\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*}

The Stacks project

48.11 Finiteness of pushforwards, II

This section is the continuation of Section 48.7. In this section we reap the fruits of the labor done in Section 48.10.

Lemma 48.11.1. Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Let $n \geq 0$ be an integer. Assume

  1. $X$ is universally catenary,

  2. for every $z \in Z$ the formal fibres of $\mathcal{O}_{X, z}$ are $(S_ n)$.

In this situation the following are equivalent

  1. for $x \in \text{Supp}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ we have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \dim (\mathcal{O}_{\overline{\{ x\} }, z}) > n$,

  2. $R^ pj_*\mathcal{F}$ is coherent for $0 \leq p < n$.

Proof. The statement is local on $X$, hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = V(I)$. Let $M$ be a finite $A$-module whose associated coherent $\mathcal{O}_ X$-module restricts to $\mathcal{F}$ over $U$, see Lemma 48.7.2. This lemma also tells us that $R^ pj_*\mathcal{F}$ is coherent if and only if $H^{p + 1}_ Z(M)$ is a finite $A$-module. Observe that the minimum of the expressions $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \dim (\mathcal{O}_{\overline{\{ x\} }, z})$ is the number $s_{A, I}(M)$ of (48.10.1.1). Having said this the lemma follows from Theorem 48.10.6 as elucidated by Remark 48.10.7. $\square$

Lemma 48.11.2. Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $n \geq 0$ be an integer. If $R^ pj_*\mathcal{O}_ U$ is coherent for $0 \leq p < n$, then the same is true for $R^ pj_*\mathcal{F}$, $0 \leq p < n$ for any finite locally free $\mathcal{O}_ U$-module $\mathcal{F}$.

Proof. The question is local on $X$, hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = V(I)$. Via Lemma 48.7.2 our lemma follows from Lemma 48.6.4. $\square$

reference

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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