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. $\square$

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