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 20.42.9 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 dimension1. 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 anti-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 20.42.9 for the second arrow) is an isomorphism for all $L$ because this is true on affines by Dualizing Complexes, Lemma 47.15.32 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 20.52.2 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 $\mathcal{E}_ x^{i + \delta (x)} \not= 0$, 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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