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 $d$,

  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 $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$.

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