Prove
Lemma 19 ([Two comparison surjections]). The following initial-page tests give…Proposition 18 ([All later obstruction candidates]). For k≥2 the complete…Proposition
The inclusion of the Frobenius subring makes P_r free of rank p^r , with basis t_1^ e_1 ⋯t_r^ e_r , 0≤e_i<p . It is therefore faithfully flat. Its kernel has coordinate algebra O(K_r)=𝔽_p[e_1,…,e_r]/(e_1^p,…,e_r^p). All the mixed terms in its coproduct vanish, so each e_i is primitive. This proves the description of the kernel and the exactness of . Here is the acyclicity calculation needed to derive invariants. Right multiplication by a kernel point is, by , simply t_i↦t_i+e_i : every mixed term contains a positive p -power of a kernel coordinate. Hence multiplication gives an isomorphism of K_r -comodules O(Q_r)⊗O(K_r) ≅O(G_r), f⊗e_1^ j_1 ⋯e_r^ j_r f t_1^ j_1 ⋯t_r^ j_r , 0≤j_i<p. The first tensor factor has trivial K_r -coaction. The second is the regular comodule. In particular, O(G_r)^ K_r =O(Q_r) , with its regular quotient coaction. A cofree comodule is injective: its cofree functor is right adjoint to the exact forgetful functor to vector spaces. Equation shows that every cofree G_r -comodule is K_r -acyclic and that its K_r -invariants are cofree over Q_r . Every injective is a summand of a cofree comodule, so these assertions also hold for injectives. Finally, M^ G_r =(M^ K_r )^ Q_r . Resolving these two invariants functors successively, using the indicated cofree resolutions, gives . This argument derives K_r -invariants; it does not assert their exactness on arbitrary modules. The spectral sequence is first quadrant, so each fixed total cohomological degree has a finite diagonal. All coordinate weights are positive and each weight component is finite dimensional. These observations give the asserted convergence, also weight by weight.