The Stacks project

53.4 Duality

In this section we work out the consequences of the very general material on dualizing complexes and duality for proper $1$-dimensional schemes over fields. If you are interested in the analogous discussion for higher dimension proper schemes over fields, see Duality for Schemes, Section 48.27.

Lemma 53.4.1. Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. There exists a dualizing complex $\omega _ X^\bullet $ with the following properties

  1. $H^ i(\omega _ X^\bullet )$ is nonzero only for $i = -1, 0$,

  2. $\omega _ X = H^{-1}(\omega _ X^\bullet )$ is a coherent Cohen-Macaulay module whose support is the irreducible components of dimension $1$,

  3. for $x \in X$ closed, the module $H^0(\omega _{X, x}^\bullet )$ is nonzero if and only if either

    1. $\dim (\mathcal{O}_{X, x}) = 0$ or

    2. $\dim (\mathcal{O}_{X, x}) = 1$ and $\mathcal{O}_{X, x}$ is not Cohen-Macaulay,

  4. for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ there are functorial isomorphisms1

    \[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, K), k) \]

    compatible with shifts and distinguished triangles,

  5. there are functorial isomorphisms $\mathop{\mathrm{Hom}}\nolimits (\mathcal{F}, \omega _ X) = \mathop{\mathrm{Hom}}\nolimits _ k(H^1(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$,

  6. if $X \to \mathop{\mathrm{Spec}}(k)$ is smooth of relative dimension $1$, then $\omega _ X \cong \Omega _{X/k}$.

Proof. Denote $f : X \to \mathop{\mathrm{Spec}}(k)$ the structure morphism. We start with the relative dualizing complex

\[ \omega _ X^\bullet = \omega _{X/k}^\bullet = a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}) \]

as described in Duality for Schemes, Remark 48.12.5. Then property (4) holds by construction as $a$ is the right adjoint for $f_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$. Since $f$ is proper we have $f^!(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}) = a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$ by definition, see Duality for Schemes, Section 48.16. Hence $\omega _ X^\bullet $ and $\omega _ X$ are as in Duality for Schemes, Example 48.22.1 and as in Duality for Schemes, Example 48.22.2. Parts (1) and (2) follow from Duality for Schemes, Lemma 48.22.4. For a closed point $x \in X$ we see that $\omega _{X, x}^\bullet $ is a normalized dualizing complex over $\mathcal{O}_{X, x}$, see Duality for Schemes, Lemma 48.21.1. Assertion (3) then follows from Dualizing Complexes, Lemma 47.20.2. Assertion (5) follows from Duality for Schemes, Lemma 48.22.5 for coherent $\mathcal{F}$ and in general by unwinding (4) for $K = \mathcal{F}[0]$ and $i = -1$. Assertion (6) follows from Duality for Schemes, Lemma 48.15.7. $\square$

Lemma 53.4.2. Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $1$. The module $\omega _ X$ of Lemma 53.4.1 has the following properties

  1. $\omega _ X$ is a dualizing module on $X$ (Duality for Schemes, Section 48.22),

  2. $\omega _ X$ is a coherent Cohen-Macaulay module whose support is $X$,

  3. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X[1]) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, K), k)$ compatible with shifts for $K \in D_\mathit{QCoh}(X)$,

  4. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^{1 + i}(\mathcal{F}, \omega _ X) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$.

Proof. Recall from the proof of Lemma 53.4.1 that $\omega _ X$ is as in Duality for Schemes, Example 48.22.1 and hence is a dualizing module. The other statements follow from Lemma 53.4.1 and the fact that $\omega _ X^\bullet = \omega _ X[1]$ as $X$ is Cohen-Macualay (Duality for Schemes, Lemma 48.23.1). $\square$

Remark 53.4.3. Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $\omega _ X^\bullet $ and $\omega _ X$ be as in Lemma 53.4.1. If $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee $ then we have canonical isomorphisms

\[ \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, \mathcal{E}), k) = H^ i(X, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _ X^\bullet ) \]

This follows from the lemma and Cohomology, Lemma 20.50.5. If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then we have canonical isomorphisms

\[ \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, \mathcal{E}), k) = H^{1 + i}(X, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \omega _ X) \]

by Lemma 53.4.2. In particular if $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module, then we have

\[ \dim _ k H^0(X, \mathcal{L}) = \dim _ k H^1(X, \mathcal{L}^{\otimes -1} \otimes _{\mathcal{O}_ X} \omega _ X) \]

and

\[ \dim _ k H^1(X, \mathcal{L}) = \dim _ k H^0(X, \mathcal{L}^{\otimes -1} \otimes _{\mathcal{O}_ X} \omega _ X) \]

Here is a sanity check for the dualizing complex.

Lemma 53.4.4. Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $\omega _ X^\bullet $ and $\omega _ X$ be as in Lemma 53.4.1.

  1. If $X \to \mathop{\mathrm{Spec}}(k)$ factors as $X \to \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$ for some field $k'$, then $\omega _ X^\bullet $ and $\omega _ X$ satisfy properties (4), (5), (6) with $k$ replaced with $k'$.

  2. If $K/k$ is a field extension, then the pullback of $\omega _ X^\bullet $ and $\omega _ X$ to the base change $X_ K$ are as in Lemma 53.4.1 for the morphism $X_ K \to \mathop{\mathrm{Spec}}(K)$.

Proof. Denote $f : X \to \mathop{\mathrm{Spec}}(k)$ the structure morphism. Assertion (1) really means that $\omega _ X^\bullet $ and $\omega _ X$ are as in Lemma 53.4.1 for the morphism $f' : X \to \mathop{\mathrm{Spec}}(k')$. In the proof of Lemma 53.4.1 we took $\omega _ X^\bullet = a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$ where $a$ be is the right adjoint of Duality for Schemes, Lemma 48.3.1 for $f$. Thus we have to show $a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}) \cong a'(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$ where $a'$ be is the right adjoint of Duality for Schemes, Lemma 48.3.1 for $f'$. Since $k' \subset H^0(X, \mathcal{O}_ X)$ we see that $k'/k$ is a finite extension (Cohomology of Schemes, Lemma 30.19.2). By uniqueness of adjoints we have $a = a' \circ b$ where $b$ is the right adjoint of Duality for Schemes, Lemma 48.3.1 for $g : \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$. Another way to say this: we have $f^! = (f')^! \circ g^!$. Thus it suffices to show that $\mathop{\mathrm{Hom}}\nolimits _ k(k', k) \cong k'$ as $k'$-modules, see Duality for Schemes, Example 48.3.2. This holds because these are $k'$-vector spaces of the same dimension (namely dimension $1$).

Proof of (2). This holds because we have base change for $a$ by Duality for Schemes, Lemma 48.6.2. See discussion in Duality for Schemes, Remark 48.12.5. $\square$

Lemma 53.4.5. Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $i : Y \to X$ be a closed immersion. Let $\omega _ X^\bullet $, $\omega _ X$, $\omega _ Y^\bullet $, $\omega _ Y$ be as in Lemma 53.4.1. Then

  1. $\omega _ Y^\bullet = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Y, \omega _ X^\bullet )$,

  2. $\omega _ Y = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Y, \omega _ X)$ and $i_*\omega _ Y = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{O}_ Y, \omega _ X)$.

Proof. Denote $g : Y \to \mathop{\mathrm{Spec}}(k)$ and $f : X \to \mathop{\mathrm{Spec}}(k)$ the structure morphisms. Then $g = f \circ i$. Denote $a, b, c$ the right adjoint of Duality for Schemes, Lemma 48.3.1 for $f, g, i$. Then $b = c \circ a$ by uniqueness of right adjoints and because $Rg_* = Rf_* \circ Ri_*$. In the proof of Lemma 53.4.1 we set $\omega _ X^\bullet = a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$ and $\omega _ Y^\bullet = b(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$. Hence $\omega _ Y^\bullet = c(\omega _ X^\bullet )$ which implies (1) by Duality for Schemes, Lemma 48.9.7. Since $\omega _ X = H^{-1}(\omega _ X^\bullet )$ and $\omega _ Y = H^{-1}(\omega _ Y^\bullet )$ we conclude that $\omega _ Y = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Y, \omega _ X)$. This implies $i_*\omega _ Y = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{O}_ Y, \omega _ X)$ by Duality for Schemes, Lemma 48.9.3. $\square$

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. $\square$

[1] This property characterizes $\omega _ X^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ up to unique isomorphism by the Yoneda lemma. Since $\omega _ X^\bullet $ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ in fact it suffices to consider $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$.

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 0E31. Beware of the difference between the letter 'O' and the digit '0'.



View Section 53.4 as pdf