84.2 Dualizing complexes on algebraic spaces

Let $U$ be a locally Noetherian scheme. Let $\mathcal{O}_{\acute{e}tale}$ be the structure sheaf of $U$ on the small étale site of $U$. We will say an object $K \in D_\mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$ is a dualizing complex on $U$ if $K = \epsilon ^*(\omega _ U^\bullet )$ for some dualizing complex $\omega _ U^\bullet$ in the sense of Duality for Schemes, Section 48.2. Here $\epsilon ^* : D_\mathit{QCoh}(\mathcal{O}_ U) \to D_\mathit{QCoh}(\mathcal{O}_{\acute{e}tale})$ is the equivalence of Derived Categories of Spaces, Lemma 73.4.2. Most of the properties of $\omega _ U^\bullet$ studied in Duality for Schemes, Section 48.2 are inherited by $K$ via the discussion in Derived Categories of Spaces, Sections 73.4 and 73.13.

We define a dualizing complex on a locally Noetherian algebraic space to be a complex which étale locally comes from a dualizing complex on the corresponding scheme.

Lemma 84.2.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent

1. For every étale morphism $U \to X$ where $U$ is a scheme the restriction $K|_ U$ is a dualizing complex for $U$ (as discussed above).

2. There exists a surjective étale morphism $U \to X$ where $U$ is a scheme such that $K|_ U$ is a dualizing complex for $U$.

Proof. Assume $U \to X$ is surjective étale where $U$ is a scheme. Let $V \to X$ be an étale morphism where $V$ is a scheme. Then

$U \leftarrow U \times _ X V \rightarrow V$

are étale morphisms of schemes with the arrow to $V$ surjective. Hence we can use Duality for Schemes, Lemma 48.26.1 to see that if $K|_ U$ is a dualizing complex for $U$, then $K|_ V$ is a dualizing complex for $V$. $\square$

Definition 84.2.2. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. An object $K$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is called a dualizing complex if $K$ satisfies the equivalent conditions of Lemma 84.2.1.

Lemma 84.2.3. Let $A$ be a Noetherian ring and let $X = \mathop{\mathrm{Spec}}(A)$. Let $\mathcal{O}_{\acute{e}tale}$ be the structure sheaf of $X$ on the small étale site of $X$. Let $K, L$ be objects of $D(A)$. If $K \in D_{\textit{Coh}}(A)$ and $L$ has finite injective dimension, then

$\epsilon ^*\widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(\epsilon ^*\widetilde{K}, \epsilon ^*\widetilde{L})$

in $D(\mathcal{O}_{\acute{e}tale})$ where $\epsilon : (X_{\acute{e}tale}, \mathcal{O}_{\acute{e}tale}) \to (X, \mathcal{O}_ X)$ is as in Derived Categories of Spaces, Section 73.4.

Proof. By Duality for Schemes, Lemma 48.2.3 we have a canonical isomorphism

$\widetilde{R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)} = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L})$

in $D(\mathcal{O}_ X)$. There is a canonical map

$\epsilon ^*R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L}) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(\epsilon ^*\widetilde{K}, \epsilon ^*\widetilde{L})$

in $D(\mathcal{O}_{\acute{e}tale})$, see Cohomology on Sites, Remark 21.34.11. We will show the left and right hand side of this arrow have isomorphic cohomology sheaves, but we will omit the verification that the isomorphism is given by this arrow.

We may assume that $L$ is given by a finite complex $I^\bullet$ of injective $A$-modules. By induction on the length of $I^\bullet$ and compatibility of the constructions with distinguished triangles, we reduce to the case that $L = I[0]$ where $I$ is an injective $A$-module. Recall that the cohomology sheaves of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(\epsilon ^*\widetilde{K}, \epsilon ^*\widetilde{L}))$ are the sheafifications of the presheaf sending $U$ étale over $X$ to the $i$th ext group between the restrictions of $\epsilon ^*\widetilde{K}$ and $\epsilon ^*\widetilde{L}$ to $U_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.34.1. If $U = \mathop{\mathrm{Spec}}(B)$ is affine, then this ext group is equal to $\text{Ext}^ i_ B(K \otimes _ A B, L \otimes _ A B)$ by the equivalence of Derived Categories of Spaces, Lemma 73.4.2 and Derived Categories of Schemes, Lemma 36.3.5 (this also uses the compatibilities detailed in Derived Categories of Spaces, Remark 73.6.3). Since $A \to B$ is étale, we see that $I \otimes _ A B$ is an injective $B$-module by Dualizing Complexes, Lemma 47.26.4. Hence we see that

\begin{align*} \mathop{\mathrm{Ext}}\nolimits ^ n_ B(K \otimes _ A B, I \otimes _ A B) & = \mathop{\mathrm{Hom}}\nolimits _ B(H^{-n}(K \otimes _ A B), I \otimes _ A B) \\ & = \mathop{\mathrm{Hom}}\nolimits _{A_ f}(H^{-n}(K) \otimes _ A B, I \otimes _ A B) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(H^{-n}(K), I) \otimes _ A B \\ & = \text{Ext}^ n_ A(K, I) \otimes _ A B \end{align*}

The penultimate equality because $H^{-n}(K)$ is a finite $A$-module, see More on Algebra, Remark 15.62.21. Therefore the cohomology sheaves of the left and right hand side of the equality in the lemma are the same. $\square$

Lemma 84.2.4. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $K$ be a dualizing complex on $X$. Then $K$ is an object of $D_{\textit{Coh}}(\mathcal{O}_ X)$ and $D = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(-, K)$ induces an anti-equivalence

$D : D_{\textit{Coh}}(\mathcal{O}_ X) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_ X)$

which comes equipped with a canonical isomorphism $\text{id} \to D \circ D$. If $X$ is quasi-compact, then $D$ exchanges $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ and $D^-_{\textit{Coh}}(\mathcal{O}_ X)$ and induces an equivalence $D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ X)$.

Proof. Let $U \to X$ be an étale morphism with $U$ affine. Say $U = \mathop{\mathrm{Spec}}(A)$ and let $\omega _ A^\bullet$ be a dualizing complex for $A$ corresponding to $K|_ U$ as in Lemma 84.2.1 and Duality for Schemes, Lemma 48.2.1. By Lemma 84.2.3 the diagram

$\xymatrix{ D_{\textit{Coh}}(A) \ar[r] \ar[d]_{R\mathop{\mathrm{Hom}}\nolimits _ A(-, \omega _ A^\bullet )} & D_{\textit{Coh}}(\mathcal{O}_{\acute{e}tale}) \ar[d]^{R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(-, K|_ U)} \\ D_{\textit{Coh}}(A) \ar[r] & D(\mathcal{O}_{\acute{e}tale}) }$

commutes where $\mathcal{O}_{\acute{e}tale}$ is the structure sheaf of the small étale site of $U$. Since formation of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits$ commutes with restriction, we conclude that $D$ sends $D_{\textit{Coh}}(\mathcal{O}_ X)$ into $D_{\textit{Coh}}(\mathcal{O}_ X)$. Moreover, the canonical map

$L \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, K), K)$

(Cohomology on Sites, Lemma 21.34.5) is an isomorphism for all $L$ in $D_{\textit{Coh}}(\mathcal{O}_ X)$ because this is true over all $U$ as above by Dualizing Complexes, Lemma 47.15.2. The statement on boundedness properties of the functor $D$ in the quasi-compact case also follows from the corresponding statements of Dualizing Complexes, Lemma 47.15.2. $\square$

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Recall that an object $L$ of $D(\mathcal{O})$ is invertible if it is an invertible object for the symmetric monoidal structure on $D(\mathcal{O}_ X)$ given by derived tensor product. In Cohomology on Sites, Lemma 21.47.2 we we have seen this means $L$ is perfect and if $(\mathcal{C}, \mathcal{O})$ is a locally ringed site, then for every object $U$ of $\mathcal{C}$ there is a covering $\{ U_ i \to U\}$ of $U$ in $\mathcal{C}$ such that $L|_{U_ i} \cong \mathcal{O}_{U_ i}[-n_ i]$ for some integers $n_ i$.

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. If $L$ in $D(\mathcal{O}_ X)$ is invertible, then there is a disjoint union decomposition $X = \coprod _{n \in \mathbf{Z}} X_ n$ such that $L|_{X_ n}$ is an invertible module sitting in degree $n$. In particular, it follows that $L = \bigoplus H^ n(L)[-n]$ which gives a well defined complex of $\mathcal{O}_ X$-modules (with zero differentials) representing $L$.

Lemma 84.2.5. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. If $K$ and $K'$ are dualizing complexes on $X$, then $K'$ is isomorphic to $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ for some invertible object $L$ of $D(\mathcal{O}_ X)$.

Proof. Set

$L = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, K')$

This is an invertible object of $D(\mathcal{O}_ X)$, because affine locally this is true. Use Lemma 84.2.3 and Dualizing Complexes, Lemma 47.15.5 and its proof. The evaluation map $L \otimes _{\mathcal{O}_ X}^\mathbf {L} K \to K'$ is an isomorphism for the same reason. $\square$

Lemma 84.2.6. Let $S$ be a scheme. Let $X$ be a locally Noetherian quasi-separated algebraic space over $S$. Let $\omega _ X^\bullet$ be a dualizing complex on $X$. Then $X$ the function $|X| \to \mathbf{Z}$ defined by

$x \longmapsto \delta (x)\text{ such that } \omega _{X, \overline{x}}^\bullet [-\delta (x)] \text{ is a normalized dualizing complex over } \mathcal{O}_{X, \overline{x}}$

is a dimension function on $|X|$.

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. Let $\omega _ U^\bullet$ be the dualizing complex on $U$ associated to $\omega _ X^\bullet |_ U$. If $u \in U$ maps to $x \in |X|$, then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{U, u}$. By Dualizing Complexes, Lemma 47.22.1 we see that if $\omega ^\bullet$ is a normalized dualizing complex for $\mathcal{O}_{U, u}$, then $\omega ^\bullet \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}}$ is a normalized dualizing complex for $\mathcal{O}_{X, \overline{x}}$. Hence we see that the dimension function $U \to \mathbf{Z}$ of Duality for Schemes, Lemma 48.2.6 for the scheme $U$ and the complex $\omega _ U^\bullet$ is equal to the composition of $U \to |X|$ with $\delta$. Using the specializations in $|X|$ lift to specializations in $U$ and that nontrivial specializations in $U$ map to nontrivial specializations in $X$ (Decent Spaces, Lemmas 66.12.2 and 66.12.1) an easy topological argument shows that $\delta$ is a dimension function on $|X|$. $\square$

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