Cartier Relation Symbols and Frattini Transgression in Special Nottingham Groups

Abstract

Let k = F_p, let V = k^n with n >= 3, and let T_N be a finite congruence quotient of the tangent-to-identity Jacobian-one automorphism group of k[[x_1,...,x_n]]. Put d_r = (p - 1)(n - 1) + pr, and let L_s be the coefficient-degree-s divergence-free polynomial vector fields. Cartier descent identifies the nonexact homogeneous layers of the associated divergence-free Lie algebra, after restriction to GL_n(F_p), with Frobenius twists of W_r = Sym^r(V*) tensor V. For p >= 5, we identify the target-three component of its first Chevalley-Eilenberg homology: H_1(L_{>=2}, direct sum_{r>=0} W_r)_3 is isomorphic to K_3 := ker(L_2 tensor V* -> Sym^2(V*)). The canonical cross-term copy of K_3 in the homology of the Frattini quotient survives in the nonlinear group: K_3 is contained in the image of H_2(T_N,k) -> H_2(T_N/Phi(T_N),k), for N > d_3. Consequently, dim_k H^2(T_N,k) >= n(n^3 + n^2 - 3n - 1)/2. The Lie-theoretic identification is obtained from an exact integral mapping-kernel complex over Z[1/6]; the group-theoretic survival follows from Cartier flux and filtered cancellation in the Frattini extension.