51.12 Finiteness of pushforwards, II

This section is the continuation of Section 51.8. In this section we reap the fruits of the labor done in Section 51.11.

Lemma 51.12.1. 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 51.8.2. 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 (51.11.1.1). Having said this the lemma follows from Theorem 51.11.6 as elucidated by Remark 51.11.7. $\square$

Lemma 51.12.2. 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 51.8.2 our lemma follows from Lemma 51.7.4. $\square$

Lemma 51.12.3. 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 28.17.3 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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