The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 21.23.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_ n)$ be an inverse system in $D(\mathcal{O})$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $m \in \mathbf{Z}$. Assume there exist an integer $n(V)$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that for $\{ V_ i \to V\} \in \text{Cov}_ V$

  1. $R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(V_ i, K_ n) = 0$, and

  2. $H^ m(V_ i, K_ n) \to H^ m(V_ i, K_{n(V)})$ is injective for $n \geq n(V)$.

Then the map on sections $H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)(V) \to H^ m(K_{n(V)})(V)$ is injective.

Proof. Let $\gamma \in H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)(V)$ map to zero in $H^ m(K_{n(V)})(V)$. Since $H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)$ is the sheafification of $U \mapsto H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n)$ (by Lemma 21.21.3) we can choose $\{ V_ i \to V\} \in \text{Cov}_ V$ and elements $\tilde\gamma _ i \in H^ m(V_ i, R\mathop{\mathrm{lim}}\nolimits K_ n)$ mapping to $\gamma |_{V_ i}$. Then $\tilde\gamma _ i$ maps to $\tilde\gamma _{i, n(V)} \in H^ m(V_ i, K_{n(V)})$. Using that $H^ m(K_{n(V)})$ is the sheafification of $U \mapsto H^ m(U, K_{n(V)})$ (by Lemma 21.21.3 again) we see that after replacing $\{ V_ i \to V\} $ by a refinement we may assume that $\tilde\gamma _{i, n(V)} = 0$ for all $i$. For this covering we consider the short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(V_ i, K_ n) \to H^ m(V_ i, R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ m(V_ i, K_ n) \to 0 \]

of Lemma 21.23.2. By assumption (1) the group on the left is zero and by assumption (2) the group on the right maps injectively into $H^ m(V_ i, K_{n(V)})$. We conclude $\tilde\gamma _ i = 0$ and hence $\gamma = 0$ as desired. $\square$


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