The Stacks project

Remark 20.36.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $f \in \Gamma (X, \mathcal{O}_ X)$. Let $\mathcal{F}$ be $\mathcal{O}_ X$-module. If $\mathcal{F}$ is $f$-torsion free, then for every $p \geq 0$ we have a short exact sequence of inverse systems

\[ 0 \to \{ H^ p(X, \mathcal{F})/f^ nH^ p(X, \mathcal{F})\} \to \{ H^ p(X, \mathcal{F}/f^ n\mathcal{F})\} \to \{ H^{p + 1}(X, \mathcal{F})[f^ n]\} \to 0 \]

Since the first inverse system has the Mittag-Leffler condition (ML) we learn three things from this:

  1. There is a short exact sequence

    \[ 0 \to \widehat{H^ p(X, \mathcal{F})} \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/f^ n\mathcal{F}) \to T_ f(H^{p + 1}(X, \mathcal{F})) \to 0 \]

    where $\widehat{\ }$ denotes the usual $f$-adic completion and $T_ f( - )$ denotes the $f$-adic Tate module from More on Algebra, Example 15.93.5.

  2. We have $R^1\mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/f^ n\mathcal{F}) = R^1\mathop{\mathrm{lim}}\nolimits H^{p + 1}(X, \mathcal{F})[f^ n]$.

  3. The system $\{ H^{p + 1}(X, \mathcal{F})[f^ n]\} $ is ML if and only if $\{ H^ p(X, \mathcal{F}/f^ n\mathcal{F})\} $ is ML.

See Homology, Lemma 12.31.3 and More on Algebra, Lemmas 15.86.2 and 15.86.13.

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