# The Stacks Project

## Tag 0BJX

### 48.11. Finiteness of pushforwards, II

This section is the continuation of Section 48.6. 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. (a)    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. (b)    $R^pj_*\mathcal{F}$ 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 48.6.2. 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 (48.10.0.1). Having said this the lemma follows from Theorem 48.10.4 as elucidated by Remark 48.10.5. $\square$

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 48.6.2 our lemma follows from Lemma 48.5.4. $\square$

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. $\square$

The code snippet corresponding to this tag is a part of the file local-cohomology.tex and is located in lines 2665–2777 (see updates for more information).

\section{Finiteness of pushforwards, II}
\label{section-finiteness-pushforward-II}

\noindent
This section is the continuation of
Section \ref{section-finiteness-pushforward}.
In this section we reap the fruits of the labor done in
Section \ref{section-finiteness-II}.

\begin{lemma}
\label{lemma-finiteness-Rjstar}
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
\begin{enumerate}
\item $X$ is universally catenary,
\item for every $z \in Z$ the formal fibres of
$\mathcal{O}_{X, z}$ are $(S_n)$.
\end{enumerate}
In this situation the following are equivalent
\begin{enumerate}
\item[(a)] for $x \in \text{Supp}(\mathcal{F})$ and
$z \in Z \cap \overline{\{x\}}$ we have
$\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \dim(\mathcal{O}_{\overline{\{x\}}, z}) > n$,
\item[(b)] $R^pj_*\mathcal{F}$ is coherent for $0 \leq p < n$.
\end{enumerate}
\end{lemma}

\begin{proof}
The statement is local on $X$, hence we may assume $X$ is affine.
Say $X = \Spec(A)$ 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 \ref{lemma-finiteness-pushforwards-and-H1-local}.
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 (\ref{equation-cutoff}).
Having said this the lemma follows from
Theorem \ref{theorem-finiteness}
\end{proof}

\begin{lemma}
\label{lemma-finiteness-for-finite-locally-free}
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 $\mathcal{F}$.
\end{lemma}

\begin{proof}
The question is local on $X$, hence we may assume $X$ is affine.
Say $X = \Spec(A)$ and $Z = V(I)$. Via
Lemma \ref{lemma-finiteness-pushforwards-and-H1-local}
our lemma follows from
Lemma \ref{lemma-local-finiteness-for-finite-locally-free}.
\end{proof}

\begin{lemma}
\label{lemma-annihilate-Hp}
\begin{reference}
\cite[Lemma 1.9]{Bhatt-local}
\end{reference}
Let $A$ be a ring and let $J \subset I \subset A$ be finitely generated ideals.
Let $p \geq 0$ be an integer. Set $U = \Spec(A) \setminus V(I)$. 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 $\mathcal{F}$.
\end{lemma}

\begin{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
\ref{properties-lemma-section-maps-backwards}
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., $f^{2N}J^n \subset \mathfrak a$.

\medskip\noindent
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.
\end{proof}

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