## 50.15 Improving coherent modules

Similar constructions can be found in [EGA] and more recently in and .

Lemma 50.15.1. Let $X$ be a Noetherian scheme. Let $T \subset X$ be a subset stable under specialization. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then there is a unique map $\mathcal{F} \to \mathcal{F}'$ of coherent $\mathcal{O}_ X$-modules such that

1. $\mathcal{F} \to \mathcal{F}'$ is surjective,

2. $\mathcal{F}_ x \to \mathcal{F}'_ x$ is an isomorphism for $x \not\in T$,

3. $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_ x) \geq 1$ for $x \in T$.

If $f : Y \to X$ is a flat morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}'$ is the corresponding quotient for $f^{-1}(T) \subset Y$ and $f^*\mathcal{F}$.

Proof. Condition (3) just means that $\text{Ass}(\mathcal{F}') \cap T = \emptyset$. Thus $\mathcal{F} \to \mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections whose support is contained in $T$. This proves uniqueness. The statement on pullbacks follows from Divisors, Lemma 30.3.1 and the uniqueness.

Existence of $\mathcal{F} \to \mathcal{F}'$. By the uniqueness it suffices to prove the existence and uniqueness locally on $X$; small detail omitted. Thus we may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{F}$ is the coherent module associated to the finite $A$-module $M$. Set $M' = M / H^0_ T(M)$ with $H^0_ T(M)$ as in Section 50.5. Then $M_\mathfrak p = M'_\mathfrak p$ for $\mathfrak p \not\in T$ which proves (1). On the other hand, we have $H^0_ T(M) = \mathop{\mathrm{colim}}\nolimits H^0_ Z(M)$ where $Z$ runs over the closed subsets of $X$ contained in $T$. Thus by Dualizing Complexes, Lemmas 47.11.6 we have $H^0_ T(M') = 0$, i.e., no associated prime of $M'$ is in $T$. Therefore $\text{depth}(M'_\mathfrak p) \geq 1$ for $\mathfrak p \in T$. $\square$

Lemma 50.15.2. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $\mathcal{F}' = j_*(\mathcal{F}|_ U)$ is coherent. Then $\mathcal{F} \to \mathcal{F}'$ is the unique map of coherent $\mathcal{O}_ X$-modules such that

1. $\mathcal{F}|_ U \to \mathcal{F}'|_ U$ is an isomorphism,

2. $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_ x) \geq 2$ for $x \in X$, $x \not\in U$.

If $f : Y \to X$ is a flat morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}'$ is the corresponding map for $f^{-1}(U) \subset Y$.

Proof. We have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_ x) \geq 2$ by Divisors, Lemma 30.6.6 part (3). The uniqueness of $\mathcal{F} \to \mathcal{F}'$ follows from Divisors, Lemma 30.5.11. The compatibility with flat pullbacks follows from flat base change, see Cohomology of Schemes, Lemma 29.5.2. $\square$

Lemma 50.15.3. Let $X$ be a Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $X$ is universally catenary and the formal fibres of local rings have $(S_1)$. Then there exists a unique map $\mathcal{F} \to \mathcal{F}''$ of coherent $\mathcal{O}_ X$-modules such that

1. $\mathcal{F}_ x \to \mathcal{F}''_ x$ is an isomorphism for $x \in X \setminus Z$,

2. $\mathcal{F}_ x \to \mathcal{F}''_ x$ is surjective and $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) = 1$ for $x \in Z$ such that there exists an immediate specialization $x' \leadsto x$ with $x' \not\in Z$ and $x' \in \text{Ass}(\mathcal{F})$,

3. $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) \geq 2$ for the remaining $x \in Z$.

If $f : Y \to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(Z) \subset Y$.

Proof. Let $\mathcal{F} \to \mathcal{F}'$ be the map constructed in Lemma 50.15.1 for the subset $Z$ of $X$. Recall that $\mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections supported on $Z$.

We first prove uniqueness. Let $\mathcal{F} \to \mathcal{F}''$ be as in the lemma. We get a factorization $\mathcal{F} \to \mathcal{F}' \to \mathcal{F}''$ since $\text{Ass}(\mathcal{F}'') \cap Z = \emptyset$ by conditions (2) and (3). Let $U \subset X$ be a maximal open subscheme such that $\mathcal{F}'|_ U \to \mathcal{F}''|_ U$ is an isomorphism. We see that $U$ contains all the points as in (2). Then by Divisors, Lemma 30.5.11 we conclude that $\mathcal{F}'' = j_*(\mathcal{F}'|_ U)$. In this way we get uniqueness (small detail: if we have two of these $\mathcal{F}''$ then we take the intersection of the opens $U$ we get from either).

Proof of existence. Recall that $\text{Ass}(\mathcal{F}') = \{ x_1, \ldots , x_ n\}$ is finite and $x_ i \not\in Z$. Let $Y_ i$ be the closure of $\{ x_ i\}$. Let $Z_{i, j}$ be the irreducible components of $Z \cap Y_ i$. Observe that $\text{Supp}(\mathcal{F}') \cap Z = \bigcup Z_{i, j}$. Let $z_{i, j} \in Z_{i, j}$ be the generic point. Let

$d_{i, j} = \dim (\mathcal{O}_{\overline{\{ x_ i\} }, z_{i, j}})$

If $d_{i, j} = 1$, then $z_{i, j}$ is one of the points as in (2). Thus we do not need to modify $\mathcal{F}'$ at these points. Furthermore, still assuming $d_{i, j} = 1$, using Lemma 50.9.2 we can find an open neighbourhood $z_{i, j} \in V_{i, j} \subset X$ such that $\text{depth}_{\mathcal{O}_{X, z}}(\mathcal{F}'_ z) \geq 2$ for $z \in Z_{i, j} \cap V_{i, j}$, $z \not= z_{i, j}$. Set

$Z' = X \setminus \left( X \setminus Z \cup \bigcup \nolimits _{d_{i, j} = 1} V_{i, j}) \right)$

Denote $j' : X \setminus Z' \to X$. By our choice of $Z'$ the assumptions of Lemma 50.8.9 are satisfied. We conclude by setting $\mathcal{F}'' = j'_*(\mathcal{F}'|_{X \setminus Z'})$ and applying Lemma 50.15.2.

The final statement follows from the formula for the change in depth along a flat local homomorphism, see Algebra, Lemma 10.157.1 and the assumption on the fibres of $f$ inherent in $f$ being Cohen-Macaulay. Details omitted. $\square$

Lemma 50.15.4. Let $X$ be a Noetherian scheme which locally has a dualizing complex. Let $T' \subset X$ be a subset stable under specialization. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume that if $x \leadsto x'$ is an immediate specialization of points in $X$ with $x' \in T'$ and $x \not\in T'$, then $\text{depth}(\mathcal{F}_ x) \geq 1$. Then there exists a unique map $\mathcal{F} \to \mathcal{F}''$ of coherent $\mathcal{O}_ X$-modules such that

1. $\mathcal{F}_ x \to \mathcal{F}''_ x$ is an isomorphism for $x \not\in T'$,

2. $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) \geq 2$ for $x \in T'$.

If $f : Y \to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(T') \subset Y$.

Proof. Let $\mathcal{F} \to \mathcal{F}'$ be the quotient of $\mathcal{F}$ constructed in Lemma 50.15.1 using $T'$. Recall that $\mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections supported on $T'$.

Proof of uniqueness. Let $\mathcal{F} \to \mathcal{F}''$ be as in the lemma. We get a factorization $\mathcal{F} \to \mathcal{F}' \to \mathcal{F}''$ since $\text{Ass}(\mathcal{F}'') \cap T' = \emptyset$ by condition (2). Let $U \subset X$ be a maximal open subscheme such that $\mathcal{F}'|_ U \to \mathcal{F}''|_ U$ is an isomorphism. We see that $U$ contains all the points of $T'$. Then by Divisors, Lemma 30.5.11 we conclude that $\mathcal{F}'' = j_*(\mathcal{F}'|_ U)$. In this way we get uniqueness (small detail: if we have two of these $\mathcal{F}''$ then we take the intersection of the opens $U$ we get from either).

Proof of existence. We will define

$\mathcal{F}'' = \mathop{\mathrm{colim}}\nolimits j_*(\mathcal{F}'|_ V)$

where $j : V \to X$ runs over the open subschemes such that $X \setminus V \subset T'$. Observe that the colimit is filtered as $T'$ is stable under specialization. Each of the maps $\mathcal{F}' \to j_*(\mathcal{F}'|_ V)$ is injective as $\text{Ass}(\mathcal{F}')$ is disjoint from $T'$. Thus $\mathcal{F}' \to \mathcal{F}''$ is injective.

Suppose $X = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{F}$ corresponds to the finite $A$-module $M$. Then $\mathcal{F}'$ corresponds to $M' = M / H^0_{T'}(M)$, see proof of Lemma 50.15.1. Applying Lemmas 50.2.2 and 50.5.3 we see that $\mathcal{F}''$ corresponds to an $A$-module $M''$ which fits into the short exact sequence

$0 \to M' \to M'' \to H^1_{T'}(M') \to 0$

By Proposition 50.11.1 and our condition on immediate specializations in the statement of the lemma we see that $M''$ is a finite $A$-module. In this way we see that $\mathcal{F}''$ is coherent.

The final statement follows from the formula for the change in depth along a flat local homomorphism, see Algebra, Lemma 10.157.1 and the assumption on the fibres of $f$ inherent in $f$ being Cohen-Macaulay. Details omitted. $\square$

Lemma 50.15.5. Let $X$ be a Noetherian scheme which locally has a dualizing complex. Let $T' \subset T \subset X$ be subsets stable under specialization such that if $x \leadsto x'$ is an immediate specialization of points in $X$ and $x' \in T'$, then $x \in T$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then there exists a unique map $\mathcal{F} \to \mathcal{F}''$ of coherent $\mathcal{O}_ X$-modules such that

1. $\mathcal{F}_ x \to \mathcal{F}''_ x$ is an isomorphism for $x \not\in T$,

2. $\mathcal{F}_ x \to \mathcal{F}''_ x$ is surjective and $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) \geq 1$ for $x \in T$, $x \not\in T'$, and

3. $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) \geq 2$ for $x \in T'$.

If $f : Y \to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(T') \subset f^{-1}(T) \subset Y$.

Proof. First, let $\mathcal{F} \to \mathcal{F}'$ be the quotient of $\mathcal{F}$ constructed in Lemma 50.15.1 using $T$. Second, let $\mathcal{F}' \to \mathcal{F}''$ be the unique map of coherent modules construction in Lemma 50.15.4 using $T'$. Then $\mathcal{F} \to \mathcal{F}''$ is as desired. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DX2. Beware of the difference between the letter 'O' and the digit '0'.