Lemma 53.4.1 (0BS2)—The Stacks project

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

$H^ i(\omega _ X^\bullet )$ is nonzero only for $i = -1, 0$,
$\omega _ X = H^{-1}(\omega _ X^\bullet )$ is a coherent Cohen-Macaulay module whose support is the irreducible components of dimension $1$,
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,
for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ there are functorial isomorphisms ¹

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

compatible with shifts and distinguished triangles,
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 $X$,
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. $\square$

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

The Stacks project

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