## 48.2 Dualizing complexes on schemes

We define a dualizing complex on a locally Noetherian scheme to be a complex which affine locally comes from a dualizing complex on the corresponding ring. This is not completely standard but agrees with all definitions in the literature on Noetherian schemes of finite dimension.

Lemma 48.2.1. Let $X$ be a locally Noetherian scheme. Let $K$ be an object of $D(\mathcal{O}_ X)$. The following are equivalent

For every affine open $U = \mathop{\mathrm{Spec}}(A) \subset X$ there exists a dualizing complex $\omega _ A^\bullet $ for $A$ such that $K|_ U$ is isomorphic to the image of $\omega _ A^\bullet $ by the functor $\widetilde{} : D(A) \to D(\mathcal{O}_ U)$.

There is an affine open covering $X = \bigcup U_ i$, $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ such that for each $i$ there exists a dualizing complex $\omega _ i^\bullet $ for $A_ i$ such that $K|_{U_ i}$ is isomorphic to the image of $\omega _ i^\bullet $ by the functor $\widetilde{} : D(A_ i) \to D(\mathcal{O}_{U_ i})$.

**Proof.**
Assume (2) and let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$. Since condition (2) implies that $K$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we find an object $\omega _ A^\bullet $ in $D(A)$ whose associated complex of quasi-coherent sheaves is isomorphic to $K|_ U$, see Derived Categories of Schemes, Lemma 36.3.5. We will show that $\omega _ A^\bullet $ is a dualizing complex for $A$ which will finish the proof.

Since $X = \bigcup U_ i$ is an open covering, we can find a standard open covering $U = D(f_1) \cup \ldots \cup D(f_ m)$ such that each $D(f_ j)$ is a standard open in one of the affine opens $U_ i$, see Schemes, Lemma 26.11.5. Say $D(f_ j) = D(g_ j)$ for $g_ j \in A_{i_ j}$. Then $A_{f_ j} \cong (A_{i_ j})_{g_ j}$ and we have

\[ (\omega _ A^\bullet )_{f_ j} \cong (\omega _ i^\bullet )_{g_ j} \]

in the derived category by Derived Categories of Schemes, Lemma 36.3.5. By Dualizing Complexes, Lemma 47.15.6 we find that the complex $(\omega _ A^\bullet )_{f_ j}$ is a dualizing complex over $A_{f_ j}$ for $j = 1, \ldots , m$. This implies that $\omega _ A^\bullet $ is dualizing by Dualizing Complexes, Lemma 47.15.7.
$\square$

Definition 48.2.2. Let $X$ be a locally Noetherian scheme. An object $K$ of $D(\mathcal{O}_ X)$ is called a *dualizing complex* if $K$ satisfies the equivalent conditions of Lemma 48.2.1.

Please see remarks made at the beginning of this section.

Lemma 48.2.3. Let $A$ be a Noetherian ring and let $X = \mathop{\mathrm{Spec}}(A)$. Let $K, L$ be objects of $D(A)$. If $K \in D_{\textit{Coh}}(A)$ and $L$ has finite injective dimension, then

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

in $D(\mathcal{O}_ X)$.

**Proof.**
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. In this case, Derived Categories of Schemes, Lemma 36.10.8, tells us that the $n$th cohomology sheaf of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\widetilde{K}, \widetilde{L})$ is the sheaf associated to the presheaf

\[ D(f) \longmapsto \mathop{\mathrm{Ext}}\nolimits ^ n_{A_ f}(K \otimes _ A A_ f, I \otimes _ A A_ f) \]

Since $A$ is Noetherian, the $A_ f$-module $I \otimes _ A A_ f$ is injective (Dualizing Complexes, Lemma 47.3.8). Hence we see that

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

The last equality because $H^{-n}(K)$ is a finite $A$-module, see Algebra, Lemma 10.10.2. This proves that the canonical map

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

is a quasi-isomorphism in this case and the proof is done.
$\square$

Lemma 48.2.4. Let $X$ be a Noetherian scheme. Let $K, L, M \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Then the map

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} K \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L), M) \]

of Cohomology, Lemma 20.39.9 is an isomorphism in the following two cases

$K \in D^-_{\textit{Coh}}(\mathcal{O}_ X)$, $L \in D^+_{\textit{Coh}}(\mathcal{O}_ X)$, and $M$ affine locally has finite injective dimension (see proof), or

$K$ and $L$ are in $D_{\textit{Coh}}(\mathcal{O}_ X)$, the object $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)$ has finite tor dimension, and $L$ and $M$ affine locally have finite injective dimension (in particular $L$ and $M$ are bounded).

**Proof.**
Proof of (1). We say $M$ has affine locally finite injective dimension if $X$ has an open covering by affines $U = \mathop{\mathrm{Spec}}(A)$ such that the object of $D(A)$ corresponding to $M|_ U$ (Derived Categories of Schemes, Lemma 36.3.5) has finite injective dimension^{1}. To prove the lemma we may replace $X$ by $U$, i.e., we may assume $X = \mathop{\mathrm{Spec}}(A)$ for some Noetherian ring $A$. Observe that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.5. Moreover, the formation of the left and right hand side of the arrow commutes with the functor $D(A) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 48.2.3 and Derived Categories of Schemes, Lemma 36.10.8 (to be sure this uses the assumptions on $K$, $L$, $M$ and what we just proved about $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$). Then finally the arrow is an isomorphism by More on Algebra, Lemmas 15.98.1 part (2).

Proof of (2). We argue as above. A small change is that here we get $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ in $D_{\textit{Coh}}(\mathcal{O}_ X)$ because affine locally (which is allowable by Lemma 48.2.3) we may appeal to Dualizing Complexes, Lemma 47.15.2. Then we finally conclude by More on Algebra, Lemma 15.98.2.
$\square$

Lemma 48.2.5. Let $K$ be a dualizing complex on a locally Noetherian scheme $X$. 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 $U \subset X$ be an affine open. Say $U = \mathop{\mathrm{Spec}}(A)$ and let $\omega _ A^\bullet $ be a dualizing complex for $A$ corresponding to $K|_ U$ as in Lemma 48.2.1. By Lemma 48.2.3 the diagram

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

commutes. We conclude that $D$ sends $D_{\textit{Coh}}(\mathcal{O}_ X)$ into $D_{\textit{Coh}}(\mathcal{O}_ X)$. Moreover, the canonical map

\[ L \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} L \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, K), K) \]

(using Cohomology, Lemma 20.39.9 for the second arrow) is an isomorphism for all $L$ because this is true on affines by Dualizing Complexes, Lemma 47.15.3^{2} and we have already seen on affines that we recover what happens in algebra. The statement on boundedness properties of the functor $D$ in the quasi-compact case also follows from the corresponding statements of Dualizing Complexes, Lemma 47.15.3.
$\square$

Let $X$ be a locally ringed space. Recall that an object $L$ of $D(\mathcal{O}_ X)$ 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, Lemma 20.49.2 we have seen this means $L$ is perfect and there is an open covering $X = \bigcup U_ i$ such that $L|_{U_ i} \cong \mathcal{O}_{U_ i}[-n_ i]$ for some integers $n_ i$. In this case, the function

\[ x \mapsto n_ x,\quad \text{where }n_ x\text{ is the unique integer such that } H^{n_ x}(L_ x) \not= 0 \]

is locally constant on $X$. In particular, we have $L = \bigoplus H^ n(L)[-n]$ which gives a well defined complex of $\mathcal{O}_ X$-modules (with zero differentials) representing $L$.

Lemma 48.2.6. Let $X$ be a locally Noetherian scheme. 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, see 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 48.2.7. Let $X$ be a locally Noetherian scheme. Let $\omega _ X^\bullet $ be a dualizing complex on $X$. Then $X$ is universally catenary and the function $X \to \mathbf{Z}$ defined by

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

is a dimension function.

**Proof.**
Immediate from the affine case Dualizing Complexes, Lemma 47.17.3 and the definitions.
$\square$

Lemma 48.2.8. Let $X$ be a locally Noetherian scheme. Let $\omega _ X^\bullet $ be a dualizing complex on $X$ with associated dimension function $\delta $. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Set $\mathcal{E}^ i = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^{-i}_{\mathcal{O}_ X}(\mathcal{F}, \omega _ X^\bullet )$. Then $\mathcal{E}^ i$ is a coherent $\mathcal{O}_ X$-module and for $x \in X$ we have

$\mathcal{E}^ i_ x$ is nonzero only for $\delta (x) \leq i \leq \delta (x) + \dim (\text{Supp}(\mathcal{F}_ x))$,

$\dim (\text{Supp}(\mathcal{E}^{i + \delta (x)}_ x)) \leq i$,

$\text{depth}(\mathcal{F}_ x)$ is the smallest integer $i \geq 0$ such that $\mathcal{E}^{i + \delta (x)} \not= 0$, and

we have $x \in \text{Supp}(\bigoplus _{j \leq i} \mathcal{E}^ j) \Leftrightarrow \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \delta (x) \leq i$.

**Proof.**
Lemma 48.2.5 tells us that $\mathcal{E}^ i$ is coherent. Choosing an affine neighbourhood of $x$ and using Derived Categories of Schemes, Lemma 36.10.8 and More on Algebra, Lemma 15.99.2 part (3) we have

\[ \mathcal{E}^ i_ x = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^{-i}_{\mathcal{O}_ X}(\mathcal{F}, \omega _ X^\bullet )_ x = \mathop{\mathrm{Ext}}\nolimits ^{-i}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \omega _{X, x}^\bullet ) = \mathop{\mathrm{Ext}}\nolimits ^{\delta (x) - i}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \omega _{X, x}^\bullet [-\delta (x)]) \]

By construction of $\delta $ in Lemma 48.2.7 this reduces parts (1), (2), and (3) to Dualizing Complexes, Lemma 47.16.5. Part (4) is a formal consequence of (3) and (1).
$\square$

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