## 48.27 Duality for proper schemes over fields

In this section we work out the consequences of the very general material above on dualizing complexes and duality for proper schemes over fields.

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 $d$,

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

**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.
$\square$

Lemma 48.27.4. Let $k$, $X$, and $\omega _ X^\bullet $ be as in Lemma 48.27.1. Let $t : H^0(X, \omega _ X^\bullet ) \to k$ be as in Remark 48.27.2. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then the pairings

\[ H^ i(X, \omega _ X^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} E^\vee ) \times H^{-i}(X, E) \longrightarrow k, \quad (\xi , \eta ) \longmapsto t((1_{\omega _ X^\bullet } \otimes \epsilon )(\xi \cup \eta )) \]

are perfect for all $i$. Here $\cup $ denotes the cupproduct of Cohomology, Section 20.31 and $\epsilon : E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E \to \mathcal{O}_ X$ is as in Cohomology, Example 20.47.7.

**Proof.**
By replacing $E$ with $E[-i]$ this reduces to the case $i = 0$. By Cohomology, Lemma 20.48.2 we see that the pairing is the same as the one discussed in Remark 48.27.2 whence the result by the discussion in that remark.
$\square$

Lemma 48.27.5. Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $d$. The module $\omega _ X$ of Lemma 48.27.1 has the following properties

$\omega _ X$ is a dualizing module on $X$ (Section 48.22),

$\omega _ X$ is a coherent Cohen-Macaulay module whose support is $X$,

there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X[d]) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, K), k)$ compatible with shifts and distinguished triangles for $K \in D_\mathit{QCoh}(X)$,

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

**Proof.**
It is clear from Lemma 48.27.1 that $\omega _ X$ is a dualizing module (as it is the left most nonvanishing cohomology sheaf of a dualizing complex). We have $\omega _ X^\bullet = \omega _ X[d]$ and $\omega _ X$ is Cohen-Macaulay as $X$ is Cohen-Macualay, see Lemma 48.23.1. The other statements follow from this combined with the corresponding statements of Lemma 48.27.1.
$\square$

Here is a sanity check for the dualizing complex.

Lemma 48.27.7. Let $X$ be a proper scheme over a field $k$. Let $\omega _ X^\bullet $ and $\omega _ X$ be as in Lemma 48.27.1.

If $X \to \mathop{\mathrm{Spec}}(k)$ factors as $X \to \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$ for some field $k'$, then $\omega _ X^\bullet $ and $\omega _ X$ are as in Lemma 48.27.1 for the morphism $X \to \mathop{\mathrm{Spec}}(k')$.

If $K/k$ is a field extension, then the pullback of $\omega _ X^\bullet $ and $\omega _ X$ to the base change $X_ K$ are as in Lemma 48.27.1 for the morphism $X_ K \to \mathop{\mathrm{Spec}}(K)$.

**Proof.**
Denote $f : X \to \mathop{\mathrm{Spec}}(k)$ the structure morphism and denote $f' : X \to \mathop{\mathrm{Spec}}(k')$ the given factorization. In the proof of Lemma 48.27.1 we took $\omega _ X^\bullet = a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$ where $a$ be is the right adjoint of Lemma 48.3.1 for $f$. Thus we have to show $a(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)}) \cong a'(\mathcal{O}_{\mathop{\mathrm{Spec}}(k)})$ where $a'$ be is the right adjoint of Lemma 48.3.1 for $f'$. Since $k' \subset H^0(X, \mathcal{O}_ X)$ we see that $k'/k$ is a finite extension (Cohomology of Schemes, Lemma 30.19.2). By uniqueness of adjoints we have $a = a' \circ b$ where $b$ is the right adjoint of Lemma 48.3.1 for $g : \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$. Another way to say this: we have $f^! = (f')^! \circ g^!$. Thus it suffices to show that $\mathop{\mathrm{Hom}}\nolimits _ k(k', k) \cong k'$ as $k'$-modules, see Example 48.3.2. This holds because these are $k'$-vector spaces of the same dimension (namely dimension $1$).

Proof of (2). This holds because we have base change for $a$ by Lemma 48.6.2. See discussion in Remark 48.12.5.
$\square$

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