Cartier Obstructions and Frattini Quotients of Jacobian-One Formal Automorphisms

Generation in congruence quotients of Jacobian-one formal automorphism groups in positive characteristic is described in two settings. In the plane over a field k of characteristic p > 0, generation is measured relative to the elementary spine. If C_q = Lqp-1/Eqp-1 is the 2q-dimensional Cartier obstruction, K<q is generated by the spine and the lower Cartier blocks, and Gamma_q is the image of K<q intersect Gqp-1 in C_q, then dim Gamma_2 = 3 < 4 = dim C_2, dim Gamma_3 = 4 < 6 = dim C_3, and Gamma_q = C_q for q >= 4. All blocks of index at least four may therefore be deleted simultaneously. The two remaining blocks can be replaced by one mixed parameterized family, so the Cartier seed-family rank relative to the spine is one. Over the prime field F_p, in every dimension n >= 2, the Frattini quotient of T_n<N = G_1/G_N is determined explicitly. Outside the two plane exceptions (n,p) = (2,2) and (2,3), the surviving Cartier coinvariants are the constant, trace, and divergence quotients in the first three layers. In every case, all higher Cartier layers lie in the Frattini subgroup. Filtered collection and a de Rham flux homomorphism give matching upper and lower bounds. The resulting dimensions are the minimal numbers of generators of the finite prime-field tangent-jet groups.