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

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

$\dim (\mathcal{O}_{X, x}) = 1$ 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 $\omega _ X \cong \Omega _{X/k}$.

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