## 57.13 Special functors

In this section we prove some results on functors of a special type that we will use later in this chapter.

Definition 57.13.1. Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Recall that $D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D^ b(\textit{Coh}(\mathcal{O}_ X))$ by Derived Categories of Schemes, Proposition 36.11.2. We say two $k$-linear exact functors

\[ F, F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D^ b(\textit{Coh}(\mathcal{O}_ X)) \longrightarrow D^ b_{\textit{Coh}}(\mathcal{O}_ Y) \]

are *siblings*, or we say $F'$ is a *sibling* of $F$ if $F$ and $F'$ are siblings in the sense of Definition 57.11.1 with abelian category being $\textit{Coh}(\mathcal{O}_ X)$. If $X$ is regular then $D_{perf}(\mathcal{O}_ X) = D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.6 and we use the same terminology for $k$-linear exact functors $F, F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$.

Lemma 57.13.2. Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. Let $F : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ be a $k$-linear exact functor sending $\textit{Coh}(\mathcal{O}_ X) \subset D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $\textit{Coh}(\mathcal{O}_ Y) \subset D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. Then there exists a Fourier-Mukai functor $F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ whose kernel is a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ flat over $X$ and with support finite over $Y$ which is a sibling of $F$.

**Proof.**
Denote $H : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ Y)$ the restriction of $F$. Since $F$ is an exact functor of triangulated categories, we see that $H$ is an exact functor of abelian categories. Of course $H$ is $k$-linear as $F$ is. By Functors and Morphisms, Lemma 56.7.5 we obtain a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ which is flat over $X$ and has support finite over $Y$. Let $F'$ be the Fourier-Mukai functor defined using $\mathcal{K}$ so that $F'$ restricts to $H$ on $ \textit{Coh}(\mathcal{O}_ X)$. The functor $F'$ sends $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ by Lemma 57.9.5. Observe that $F$ and $F'$ satisfy the first and second condition of Lemma 57.11.2 and hence are siblings.
$\square$

Lemma 57.13.4. Let $k$ be a field. Let $X$, $Y$ be proper schemes over $k$. Assume $X$ is regular. Let $F, G : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ be $k$-linear exact functors such that

$F(\mathcal{F}) \cong G(\mathcal{F})$ for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$ with $\dim (\text{Supp}(\mathcal{F})) = 0$,

$F$ is fully faithful.

Then the essential image of $G$ is contained in the essential image of $F$.

**Proof.**
Recall that $F$ and $G$ have both adjoints, see Lemma 57.8.1. In particular the essential image $\mathcal{A} \subset D_{perf}(\mathcal{O}_ Y)$ of $F$ satisfies the equivalent conditions of Derived Categories, Lemma 13.40.7. We claim that $G$ factors through $\mathcal{A}$. Since $\mathcal{A} = {}^\perp (\mathcal{A}^\perp )$ by Derived Categories, Lemma 13.40.7 it suffices to show that $\mathop{\mathrm{Hom}}\nolimits _ Y(G(M), N) = 0$ for all $M$ in $D_{perf}(\mathcal{O}_ X)$ and $N \in \mathcal{A}^\perp $. We have

\[ \mathop{\mathrm{Hom}}\nolimits _ Y(G(M), N) = \mathop{\mathrm{Hom}}\nolimits _ X(M, G_ r(N)) \]

where $G_ r$ is the right adjoint to $G$. Thus it suffices to prove that $G_ r(N) = 0$. Since $G(\mathcal{F}) \cong F(\mathcal{F})$ for $\mathcal{F}$ as in (1) we see that

\[ \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, G_ r(N)) = \mathop{\mathrm{Hom}}\nolimits _ Y(G(\mathcal{F}), N) = \mathop{\mathrm{Hom}}\nolimits _ Y(F(\mathcal{F}), N) = 0 \]

as $N$ is in the right orthogonal to the essential image $\mathcal{A}$ of $F$. Of course, the same vanishing holds for $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, G_ r(N)[i])$ for any $i \in \mathbf{Z}$. Thus $G_ r(N) = 0$ by Lemma 57.12.3 and we win.
$\square$

reference
Lemma 57.13.5. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is regular. Let $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ X)$ be a $k$-linear exact functor. Assume for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ with $\dim (\text{Supp}(\mathcal{F})) = 0$ there is an isomorphism $\mathcal{F} \cong F(\mathcal{F})$. Then there exists an automorphism $f : X \to X$ over $k$ which induces the identity on the underlying topological space^{1} and an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ such that $F$ and $F'(M) = f^*M \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{L}$ are siblings.

**Proof.**
By Lemma 57.12.6 the functor $F$ is fully faithful. By Lemma 57.13.4 the essential image of the identity functor is contained in the essential image of $F$, i.e., we see that $F$ is essentially surjective. Thus $F$ is an equivalence. Observe that the quasi-inverse $F^{-1}$ satisfies the same assumptions as $F$.

Let $M \in D_{perf}(\mathcal{O}_ X)$ and say $H^ i(M) = 0$ for $i > b$. Since $F$ is fully faithful, we see that

\[ \mathop{\mathrm{Hom}}\nolimits _ X(M, \mathcal{O}_ x[-i]) = \mathop{\mathrm{Hom}}\nolimits _ X(F(M), F(\mathcal{O}_ x)[-i]) \cong \mathop{\mathrm{Hom}}\nolimits _ X(F(M), \mathcal{O}_ x[-i]) \]

for any $i \in \mathbf{Z}$ for any closed point $x$ of $X$. Thus by Lemma 57.12.4 we see that $F(M)$ has vanishing cohomology sheaves in degrees $> b$.

Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. By the above $F(\mathcal{F})$ has nonzero cohomology sheaves only in degrees $\leq 0$. Set $\mathcal{G} = H^0(F(\mathcal{F}))$. Choose a distinguished triangle

\[ K \to F(\mathcal{F}) \to \mathcal{G} \to K[1] \]

Then $K$ has nonvanishing cohomology sheaves only in degrees $\leq -1$. Applying $F^{-1}$ we obtain a distinguished triangle

\[ F^{-1}(K) \to \mathcal{F} \to F^{-1}(\mathcal{G}) \to F^{-1}(K')[1] \]

Since $F^{-1}(K)$ has nonvanishing cohomology sheaves only in degrees $\leq -1$ (by the previous paragraph applied to $F^{-1}$) we see that the arrow $F^{-1}(K) \to \mathcal{F}$ is zero (Derived Categories, Lemma 13.27.3). Hence $K \to F(\mathcal{F})$ is zero, which implies that $F(\mathcal{F}) = \mathcal{G}$ by our choice of the first distinguished triangle.

From the preceding paragraph, we deduce that $F$ preserves $\textit{Coh}(\mathcal{O}_ X)$ and indeed defines an equivalence $H : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ X)$. By Functors and Morphisms, Lemma 56.7.8 we get an automorphism $f : X \to X$ over $k$ and an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ such that $H(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$. Set $F'(M) = f^*M \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{L}$. Using Lemma 57.11.2 we see that $F$ and $F'$ are siblings. To see that $f$ is the identity on the underlying topological space of $X$, we use that $F(\mathcal{O}_ x) \cong \mathcal{O}_ x$ and that the support of $\mathcal{O}_ x$ is $\{ x\} $. This finishes the proof.
$\square$

Lemma 57.13.6. Let $k$ be a field. Let $X$, $Y$ be proper schemes over $k$. Assume $X$ regular. Let $F, G : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ be $k$-linear exact functors such that

$F(\mathcal{F}) \cong G(\mathcal{F})$ for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$ with $\dim (\text{Supp}(\mathcal{F})) = 0$,

$F$ is fully faithful, and

$G$ is a Fourier-Mukai functor whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$.

Then there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ such that $F$ and $F'$ are siblings.

**Proof.**
The essential image of $G$ is contained in the essential image of $F$ by Lemma 57.13.4. Consider the functor $H = F^{-1} \circ G$ which makes sense as $F$ is fully faithful. By Lemma 57.13.5 we obtain an automorphism $f : X \to X$ and an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ such that the functor $H' : K \mapsto f^*K \otimes \mathcal{L}$ is a sibling of $H$. In particular $H$ is an auto-equivalence by Lemma 57.11.3 and $H$ induces an auto-equivalence of $\textit{Coh}(\mathcal{O}_ X)$ (as this is true for its sibling functor $H'$). Thus the quasi-inverses $H^{-1}$ and $(H')^{-1}$ exist, are siblings (small detail omitted), and $(H')^{-1}$ sends $M$ to $(f^{-1})^*(M \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{L}^{\otimes -1})$ which is a Fourier-Mukai functor (details omitted). Then of course $F = G \circ H^{-1}$ is a sibling of $G \circ (H')^{-1}$. Since compositions of Fourier-Mukai functors are Fourier-Mukai by Lemma 57.9.3 we conclude.
$\square$

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