The Stacks project

Lemma 48.27.1. Let $X$ be a proper scheme 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 $i \in [-\dim (X), 0]$,

  2. $\omega _ X = H^{-\dim (X)}(\omega _ X^\bullet )$ is a coherent $(S_2)$-module whose support is the irreducible components of dimension $\dim (X)$,

  3. the dimension of the support of $H^ i(\omega _ X^\bullet )$ is at most $-i$,

  4. for $x \in X$ closed the module $H^ i(\omega _{X, x}^\bullet ) \oplus \ldots \oplus H^0(\omega _{X, x}^\bullet )$ is nonzero if and only if $\text{depth}(\mathcal{O}_{X, x}) \leq -i$,

  5. 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,

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

  7. if $X \to \mathop{\mathrm{Spec}}(k)$ is smooth of relative dimension $d$, then $\omega _ X^\bullet \cong \wedge ^ d\Omega _{X/k}[d]$ and $\omega _ X \cong \wedge ^ d\Omega _{X/k}$.

Proof. Denote $f : X \to \mathop{\mathrm{Spec}}(k)$ the structure morphism. Let $a$ be the right adjoint of pushforward of this morphism, see Lemma 48.3.1. Consider the relative dualizing complex

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

Compare with Remark 48.12.5. Since $f$ is proper we have $f^!(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}) = a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$ by definition, see Section 48.16. Applying Lemma 48.17.7 we find that $\omega _ X^\bullet $ is a dualizing complex. Moreover, we see that $\omega _ X^\bullet $ and $\omega _ X$ are as in Example 48.22.1 and as in Example 48.22.2.

Parts (1), (2), and (3) follow from 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 Lemma 48.21.1. Part (4) then follows from Dualizing Complexes, Lemma 47.20.1.

Part (5) holds by construction as $a$ is the right adjoint to $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}) = D(k)$ which we can identify with $K \mapsto R\Gamma (X, K)$. We also use that the derived category $D(k)$ of $k$-modules is the same as the category of graded $k$-vector spaces.

Part (6) follows from Lemma 48.22.5 for coherent $\mathcal{F}$ and in general by unwinding (5) for $K = \mathcal{F}[0]$ and $i = -\dim (X)$.

Part (7) follows from Lemma 48.15.7. $\square$

[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 (2)

Comment #7945 by Haohao Liu on

In item (2), should be ?

There are also:

  • 2 comment(s) on Section 48.27: Duality for proper schemes over fields

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