The Stacks project

Lemma 53.4.6. Let $X$ be a proper scheme over a field $k$ which is Gorenstein, reduced, and equidimensional of dimension $1$. Let $i : Y \to X$ be a reduced closed subscheme equidimensional of dimension $1$. Let $j : Z \to X$ be the scheme theoretic closure of $X \setminus Y$. Then

  1. $Y$ and $Z$ are Cohen-Macaulay,

  2. if $\mathcal{I} \subset \mathcal{O}_ X$, resp. $\mathcal{J} \subset \mathcal{O}_ X$ is the ideal sheaf of $Y$, resp. $Z$ in $X$, then

    \[ \mathcal{I} = i_*\mathcal{I}' \quad \text{and}\quad \mathcal{J} = j_*\mathcal{J}' \]

    where $\mathcal{I}' \subset \mathcal{O}_ Z$, resp. $\mathcal{J}' \subset \mathcal{O}_ Y$ is the ideal sheaf of $Y \cap Z$ in $Z$, resp. $Y$,

  3. $\omega _ Y = \mathcal{J}'(i^*\omega _ X)$ and $i_*(\omega _ Y) = \mathcal{J}\omega _ X$,

  4. $\omega _ Z = \mathcal{I}'(i^*\omega _ X)$ and $i_*(\omega _ Z) = \mathcal{I}\omega _ X$,

  5. we have the following short exact sequences

    \begin{align*} 0 \to \omega _ X \to i_*i^*\omega _ X \oplus j_*j^*\omega _ X \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to i_*\omega _ Y \to \omega _ X \to j_*j^*\omega _ X \to 0 \\ 0 \to j_*\omega _ Z \to \omega _ X \to i_*i^*\omega _ X \to 0 \\ 0 \to i_*\omega _ Y \oplus j_*\omega _ Z \to \omega _ X \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to \omega _ Y \to i^*\omega _ X \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to \omega _ Z \to j^*\omega _ X \to \mathcal{O}_{Y \cap Z} \to 0 \end{align*}

Here $\omega _ X$, $\omega _ Y$, $\omega _ Z$ are as in Lemma 53.4.1.

Proof. A reduced $1$-dimensional Noetherian scheme is Cohen-Macaulay, so (1) is true. Since $X$ is reduced, we see that $X = Y \cup Z$ scheme theoretically. With notation as in Morphisms, Lemma 29.4.6 and by the statement of that lemma we have a short exact sequence

\[ 0 \to \mathcal{O}_ X \to \mathcal{O}_ Y \oplus \mathcal{O}_ Z \to \mathcal{O}_{Y \cap Z} \to 0 \]

Since $\mathcal{J} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to \mathcal{O}_ Z)$, $\mathcal{J}' = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to \mathcal{O}_{Y \cap Z})$, $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to \mathcal{O}_ Y)$, and $\mathcal{I}' = \mathop{\mathrm{Ker}}(\mathcal{O}_ Z \to \mathcal{O}_{Y \cap Z})$ a diagram chase implies (2). Observe that $\mathcal{I} + \mathcal{J}$ is the ideal sheaf of $Y \cap Z$ and that $\mathcal{I} \cap \mathcal{J} = 0$. Hence we have the following exact sequences

\begin{align*} 0 \to \mathcal{O}_ X \to \mathcal{O}_ Y \oplus \mathcal{O}_ Z \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to \mathcal{J} \to \mathcal{O}_ X \to \mathcal{O}_ Z \to 0 \\ 0 \to \mathcal{I} \to \mathcal{O}_ X \to \mathcal{O}_ Y \to 0 \\ 0 \to \mathcal{J} \oplus \mathcal{I} \to \mathcal{O}_ X \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to \mathcal{J}' \to \mathcal{O}_ Y \to \mathcal{O}_{Y \cap Z} \to 0 \\ 0 \to \mathcal{I}' \to \mathcal{O}_ Z \to \mathcal{O}_{Y \cap Z} \to 0 \end{align*}

Since $X$ is Gorenstein $\omega _ X$ is an invertible $\mathcal{O}_ X$-module (Duality for Schemes, Lemma 48.24.4). Since $Y \cap Z$ has dimension $0$ we have $\omega _ X|_{Y \cap Z} \cong \mathcal{O}_{Y \cap Z}$. Thus if we prove (3) and (4), then we obtain the short exact sequences of the lemma by tensoring the above short exact sequence with the invertible module $\omega _ X$. By symmetry it suffices to prove (3) and by (2) it suffices to prove $i_*(\omega _ Y) = \mathcal{J}\omega _ X$.

We have $i_*\omega _ Y = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{O}_ Y, \omega _ X)$ by Lemma 53.4.5. Again using that $\omega _ X$ is invertible we finally conclude that it suffices to show $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{O}_ X/\mathcal{I}, \mathcal{O}_ X)$ maps isomorphically to $\mathcal{J}$ by evaluation at $1$. In other words, that $\mathcal{J}$ is the annihilator of $\mathcal{I}$. This follows from the above. $\square$


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 0E34. Beware of the difference between the letter 'O' and the digit '0'.