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 1.9, Bhatt-local]

Lemma 48.11.3. Let $A$ be a ring and let $J \subset I \subset A$ be finitely generated ideals. Let $p \geq 0$ be an integer. Set $U = \mathop{\mathrm{Spec}}(A) \setminus V(I)$. If $H^ p(U, \mathcal{O}_ U)$ is annihilated by $J^ n$ for some $n$, then $H^ p(U, \mathcal{F})$ annihilated by $J^ m$ for some $m = m(\mathcal{F})$ for every finite locally free $\mathcal{O}_ U$-module $\mathcal{F}$.

Proof. Consider the annihilator $\mathfrak a$ of $H^ p(U, \mathcal{F})$. Let $u \in U$. There exists an open neighbourhood $u \in U' \subset U$ and an isomorphism $\varphi : \mathcal{O}_{U'}^{\oplus r} \to \mathcal{F}|_{U'}$. Pick $f \in A$ such that $u \in D(f) \subset U'$. There exist maps

\[ a : \mathcal{O}_ U^{\oplus r} \longrightarrow \mathcal{F} \quad \text{and}\quad b : \mathcal{F} \longrightarrow \mathcal{O}_ U^{\oplus r} \]

whose restriction to $D(f)$ are equal to $f^ N \varphi $ and $f^ N \varphi ^{-1}$ for some $N$. Moreover we may assume that $a \circ b$ and $b \circ a$ are equal to multiplication by $f^{2N}$. This follows from Properties, Lemma 27.17.3 since $U$ is quasi-compact ($I$ is finitely generated), separated, and $\mathcal{F}$ and $\mathcal{O}_ U^{\oplus r}$ are finitely presented. Thus we see that $H^ p(U, \mathcal{F})$ is annihilated by $f^{2N}J^ n$, i.e., $f^{2N}J^ n \subset \mathfrak a$.

As $U$ is quasi-compact we can find finitely many $f_1, \ldots , f_ t$ and $N_1, \ldots , N_ t$ such that $U = \bigcup D(f_ i)$ and $f_ i^{2N_ i}J^ n \subset \mathfrak a$. Then $V(I) = V(f_1, \ldots , f_ t)$ and since $I$ is finitely generated we conclude $I^ M \subset (f_1, \ldots , f_ t)$ for some $M$. All in all we see that $J^ m \subset \mathfrak a$ for $m \gg 0$, for example $m = M (2N_1 + \ldots + 2N_ t) n$ will do. $\square$


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