The Stacks project

Lemma 52.4.5. Let $I$ be an ideal of a Noetherian ring $A$. Let $X$ be a Noetherian 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 coherent $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$. If $\text{cd}(A, I) = 1$, then for all $p \in \mathbf{Z}$ the limit topology on $\mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ is $I$-adic.

Proof. First it is clear that $I^ t \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ maps to zero in $H^ p(X, \mathcal{F}_ t)$. Thus the $I$-adic topology is finer than the limit topology. For the converse we set $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$, we pick generators $f_1, \ldots , f_ r$ of $I$, we pick $c \geq 1$, and we choose $\Phi _\mathcal {F}$ as in Lemma 52.4.4. We will use the results of Lemma 52.2.4 without further mention. In particular we have a short exact sequence

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}_ n) \to H^ p(X, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n) \to 0 \]

Thus we can lift any element $\xi $ of $\mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ to an element $\xi ' \in H^ p(X, \mathcal{F})$. Suppose $\xi $ maps to zero in $H^ p(X, \mathcal{F}_{nc})$ for some $n$, in other words, suppose $\xi $ is “small” in the limit topology. We have a short exact sequence

\[ 0 \to I^{nc}\mathcal{F} \to \mathcal{F} \to \mathcal{F}_{nc} \to 0 \]

and hence the assumption means we can lift $\xi '$ to an element $\xi '' \in H^ p(X, I^{nc}\mathcal{F})$. Applying $\Phi _\mathcal {F}$ we get

\[ \Phi _\mathcal {F}(\xi '') = \sum \nolimits _{e_1 + \ldots + e_ r = n} \xi '_{e_1, \ldots , e_ r} \cdot T_1^{e_1} \ldots T_ r^{e_ r} \]

for some $\xi '_{e_1, \ldots , e_ r} \in H^ p(X, \mathcal{F})$. Letting $\xi _{e_1, \ldots , e_ r} \in \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ be the images and using the final assertion of Lemma 52.4.4 we conclude that

\[ \xi = \sum f_1^{e_1} \ldots f_ r^{e_ r} \xi _{e_1, \ldots , e_ r} \]

is in $I^ n \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$ as desired. $\square$


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