The First Frattini–Commutator Layer and the Cartier Support Cone

Let k=F_p with p>=5, let V=k^n with n>=3, and let T_N be a finite congruence quotient of the tangent-to-the-identity Jacobian-one formal automorphism group, with Frattini subgroup Phi_N. We determine the first Frattini-commutator layer J_N=[T_N,Phi_N]/([T_N,[T_N,Phi_N]][T_N,Phi_N]^p). It is canonically the cokernel of H_2(T_N,k) -> H_2(T_N/[T_N,Phi_N],k). Write W_r=Sym^r(V*) tensor V and d_r=(p-1)(n-1)+pr. For N>d_4, the congruence-associated graded has gr^4 J_N isomorphic to L_4, gr^{d_2} J_N isomorphic to L_2^(1), and gr^{d_4} J_N isomorphic to Sym^3(V*)^(1), with all other graded pieces vanishing. Hence dim_k J_N=dim W_4+dim L_2=n*binom(n+3,4)+n(n-1)(n+2)/2, independently of N, and the truncation maps are eventually isomorphisms. Moreover J_N isomorphic to P_3/P_4, where P_1=T_N and P_{i+1}=P_i^p[P_i,T_N]. More generally, the Cartier component of index r in P_i/P_{i+1} vanishes whenever r>=i+2. Thus the lower p-central Cartier array is supported in the cone r