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

$H^ i(\omega _ X^\bullet )$ is nonzero only for $i \in [-\dim (X), 0]$,

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

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

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 $\text{depth}(\mathcal{O}_{X, x}) \leq -i$,

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

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 $\omega _ X \cong \wedge ^ d\Omega _{X/k}$.

## Comments (2)

Comment #7945 by Haohao Liu on

Comment #8185 by Stacks Project on

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