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50.8 Finiteness of pushforwards, I

In this section we discuss the easiest nontrivial case of the finiteness theorem, namely, the finiteness of the first local cohomology or what is equivalent, finiteness of $j_*\mathcal{F}$ where $j : U \to X$ is an open immersion, $X$ is locally Noetherian, and $\mathcal{F}$ is a coherent sheaf on $U$. Following a method of Kollár ([Kollar-variants] and [Kollar-local-global-hulls]) we find a necessary and sufficient condition, see Proposition 50.8.7. The reader who is interested in higher direct images or higher local cohomology groups should skip ahead to Section 50.12 or Section 50.11 (which are developed independently of the rest of this section).

Lemma 50.8.1. Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. For $x \in U$ let $i_ x : W_ x \to U$ be the integral closed subscheme with generic point $x$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. The following are equivalent

  1. for all $x \in \text{Ass}(\mathcal{F})$ the $\mathcal{O}_ X$-module $j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent,

  2. $j_*\mathcal{F}$ is coherent.

Proof. We first prove that (1) implies (2). Assume (1) holds. The statement is local on $X$, hence we may assume $X$ is affine. Then $U$ is quasi-compact, hence $\text{Ass}(\mathcal{F})$ is finite (Divisors, Lemma 30.2.5). Thus we may argue by induction on the number of associated points. Let $x \in U$ be a generic point of an irreducible component of the support of $\mathcal{F}$. By Divisors, Lemma 30.2.5 we have $x \in \text{Ass}(\mathcal{F})$. By our choice of $x$ we have $\dim (\mathcal{F}_ x) = 0$ as $\mathcal{O}_{X, x}$-module. Hence $\mathcal{F}_ x$ has finite length as an $\mathcal{O}_{X, x}$-module (Algebra, Lemma 10.61.3). Thus we may use induction on this length.

Set $\mathcal{G} = j_*i_{x, *}\mathcal{O}_{W_ x}$. This is a coherent $\mathcal{O}_ X$-module by assumption. We have $\mathcal{G}_ x = \kappa (x)$. Choose a nonzero map $\varphi _ x : \mathcal{F}_ x \to \kappa (x) = \mathcal{G}_ x$. By Cohomology of Schemes, Lemma 29.9.6 there is an open $x \in V \subset U$ and a map $\varphi _ V : \mathcal{F}|_ V \to \mathcal{G}|_ V$ whose stalk at $x$ is $\varphi _ x$. Choose $f \in \Gamma (X, \mathcal{O}_ X)$ which does not vanish at $x$ such that $D(f) \subset V$. By Cohomology of Schemes, Lemma 29.10.4 (for example) we see that $\varphi _ V$ extends to $f^ n\mathcal{F} \to \mathcal{G}|_ U$ for some $n$. Precomposing with multiplication by $f^ n$ we obtain a map $\mathcal{F} \to \mathcal{G}|_ U$ whose stalk at $x$ is nonzero. Let $\mathcal{F}' \subset \mathcal{F}$ be the kernel. Note that $\text{Ass}(\mathcal{F}') \subset \text{Ass}(\mathcal{F})$, see Divisors, Lemma 30.2.4. Since $\text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}'_ x) = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) - 1$ we may apply the induction hypothesis to conclude $j_*\mathcal{F}'$ is coherent. Since $\mathcal{G} = j_*(\mathcal{G}|_ U) = j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent, we can consider the exact sequence

\[ 0 \to j_*\mathcal{F}' \to j_*\mathcal{F} \to \mathcal{G} \]

By Schemes, Lemma 25.24.1 the sheaf $j_*\mathcal{F}$ is quasi-coherent. Hence the image of $j_*\mathcal{F}$ in $j_*(\mathcal{G}|_ U)$ is coherent by Cohomology of Schemes, Lemma 29.9.3. Finally, $j_*\mathcal{F}$ is coherent by Cohomology of Schemes, Lemma 29.9.2.

Assume (2) holds. Exactly in the same manner as above we reduce to the case $X$ affine. We pick $x \in \text{Ass}(\mathcal{F})$ and we set $\mathcal{G} = j_*i_{x, *}\mathcal{O}_{W_ x}$. Then we choose a nonzero map $\varphi _ x : \mathcal{G}_ x = \kappa (x) \to \mathcal{F}_ x$ which exists exactly because $x$ is an associated point of $\mathcal{F}$. Arguing exactly as above we may assume $\varphi _ x$ extends to an $\mathcal{O}_ U$-module map $\varphi : \mathcal{G}|_ U \to \mathcal{F}$. Then $\varphi $ is injective (for example by Divisors, Lemma 30.2.10) and we find an injective map $\mathcal{G} = j_*(\mathcal{G}|_ V) \to j_*\mathcal{F}$. Thus (1) holds. $\square$

Lemma 50.8.2. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $X = \mathop{\mathrm{Spec}}(A)$, $Z = V(I)$, $U = X \setminus Z$, and $j : U \to X$ the inclusion morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Then

  1. there exists a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction of $\widetilde{M}$ to $U$,

  2. given $M$ there is an exact sequence

    \[ 0 \to H^0_ Z(M) \to M \to H^0(U, \mathcal{F}) \to H^1_ Z(M) \to 0 \]

    and isomorphisms $H^ p(U, \mathcal{F}) = H^{p + 1}_ Z(M)$ for $p \geq 1$,

  3. given $M$ and $p \geq 0$ the following are equivalent

    1. $R^ pj_*\mathcal{F}$ is coherent,

    2. $H^ p(U, \mathcal{F})$ is a finite $A$-module,

    3. $H^{p + 1}_ Z(M)$ is a finite $A$-module,

  4. if the equivalent conditions in (3) hold for $p = 0$, we may take $M = \Gamma (U, \mathcal{F})$ in which case we have $H^0_ Z(M) = H^1_ Z(M) = 0$.

Proof. By Properties, Lemma 27.22.4 there exists a coherent $\mathcal{O}_ X$-module $\mathcal{F}'$ whose restriction to $U$ is isomorphic to $\mathcal{F}$. Say $\mathcal{F}'$ corresponds to the finite $A$-module $M$ as in (1). Note that $R^ pj_*\mathcal{F}$ is quasi-coherent (Cohomology of Schemes, Lemma 29.4.5) and corresponds to the $A$-module $H^ p(U, \mathcal{F})$. By Lemma 50.2.1 and the general facts in Cohomology, Section 20.22 we obtain an exact sequence

\[ 0 \to H^0_ Z(M) \to M \to H^0(U, \mathcal{F}) \to H^1_ Z(M) \to 0 \]

and isomorphisms $H^ p(U, \mathcal{F}) = H^{p + 1}_ Z(M)$ for $p \geq 1$. Here we use that $H^ j(X, \mathcal{F}') = 0$ for $j > 0$ as $X$ is affine and $\mathcal{F}'$ is quasi-coherent (Cohomology of Schemes, Lemma 29.2.2). This proves (2). Parts (3) and (4) are straightforward from (2); see also Lemma 50.2.2. $\square$

Lemma 50.8.3. 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. Assume

  1. $X$ is Nagata,

  2. $X$ is universally catenary, and

  3. for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ we have $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) \geq 2$.

Then $j_*\mathcal{F}$ is coherent.

Proof. By Lemma 50.8.1 it suffices to prove $j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent for $x \in \text{Ass}(\mathcal{F})$. Let $\pi : Y \to X$ be the normalization of $X$ in $\mathop{\mathrm{Spec}}(\kappa (x))$, see Morphisms, Section 28.52. By Morphisms, Lemma 28.51.14 the morphism $\pi $ is finite. Since $\pi $ is finite $\mathcal{G} = \pi _*\mathcal{O}_ Y$ is a coherent $\mathcal{O}_ X$-module by Cohomology of Schemes, Lemma 29.9.9. Observe that $W_ x = U \cap \pi (Y)$. Thus $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ factors through $i_ x : W_ x \to U$ and we obtain a canonical map

\[ i_{x, *}\mathcal{O}_{W_ x} \longrightarrow (\pi |_{\pi ^{-1}(U)})_*(\mathcal{O}_{\pi ^{-1}(U)}) = (\pi _*\mathcal{O}_ Y)|_ U = \mathcal{G}|_ U \]

This map is injective (for example by Divisors, Lemma 30.2.10). Hence $j_*i_{x, *}\mathcal{O}_{W_ x} \subset j_*\mathcal{G}|_ U$ and it suffices to show that $j_*\mathcal{G}|_ U$ is coherent.

It remains to prove that $j_*(\mathcal{G}|_ U)$ is coherent. We claim Divisors, Lemma 30.5.11 applies to

\[ \mathcal{G} \longrightarrow j_*(\mathcal{G}|_ U) \]

which finishes the proof. It suffices to show that $\text{depth}(\mathcal{G}_ z) \geq 2$ for $z \in Z$. Let $y_1, \ldots , y_ n \in Y$ be the points mapping to $z$. By Algebra, Lemma 10.71.11 it suffices to show that $\text{depth}(\mathcal{O}_{Y, y_ i}) \geq 2$ for $i = 1, \ldots , n$. If not, then by Properties, Lemma 27.12.5 we see that $\dim (\mathcal{O}_{Y, y_ i}) = 1$ for some $i$. This is impossible by the dimension formula (Morphisms, Lemma 28.50.1) for $\pi : Y \to \overline{\{ x\} }$ and assumption (3). $\square$

Lemma 50.8.4. Let $X$ be an integral locally Noetherian scheme. Let $j : U \to X$ be the inclusion of a nonempty open subscheme with complement $Z$. Assume that for all $z \in Z$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{X, z}^\wedge $ we have $\dim (\mathcal{O}_{X, z}^\wedge /\mathfrak p) \geq 2$. Then $j_*\mathcal{O}_ U$ is coherent.

Proof. We may assume $X$ is affine. Using Lemmas 50.7.2 and 50.8.2 we reduce to $X = \mathop{\mathrm{Spec}}(A)$ where $(A, \mathfrak m)$ is a Noetherian local domain and $\mathfrak m \in Z$. Then we can use induction on $d = \dim (A)$. (The base case is $d = 0, 1$ which do not happen by our assumption on the local rings.) Set $V = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\} $. Observe that the local rings of $V$ have dimension strictly smaller than $d$. Repeating the arguments for $j' : U \to V$ we and using induction we conclude that $j'_*\mathcal{O}_ U$ is a coherent $\mathcal{O}_ V$-module. Pick a nonzero $f \in A$ which vanishes on $Z$. Since $D(f) \cap V \subset U$ we find an $n$ such that multiplication by $f^ n$ on $U$ extends to a map $f^ n : j'_*\mathcal{O}_ U \to \mathcal{O}_ V$ over $V$ (for example by Cohomology of Schemes, Lemma 29.10.4). This map is injective hence there is an injective map

\[ j_*\mathcal{O}_ U = j''_* j'_* \mathcal{O}_ U \to j''_*\mathcal{O}_ V \]

on $X$ where $j'' : V \to X$ is the inclusion morphism. Hence it suffices to show that $j''_*\mathcal{O}_ V$ is coherent. In other words, we may assume that $X$ is the spectrum of a local Noetherian domain and that $Z$ consists of the closed point.

Assume $X = \mathop{\mathrm{Spec}}(A)$ with $(A, \mathfrak m)$ local and $Z = \{ \mathfrak m\} $. Let $A^\wedge $ be the completion of $A$. Set $X^\wedge = \mathop{\mathrm{Spec}}(A^\wedge )$, $Z^\wedge = \{ \mathfrak m^\wedge \} $, $U^\wedge = X^\wedge \setminus Z^\wedge $, and $\mathcal{F}^\wedge = \mathcal{O}_{U^\wedge }$. The ring $A^\wedge $ is universally catenary and Nagata (Algebra, Remark 10.154.9 and Lemma 10.156.8). Moreover, condition (3) of Lemma 50.8.3 for $X^\wedge , Z^\wedge , U^\wedge , \mathcal{F}^\wedge $ holds by assumption! Thus we see that $(U^\wedge \to X^\wedge )_*\mathcal{O}_{U^\wedge }$ is coherent. Since the morphism $c : X^\wedge \to X$ is flat we conclude that the pullback of $j_*\mathcal{O}_ U$ is $(U^\wedge \to X^\wedge )_*\mathcal{O}_{U^\wedge }$ (Cohomology of Schemes, Lemma 29.5.2). Finally, since $c$ is faithfully flat we conclude that $j_*\mathcal{O}_ U$ is coherent by Descent, Lemma 34.7.1. $\square$

Remark 50.8.5. Let $j : U \to X$ be an open immersion of locally Noetherian schemes. Let $x \in U$. Let $i_ x : W_ x \to U$ be the integral closed subscheme with generic point $x$ and let $\overline{\{ x\} }$ be the closure in $X$. Then we have a commutative diagram

\[ \xymatrix{ W_ x \ar[d]_{i_ x} \ar[r]_{j'} & \overline{\{ x\} } \ar[d]^ i \\ U \ar[r]^ j & X } \]

We have $j_*i_{x, *}\mathcal{O}_{W_ x} = i_*j'_*\mathcal{O}_{W_ x}$. As the left vertical arrow is a closed immersion we see that $j_*i_{x, *}\mathcal{O}_{W_ x}$ is coherent if and only if $j'_*\mathcal{O}_{W_ x}$ is coherent.

Remark 50.8.6. 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. If there exists an $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ such that $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) \leq 1$, then $j_*\mathcal{F}$ is not coherent. To prove this we can do a flat base change to the spectrum of $\mathcal{O}_{X, z}$. Let $X' = \overline{\{ x\} }$. The assumption implies $\mathcal{O}_{X' \cap U} \subset \mathcal{F}$. Thus it suffices to see that $j_*\mathcal{O}_{X' \cap U}$ is not coherent. This is clear because $X' = \{ x, z\} $, hence $j_*\mathcal{O}_{X' \cap U}$ corresponds to $\kappa (x)$ as an $\mathcal{O}_{X, z}$-module which cannot be finite as $x$ is not a closed point.

In fact, the converse of Lemma 50.8.4 holds true: given an open immersion $j : U \to X$ of integral Noetherian schemes and there exists a $z \in X \setminus U$ and an associated prime $\mathfrak p$ of the completion $\mathcal{O}_{X, z}^\wedge $ with $\dim (\mathcal{O}_{X, z}^\wedge /\mathfrak p) = 1$, then $j_*\mathcal{O}_ U$ is not coherent. Namely, you can pass to the local ring, you can enlarge $U$ to the punctured spectrum, you can pass to the completion, and then the argument above gives the nonfiniteness.

sloganreference

Proposition 50.8.7 (Kollár). Let $j : U \to X$ be an open immersion of locally Noetherian schemes with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. The following are equivalent

  1. $j_*\mathcal{F}$ is coherent,

  2. for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ and any associated prime $\mathfrak p$ of the completion $\mathcal{O}_{\overline{\{ x\} }, z}^\wedge $ we have $\dim (\mathcal{O}_{\overline{\{ x\} }, z}^\wedge /\mathfrak p) \geq 2$.

Proof. If (2) holds we get (1) by a combination of Lemmas 50.8.1, Remark 50.8.5, and Lemma 50.8.4. If (2) does not hold, then $j_*i_{x, *}\mathcal{O}_{W_ x}$ is not finite for some $x \in \text{Ass}(\mathcal{F})$ by the discussion in Remark 50.8.6 (and Remark 50.8.5). Thus $j_*\mathcal{F}$ is not coherent by Lemma 50.8.1. $\square$

Lemma 50.8.8. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $Z = V(I)$. Let $M$ be a finite $A$-module. The following are equivalent

  1. $H^1_ Z(M)$ is a finite $A$-module, and

  2. for all $\mathfrak p \in \text{Ass}(M)$, $\mathfrak p \not\in Z$ and all $\mathfrak q \in V(\mathfrak p + I)$ the completion of $(A/\mathfrak p)_\mathfrak q$ does not have associated primes of dimension $1$.

Proof. Follows immediately from Proposition 50.8.7 via Lemma 50.8.2. $\square$

The formulation in the following lemma has the advantage that conditions (1) and (2) are inherited by schemes of finite type over $X$. Moreover, this is the form of finiteness which we will generalize to higher direct images in Section 50.12.

Lemma 50.8.9. 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. Assume

  1. $X$ is universally catenary,

  2. for every $z \in Z$ the formal fibres of $\mathcal{O}_{X, z}$ are $(S_1)$.

In this situation the following are equivalent

  1. for $x \in \text{Ass}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ we have $\dim (\mathcal{O}_{\overline{\{ x\} }, z}) \geq 2$, and

  2. $j_*\mathcal{F}$ is coherent.

Proof. Let $x \in \text{Ass}(\mathcal{F})$. By Proposition 50.8.7 it suffices to check that $A = \mathcal{O}_{\overline{\{ x\} }, z}$ satisfies the condition of the proposition on associated primes of its completion if and only if $\dim (A) \geq 2$. Observe that $A$ is universally catenary (this is clear) and that its formal fibres are $(S_1)$ as follows from More on Algebra, Lemma 15.50.10 and Proposition 15.50.5. Let $\mathfrak p' \subset A^\wedge $ be an associated prime. As $A \to A^\wedge $ is flat, by Algebra, Lemma 10.64.3, we find that $\mathfrak p'$ lies over $(0) \subset A$. The formal fibre $A^\wedge \otimes _ A F$ is $(S_1)$ where $F$ is the fraction field of $A$. We conclude that $\mathfrak p'$ is a minimal prime, see Algebra, Lemma 10.151.2. Since $A$ is universally catenary it is formally catenary by More on Algebra, Proposition 15.97.5. Hence $\dim (A^\wedge /\mathfrak p') = \dim (A)$ which proves the equivalence. $\square$


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