The Stacks project

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. $\square$

Remark 57.13.3. If $F, F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to \mathcal{D}$ are siblings, $F$ is fully faithful, and $X$ is reduced and projective over $k$ then $F \cong F'$; this follows from Proposition 57.11.6 via the argument given in the proof of Theorem 57.14.3. However, in general we do not know whether siblings are isomorphic. Even in the situation of Lemma 57.13.2 it seems difficult to prove that the siblings $F$ and $F'$ are isomorphic functors. If $X$ is smooth and proper over $k$ and $F$ is fully faithful, then $F \cong F'$ as is shown in [Noah]. If you have a proof or a counter example in more general situations, please email stacks.project@gmail.com.

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

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

  2. $F$ 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 space1 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

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

  2. $F$ is fully faithful, and

  3. $G$ 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. $\square$

[1] This often forces $f$ to be the identity, see Varieties, Lemma 33.32.1.

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