The Stacks project

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

  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. $\square$

Remark 48.27.2. Let $k$, $X$, and $\omega _ X^\bullet $ be as in Lemma 48.27.1. The identity on the complex $\omega _ X^\bullet $ corresponds, via the functorial isomorphism in part (5), to a map

\[ t : H^0(X, \omega _ X^\bullet ) \longrightarrow k \]

For an arbitrary $K$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ the identification $\mathop{\mathrm{Hom}}\nolimits (K, \omega _ X^\bullet )$ with $H^0(X, K)^\vee $ in part (5) corresponds to the pairing

\[ \mathop{\mathrm{Hom}}\nolimits _ X(K, \omega _ X^\bullet ) \times H^0(X, K) \longrightarrow k,\quad (\alpha , \beta ) \longmapsto t(\alpha (\beta )) \]

This follows from the functoriality of the isomorphisms in (5). Similarly for any $i \in \mathbf{Z}$ we get the pairing

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X^\bullet ) \times H^{-i}(X, K) \longrightarrow k,\quad (\alpha , \beta ) \longmapsto t(\alpha (\beta )) \]

Here we think of $\alpha $ as a morphism $K[-i] \to \omega _ X^\bullet $ and $\beta $ as an element of $H^0(X, K[-i])$ in order to define $\alpha (\beta )$. Observe that if $K$ is general, then we only know that this pairing is nondegenerate on one side: the pairing induces an isomorphism of $\mathop{\mathrm{Hom}}\nolimits _ X(K, \omega _ X^\bullet )$, resp. $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X^\bullet )$ with the $k$-linear dual of $H^0(X, K)$, resp. $H^{-i}(X, K)$ but in general not vice versa. If $K$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$, then $\mathop{\mathrm{Hom}}\nolimits _ X(K, \omega _ X^\bullet )$, $\mathop{\mathrm{Ext}}\nolimits _ X(K, \omega _ X^\bullet )$, $H^0(X, K)$, and $H^ i(X, K)$ are finite dimensional $k$-vector spaces (by Derived Categories of Schemes, Lemmas 36.11.5 and 36.11.4) and the pairings are perfect in the usual sense.

Remark 48.27.3. We continue the discussion in Remark 48.27.2 and we use the same notation $k$, $X$, $\omega _ X^\bullet $, and $t$. If $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module we obtain perfect pairings

\[ \langle -, - \rangle : \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \omega _ X^\bullet ) \times H^{-i}(X,\mathcal{F}) \longrightarrow k,\quad (\alpha , \beta ) \longmapsto t(\alpha (\beta )) \]

of finite dimensional $k$-vector spaces. These pairings satisfy the following (obvious) functoriality: if $\varphi : \mathcal{F} \to \mathcal{G}$ is a homomorphism of coherent $\mathcal{O}_ X$-modules, then we have

\[ \langle \alpha \circ \varphi , \beta \rangle = \langle \alpha , \varphi (\beta ) \rangle \]

for $\alpha \in \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{G}, \omega _ X^\bullet )$ and $\beta \in H^{-i}(X, \mathcal{F})$. In other words, the $k$-linear map $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{G}, \omega _ X^\bullet ) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \omega _ X^\bullet )$ induced by $\varphi $ is, via the pairings, the $k$-linear dual of the $k$-linear map $H^{-i}(X, \mathcal{F}) \to H^{-i}(X, \mathcal{G})$ induced by $\varphi $. Formulated in this manner, this still works if $\varphi $ is a homomorphism of quasi-coherent $\mathcal{O}_ X$-modules.

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 20.48.7.

Proof. By replacing $E$ with $E[-i]$ this reduces to the case $i = 0$. By Cohomology, Lemma 20.49.2 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

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

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

  3. 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 $K \in D_\mathit{QCoh}(X)$,

  4. 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. $\square$

Remark 48.27.6. Let $X$ be a proper Cohen-Macaulay scheme over a field $k$ which is equidimensional of dimension $d$. Let $\omega _ X^\bullet $ and $\omega _ X$ be as in Lemma 48.27.1. By Lemma 48.27.5 we have $\omega _ X^\bullet = \omega _ X[d]$. Let $t : H^ d(X, \omega _ X) \to k$ be the map of Remark 48.27.2. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee $. Then we have perfect pairings

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

where $\cup $ is the cup-product and $\epsilon : \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{E} \to \mathcal{O}_ X$ is the evaluation map. This is a special case of Lemma 48.27.4.

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.

  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 $X \to \mathop{\mathrm{Spec}}(k')$.

  2. 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. $\square$

[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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