52.2 Formal sections, I
We suggest looking at Cohomology, Section 20.35 first.
Lemma 52.2.1. Let X be a scheme. Let \mathcal{I} \subset \mathcal{O}_ X be a quasi-coherent sheaf of ideals. Let
\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1
be an inverse system of quasi-coherent \mathcal{O}_ X-modules such that \mathcal{F}_ n = \mathcal{F}_{n + 1}/\mathcal{I}^ n\mathcal{F}_{n + 1}. Set \mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n. Then
\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n,
for any affine open U \subset X we have H^ p(U, \mathcal{F}) = 0 for p > 0, and
for each p there is a short exact sequence 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}_ n) \to H^ p(X, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n) \to 0.
If moreover \mathcal{I} is of finite type, then
\mathcal{F}_ n = \mathcal{F}/\mathcal{I}^ n\mathcal{F}, and
\mathcal{I}^ n \mathcal{F} = \mathop{\mathrm{lim}}\nolimits _{m \geq n} \mathcal{I}^ n\mathcal{F}_ m.
Proof.
Parts (1), (2), and (3) are general facts about inverse systems of quasi-coherent modules with surjective transition maps, see Derived Categories of Schemes, Lemma 36.3.2 and Cohomology, Lemma 20.37.1. Next, assume \mathcal{I} is of finite type. Let U \subset X be affine open. Say U = \mathop{\mathrm{Spec}}(A) and \mathcal{I}|_ U corresponds to I \subset A. Observe that I is a finitely generated ideal. By the equivalence of categories between quasi-coherent \mathcal{O}_ U-modules and A-modules (Schemes, Lemma 26.7.5) we find that M_ n = \mathcal{F}_ n(U) is an inverse system of A-modules with M_ n = M_{n + 1}/I^ nM_{n + 1}. Thus
M = \mathcal{F}(U) = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n(U) = \mathop{\mathrm{lim}}\nolimits M_ n
is an I-adically complete module with M/I^ nM = M_ n by Algebra, Lemma 10.98.2. This proves (4). Part (5) translates into the statement that \mathop{\mathrm{lim}}\nolimits _{m \geq n} I^ nM/I^ mM = I^ nM. Since I^ mM = I^{m - n} \cdot I^ nM this is just the statement that I^ mM is I-adically complete. This follows from Algebra, Lemma 10.96.3 and the fact that M is complete.
\square
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