequently, every element whose local M1--component has odd parity is non-hit, and every odd-parity linear combination of the monotone translates of zk determines a nonzero class in the motivic hit quotient. We also obtain a systematic arithmetic family. For every integer m≥ 3, set n=2r+1 and k=n-m. If r≥ m+α(m-3), then the degree d=(n-1)(2k+1-2)+k satisfies β(d)>n. Hence, for every fixed m≥ 3, these classes give infinitely many motivic Peterson-type counterexamples with k=n-m. The local parity theorem holds over every algebraically closed field of characteristic different from 2, and naturality under extension of the base field carries its non-hit consequences to every field of characteristic different from 2.