## 52.2 Formal sections, I

Let $A$ be a ring and $I \subset A$ an ideal. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. In this section we prove some general facts on inverse systems of $\mathcal{O}_ X$-modules $\{ \mathcal{F}_ n\}$ such that $\mathcal{F}_ n = \mathcal{F}_{n + 1} / I^ n \mathcal{F}_{n + 1}$. Some of these results are proved in greater generality in Cohomology, Section 20.35.

Lemma 52.2.1. Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let

$\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$

be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Assume

$\bigoplus \nolimits _{n \geq 0} H^1(X, I^ n\mathcal{F}_{n + 1})$

satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module. Then the inverse system $M_ n = \Gamma (X, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition.

Proof. This is a special case of the more general Cohomology, Lemma 20.35.1. $\square$

Lemma 52.2.2. Let $I$ be an ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let

$\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$

be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Given $n$ define

$H^1_ n = \bigcap \nolimits _{m \geq n} \mathop{\mathrm{Im}}\left( H^1(X, I^ n\mathcal{F}_{m + 1}) \to H^1(X, I^ n\mathcal{F}_{n + 1}) \right)$

If $\bigoplus H^1_ n$ satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module, then the inverse system $M_ n = \Gamma (X, \mathcal{F}_ n)$ satisfies the Mittag-Leffler condition.

Proof. This is a special case of the more general Cohomology, Lemma 20.35.2. $\square$

Lemma 52.2.3. Let $I$ be a finitely generated ideal of a ring $A$. Let $X$ be a scheme over $\mathop{\mathrm{Spec}}(A)$. Let

$\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$

be an inverse system of $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. Assume

$\bigoplus \nolimits _{n \geq 0} H^0(X, I^ n\mathcal{F}_{n + 1})$

satisfies the ascending chain condition as a graded $\bigoplus _{n \geq 0} I^ n/I^{n + 1}$-module. Then the limit topology on $M = \mathop{\mathrm{lim}}\nolimits \Gamma (X, \mathcal{F}_ n)$ is the $I$-adic topology.

Proof. This is a special case of the more general Cohomology, Lemma 20.35.3. $\square$

Lemma 52.2.4. 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

1. $\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$,

2. for any affine open $U \subset X$ we have $H^ p(U, \mathcal{F}) = 0$ for $p > 0$, and

3. 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

1. $\mathcal{F}_ n = \mathcal{F}/\mathcal{I}^ n\mathcal{F}$, and

2. $\mathcal{I}^ n \mathcal{F} = \mathop{\mathrm{lim}}\nolimits _{m \geq n} \mathcal{I}^ n\mathcal{F}_ m$.

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 20.36.1. 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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