The Stacks project

Lemma 51.15.3. 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 51.15.1 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 31.5.11 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 51.9.2 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 51.8.9 are satisfied. We conclude by setting $\mathcal{F}'' = j'_*(\mathcal{F}'|_{X \setminus Z'})$ and applying Lemma 51.15.2.

The final statement follows from the formula for the change in depth along a flat local homomorphism, see Algebra, Lemma 10.161.1 and the assumption on the fibres of $f$ inherent in $f$ being Cohen-Macaulay. Details omitted. $\square$


Comments (0)


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 0DX5. Beware of the difference between the letter 'O' and the digit '0'.