A Direct Proof of the Theorem on Formal Functions

We give a direct and elementary proof of the theorem on formal functions by studying the behaviour of the Godement resolution of a sheaf of modules under completion.


Introduction
Let π : X → Spec A be a proper scheme over a ring A. Let M be a coherent O Xmodule and Y ⊂ Spec A a closed subscheme. Let us denote by ∧ the completion along Y (respectively, along π −1 (Y )). The theorem on formal functions states that Two important corollaries of this theorem are Stein's factorization theorem and Zariski's Main Theorem ( [H] III,11.4,11.5).
Hartshorne [H] gives a proof of the theorem on formal functions for projective schemes (over a ring). Grothendieck [G] proves it for proper schemes. He first gives sufficient conditions for the commutation of the cohomology of complexes of A-modules with inverse limits (0, 13.2.3 [G]); secondly, he gives a general theorem on the commutation of the cohomology of sheaves with inverse limits (0, 13.3.1 [G]); finally, he laboriously checks that the theorem on formal functions is under the hypothesis of this general one (4.1.5 [G]).
In this paper we give the "obvious direct proof" of the theorem on formal functions. Very briefly, we prove that the completion of the Godement resolution of a coherent sheaf is a flasque resolution of the completion of the coherent sheaf and that taking sections in the Godement complex commutes with completion. Every quasi-coherent module is affinely p-acyclic. Notations: For any sheaf F , let us denote

Theorem on formal functions
Lemma 3. Let X be a scheme, p a coherent ideal and M an O X -module. Denote I = Γ(X, p) and assume that p is generated by a finite number of global sections (this holds for example when X is affine). For any open subset V ⊆ X one has Proof. If J is a finitely generated ideal of a ring A and M i is a collection of Amodules, then J · M i = (J · M i ). Now, by hypothesis p is generated by a finite number of global sections f 1 , . . . , f r . Let J = (f 1 , . . . , f r ). Then Proposition 4. Let X be a scheme and let p be a coherent ideal. For any O Xmodule M one has: (1) Proof. 1. We may assume that X is affine. Hence pC 0 M = C 0 (pM) by the previous lemma and and the isomorphism C 0 M/pC 0 M = C 0 (M/pM) it follows that M 1 /pM 1 = (M/pM) 1 and pM 1 = (pM) 1 . Consequently pC 1 M = pC 0 (M 1 ) = C 0 (pM 1 ) = C 0 ((pM) 1 ) = C 1 (pM), and analogously C 1 M/pC 1 M = C 1 (M/pM). Repeating this argument one concludes 1.
2. Denote N = C 0 M. By (1), N /p n N is acyclic on any open subset. From the long exact sequence of cohomology associated to 0 → p n N → N → N /p n N → 0 and the acyclicity of p n N (by (1)) one obtains that Moreover, if U is affine Γ(U, p n N ) = p n (U )Γ(U, N ), by Lemma 3. We have concluded.
3. Let us prove that N = C 0 M is flasque. It suffices to prove that its restriction to any affine open subset is flasque, so we may assume that X is affine. Let us denote I = p(X). For any open subset V , one has as in the proof of (2) and by Lemma 3, Γ(V, p n N ) = I n Γ(V, N ). In conclusion, Γ(V, N ) = Γ(V, N ) c . One concludes that N is flasque because N is flasque and the I-adic completion preserves surjections. The same arguments prove the second part of the satement.

Proposition 5. If M is affinely p-acyclic, then C · M is a flasque resolution of M.
Proof. We already know that C · M is flasque. Let us prove now that M 1 is affinely p-acyclic. Remark 6. In the proof of the preceding proposition it has been proved that if M is affinely p-acyclic, then M is acyclic on any affine subset. Theorem 8 (on formal functions ). Let f : X → Y be a proper morphism of locally noetherian schemes, p a coherent sheaf of ideals on Y and pO X the ideal induced in X. For any coherent module M on X, the natural morphisms (where completions are made by p and pO X respectively) Proof. The question is local on Y , so we may assume that Y = Spec A is affine. It suffices to show that H i (X, M) c = H i (X, M). It is clear that pO X is generated by its global sections. As usual, we denote I = Γ(X, pO X ).
Let C · M be the Godement resolution of M. Then C · M is a flasque resolution of M (by Proposition 5) and Γ(X, C · M) = Γ(X, C · M) c (by Proposition 4, (3)). Then we have to prove that the natural map is an isomorphism. Let us denote by d i the differential of the complex Γ(X, C · M) on degree i. Completing the exact sequences because, as we shall see below, the I-adic topology of Γ(X, C i M) induces in Ker d i the I-adic topology. Hence Let us prove that the I-adic topology of Γ(X, C i M) induces the I-adic topology on Ker d i = Γ(X, M i ). Intersecting the equality I n Γ(X, C 0 M i ) = Γ(X, C 0 (p n M i )) with Γ(X, M i ), one obtains that the induced topology on Γ(X, M i ) is given by the filtration {Γ(X, p n M i )}. Hence it suffices to show that this filtration is I-stable. Since p n M i = (p n M) i (see the proof of 4 1.), it is enough to prove that the filtration {Γ(X, (p n M) i )} is I-stable; this is equivalent to show that ⊕ ∞ n=0 Γ(X, (p n M) i ) is a D I A-module generated by a finite number of homogeneous components, where H i (X, p n M) → 0 it suffices to see the statement for the first and the third members. For the first one is obvious because Γ(X, C i−1 (p n M)) = I n Γ(X, C i−1 M). For the third one, it suffices to see that it is a finite D I A-module. Let X ′ = X × A D I A, π : X ′ → X the natural projection and M ′ = ∞ ⊕ n=0 p n M the obvious O X ′ -module. Since Remark 9. Reading carefully the above proof, it is not difficult to see that one has already showed that H i (X, M) ∧ = lim ← n H i (X, M/p n M).