Let A be a ring and let I be an ideal of A. Let (X, O_X) be a space with a sheaf of A-algebras (or X could be a site). Let F be a sheaf of O_X-modules. Set F_n = F/I^{n + 1}F. Then we can ask whether the theorem on formal functions holds in the naive form that

lim H^p(X, F)/I^nH^p(X, F) = lim H^p(X, F_n)

As is shown in EGA this holds if X is a proper scheme over A, the ring A is Noetherian, and F is coherent. Let me give a proof using derived completion for those who’ve not encountered this before.

STEP 1: Cohomology commutes with derived completion. If I is finitely generated, then we have

RΓ(X, F)^ = RΓ(X, F^)

where the wedge means derived I-adic completion on both sides, see Tag 0BLX

STEP 2: Derived completion versus “usual” completion. Provided that X has enough opens that look like spectra of Noetherian rings and F is coherent, then we have that F^ = Rlim F_n. See for example Tag 0A0K. Similarly, if A is Noetherian and the cohomology modules H^p(X, F) are finite A-modules, then the cohomology modules of RΓ(X, F)^ are the usual naive I-adic completions of the H^p(X, F), see Tag 0BKH.

STEP 3: RΓ(X, -) commutes with Rlim. See for example Tag 08U1 (actually we’ve already used this result in the first step).

STEP 4: The pth cohomology of Rlim RΓ(X, F_n) is lim H^p(X, F_n). To see this it suffices to show that the inverse system of modules H^p(X, F_n) has Mittag-Leffler (for all p). By Tag 0GYQ it suffices to show that

H^p(X, F_1) ⊕ H^p(X, IF_2) ⊕ H^p(X, I^2F_3) ⊕ …

satisfies the ascending chain condition as a graded module over

Gr_I(A) = A/I ⊕ I/I^2 ⊕ I^2/I^3 ⊕ …

for all p. This holds as soon as X is proper over A Noetherian and F_0 is a coherent O_X-module (by finiteness of cohomology of coherent modules over proper schemes).

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Note that if A is Noetherian and complete wrt I and X is a formal scheme proper over Spf(A) and if we have an inverse system (F_n) of coherent O_X-modules with F_{n – 1} = F_n/I^nF_n giving rise to the coherent O_X-module F = lim F_n then we obtain the same statement with the same proof (in this case completion on the left hand side of the theorem is unnecessary). Of course, it is debatable whether in this case one should really even be interested in the “usual” cohomology groups H^p(X, F)…