Situation 57.13.1. Here $k$ is a field. We have proper smooth schemes $X$ and $Y$ over $k$. We have a $k$-linear, exact, fully faithful functor $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$.
57.13 Fully faithful functors
Our goal is to prove fully faithful functors between derived categories are siblings of Fourier-Mukai functors, following [Orlov-K3] and [Ballard].
Before reading on, it makes sense to read at least some of Derived Categories, Section 13.41.
Recall that $X$ is regular and hence has the resolution property (Varieties, Lemma 33.25.3 and Derived Categories of Schemes, Lemma 36.36.8). Thus on $X \times X$ we may choose a resolution
\[ \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0 \]
where each $\mathcal{E}_ i$ and $\mathcal{G}_ i$ is a finite locally free $\mathcal{O}_ X$-module, see Lemma 57.9.3. Using the complex
57.13.1.1
\begin{equation} \label{equiv-equation-original-complex} \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \end{equation}
in $D_{perf}(\mathcal{O}_{X \times X})$ as in Derived Categories, Example 13.41.2 if for each $n$ we denote
\[ M_ n = (\mathcal{E}_ n \boxtimes \mathcal{G}_ n \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0)[-n] \]
we obtain an infinite Postnikov system for the complex (57.13.1.1). This means the morphisms $M_0 \to M_1[1] \to M_2[2] \to \ldots $ and $M_ n \to \mathcal{E}_ n \boxtimes \mathcal{G}_ n$ and $\mathcal{E}_ n \boxtimes \mathcal{G}_ n \to M_{n - 1}$ satisfy certain conditions documented in Derived Categories, Definition 13.41.1. Set
\[ \mathcal{F}_ n = \mathop{\mathrm{Ker}}(\mathcal{E}_ n \boxtimes \mathcal{G}_ n \to \mathcal{E}_{n - 1} \boxtimes \mathcal{G}_{n - 1}) \]
Observe that since $\mathcal{O}_\Delta $ is flat over $X$ via $\text{pr}_1$ the same is true for $\mathcal{F}_ n$ for all $n$ (this is a convenient though not essential observation). We have
\[ H^ q(M_ n[n]) = \left\{ \begin{matrix} \mathcal{O}_\Delta
& \text{if}
& q = 0
\\ \mathcal{F}_ n
& \text{if}
& q = -n
\\ 0
& \text{if}
& q \not= 0, -n
\end{matrix} \right. \]
Thus for $n \geq \dim (X \times X)$ we have
\[ M_ n[n] \cong \mathcal{O}_\Delta \oplus \mathcal{F}_ n[n] \]
in $D_{perf}(\mathcal{O}_{X \times X})$ by Lemma 57.9.5.
We are interested in the complex
57.13.1.2
\begin{equation} \label{equiv-equation-complex} \ldots \to \mathcal{E}_2 \boxtimes F(\mathcal{G}_2) \to \mathcal{E}_1 \boxtimes F(\mathcal{G}_1) \to \mathcal{E}_0 \boxtimes F(\mathcal{G}_0) \end{equation}
in $D_{perf}(\mathcal{O}_{X \times Y})$ as the “totalization” of this complex should give us the kernel of the Fourier-Mukai functor we are trying to construct. For all $i, j \geq 0$ we have
\begin{align*} \mathop{\mathrm{Ext}}\nolimits ^ q_{X \times Y}(\mathcal{E}_ i \boxtimes F(\mathcal{G}_ i), \mathcal{E}_ j \boxtimes F(\mathcal{G}_ j)) & = \bigoplus \nolimits _ p \mathop{\mathrm{Ext}}\nolimits ^{q + p}_ X(\mathcal{E}_ i, \mathcal{E}_ j) \otimes _ k \mathop{\mathrm{Ext}}\nolimits ^{-p}_ Y(F(\mathcal{G}_ i), F(\mathcal{G}_ j)) \\ & = \bigoplus \nolimits _ p \mathop{\mathrm{Ext}}\nolimits ^{q + p}_ X(\mathcal{E}_ i, \mathcal{E}_ j) \otimes _ k \mathop{\mathrm{Ext}}\nolimits ^{-p}_ X(\mathcal{G}_ i, \mathcal{G}_ j) \end{align*}
The second equality holds because $F$ is fully faithful and the first by Derived Categories of Schemes, Lemma 36.25.1. We find these $\mathop{\mathrm{Ext}}\nolimits ^ q$ are zero for $q < 0$. Hence by Derived Categories, Lemma 13.41.6 we can build an infinite Postnikov system $K_0, K_1, K_2, \ldots $ in $D_{perf}(\mathcal{O}_{X \times Y})$ for the complex (57.13.1.2). Parallel to what happens with $M_0, M_1, M_2, \ldots $ this means we obtain morphisms $K_0 \to K_1[1] \to K_2[2] \to \ldots $ and $K_ n \to \mathcal{E}_ n \boxtimes F(\mathcal{G}_ n)$ and $\mathcal{E}_ n \boxtimes F(\mathcal{G}_ n) \to K_{n - 1}$ in $D_{perf}(\mathcal{O}_{X \times Y})$ satisfying certain conditions documented in Derived Categories, Definition 13.41.1.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module whose support has a finite number of points, i.e., with $\dim (\text{Supp}(\mathcal{F})) = 0$. Consider the exact functor of triangulated categories
\[ D_{perf}(\mathcal{O}_{X \times Y}) \longrightarrow D_{perf}(\mathcal{O}_ Y),\quad N \longmapsto R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times Y}} N) \]
It follows that the objects $R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times Y}} K_ i)$ form a Postnikov system for the complex in $D_{perf}(\mathcal{O}_ Y)$ with terms
\[ R\text{pr}_{2, *}( (\mathcal{F} \otimes \mathcal{E}_ i) \boxtimes F(\mathcal{G}_ i)) = \Gamma (X, \mathcal{F} \otimes \mathcal{E}_ i) \otimes _ k F(\mathcal{G}_ i) = F(\Gamma (X, \mathcal{F} \otimes \mathcal{E}_ i) \otimes _ k \mathcal{G}_ i) \]
Here we have used that $\mathcal{F} \otimes \mathcal{E}_ i$ has vanishing higher cohomology as its support has dimension $0$. On the other hand, applying the exact functor
\[ D_{perf}(\mathcal{O}_{X \times X}) \longrightarrow D_{perf}(\mathcal{O}_ Y),\quad N \longmapsto F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times X}} N)) \]
we find that the objects $F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times X}} M_ n))$ form a second infinite Postnikov system for the complex in $D_{perf}(\mathcal{O}_ Y)$ with terms
\[ F(R\text{pr}_{2, *}( (\mathcal{F} \otimes \mathcal{E}_ i) \boxtimes \mathcal{G}_ i)) = F(\Gamma (X, \mathcal{F} \otimes \mathcal{E}_ i) \otimes _ k \mathcal{G}_ i) \]
This is the same as before! By uniqueness of Postnikov systems (Derived Categories, Lemma 13.41.6) which applies because
\[ \mathop{\mathrm{Ext}}\nolimits ^ q_ Y( F(\Gamma (X, \mathcal{F} \otimes \mathcal{E}_ i) \otimes _ k \mathcal{G}_ i), F(\Gamma (X, \mathcal{F} \otimes \mathcal{E}_ j) \otimes _ k \mathcal{G}_ j)) = 0, \quad q < 0 \]
as $F$ is fully faithful, we find a system of isomorphisms
\[ F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times X}} M_ n[n])) \cong R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times Y}} K_ n[n]) \]
in $D_{perf}(\mathcal{O}_ Y)$ compatible with the morphisms in $D_{perf}(\mathcal{O}_ Y)$ induced by the morphisms
\[ M_{n - 1}[n - 1] \to M_ n[n] \quad \text{and}\quad K_{n - 1}[n - 1] \to K_ n[n] \]
\[ M_ n \to \mathcal{E}_ n \boxtimes \mathcal{G}_ n \quad \text{and}\quad K_ n \to \mathcal{E}_ n \boxtimes F(\mathcal{G}_ n) \]
\[ \mathcal{E}_ n \boxtimes \mathcal{G}_ n \to M_{n - 1} \quad \text{and}\quad \mathcal{E}_ n \boxtimes F(\mathcal{G}_ n) \to K_{n - 1} \]
which are part of the structure of Postnikov systems. For $n$ sufficiently large we obtain a direct sum decomposition
\[ F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times X}} M_ n[n])) = F(\mathcal{F}) \oplus F(R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \mathcal{F}_ n ))[n] \]
corresponding to the direct sum decomposition of $M_ n$ constructed above (we are using the flatness of $\mathcal{F}_ n$ over $X$ via $\text{pr}_1$ to write a usual tensor product in the formula above, but this isn't essential for the argument). By Lemma 57.9.9 we find there exists an integer $m \geq 0$ such that the first summand in this direct sum decomposition has nonzero cohomology sheaves only in the interval $[-m, m]$ and the second summand in this direct sum decomposition has nonzero cohomology sheaves only in the interval $[-m - n, m + \dim (X) - n]$. We conclude the system $K_0 \to K_1[1] \to K_2[2] \to \ldots $ in $D_{perf}(\mathcal{O}_{X \times Y})$ satisfies the assumptions of Lemma 57.9.10 after possibly replacing $m$ by a larger integer. We conclude we can write
\[ K_ n[n] = K \oplus C_ n \]
for $n \gg 0$ compatible with transition maps and with $C_ n$ having nonzero cohomology sheaves only in the range $[-m - n, m - n]$. Denote $G$ the Fourier-Mukai functor corresponding to $K$. Putting everything together we find
\[ \begin{matrix} G(\mathcal{F}) \oplus R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}}^\mathbf {L} C_ n) \cong
\\ R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times Y}} K_ n[n]) \cong
\\ F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes ^\mathbf {L}_{\mathcal{O}_{X \times X}} M_ n[n])) \cong
\\ F(\mathcal{F}) \oplus F(R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} \mathcal{F}_ n ))[n]
\end{matrix} \]
Looking at the degrees that objects live in we conclude that for $n \gg m$ we obtain an isomorphism
\[ F(\mathcal{F}) \cong G(\mathcal{F}) \]
Moreover, recall that this holds for every coherent $\mathcal{F}$ on $X$ whose support has dimension $0$.
Lemma 57.13.2. Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. Given a $k$-linear, exact, fully faithful functor $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is a sibling to $F$.
Proof. Apply Lemma 57.12.6 to $F$ and the functor $G$ constructed above. $\square$
The following theorem is also true without assuming $X$ is projective, see [Noah].
Theorem 57.13.3 (Orlov). Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$ with $X$ projective over $k$. Any $k$-linear fully faithful exact functor $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ is a Fourier-Mukai functor for some kernel in $D_{perf}(\mathcal{O}_{X \times Y})$.
Proof. Let $F'$ be the Fourier-Mukai functor which is a sibling of $F$ as in Lemma 57.13.2. By Proposition 57.10.6 we have $F \cong F'$ provided we can show that $\textit{Coh}(\mathcal{O}_ X)$ has enough negative objects. However, if $X = \mathop{\mathrm{Spec}}(k)$ for example, then this isn't true. Thus we first decompose $X = \coprod X_ i$ into its connected (and irreducible) components and we argue that it suffices to prove the result for each of the (fully faithful) composition functors
\[ F_ i : D_{perf}(\mathcal{O}_{X_ i}) \to D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y) \]
Details omitted. Thus we may assume $X$ is irreducible.
The case $\dim (X) = 0$. Here $X$ is the spectrum of a finite (separable) extension $k'/k$ and hence $D_{perf}(\mathcal{O}_ X)$ is equivalent to the category of graded $k'$-vector spaces such that $\mathcal{O}_ X$ corresponds to the trivial $1$-dimensional vector space in degree $0$. It is straightforward to see that any two siblings $F, F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ are isomorphic. Namely, we are given an isomorphism $F(\mathcal{O}_ X) \cong F'(\mathcal{O}_ X)$ compatible the action of the $k$-algebra $k' = \text{End}_{D_{perf}(\mathcal{O}_ X)}(\mathcal{O}_ X)$ which extends canonically to an isomorphism on any graded $k'$-vector space.
The case $\dim (X) > 0$. Here $X$ is a projective smooth variety of dimension $> 1$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. We have to show there exists a coherent module $\mathcal{N}$ such that
there is a surjection $\mathcal{N} \to \mathcal{F}$ and
$\mathop{\mathrm{Hom}}\nolimits (\mathcal{F}, \mathcal{N}) = 0$.
Choose an ample invertible $\mathcal{O}_ X$-module $\mathcal{L}$. We claim that $\mathcal{N} = (\mathcal{L}^{\otimes n})^{\oplus r}$ will work for $n \ll 0$ and $r$ large enough. Condition (1) follows from Properties, Proposition 28.26.13. Finally, we have
\[ \mathop{\mathrm{Hom}}\nolimits (\mathcal{F}, \mathcal{L}^{\otimes n}) = H^0(X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{L}^{\otimes n})) = H^0(X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{O}_ X) \otimes \mathcal{L}^{\otimes n}) \]
Since the dual $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{O}_ X)$ is torsion free, this vanishes for $n \ll 0$ by Varieties, Lemma 33.48.1. This finishes the proof. $\square$
Proposition 57.13.4. Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. If $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ is a $k$-linear exact equivalence of triangulated categories then there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is an equivalence and a sibling of $F$.
Proof. The functor $F'$ of Lemma 57.13.2 is an equivalence by Lemma 57.10.3. $\square$
Lemma 57.13.5. Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $K \in D_{perf}(\mathcal{O}_{X \times X})$. If the Fourier-Mukai functor $\Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ X)$ is isomorphic to the identity functor, then $K \cong \Delta _*\mathcal{O}_ X$ in $_{perf}(\mathcal{O}_{X \times X})$.
Proof. Let $i$ be the minimal integer such that the cohomology sheaf $H^ i(K)$ is nonzero. Let $\mathcal{E}$ and $\mathcal{G}$ be finite locally free $\mathcal{O}_ X$-modules. Then
\begin{align*} H^ i(X \times X, K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} (\mathcal{E} \boxtimes \mathcal{G})) & = H^ i(X, R\text{pr}_{2, *}(K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} (\mathcal{E} \boxtimes \mathcal{G}))) \\ & = H^ i(X, \Phi _ K(\mathcal{E}) \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}) \\ & \cong H^ i(X, \mathcal{E} \otimes \mathcal{G}) \end{align*}
which is zero if $i < 0$. On the other hand, we can choose $\mathcal{E}$ and $\mathcal{G}$ such that there is a surjection $\mathcal{E}^\vee \boxtimes \mathcal{G}^\vee \to H^ i(K)$ by Lemma 57.9.1. In this case the left hand side of the equalities is nonzero. Hence we conclude that $H^ i(K) = 0$ for $i < 0$.
Let $i$ be the maximal integer such that $H^ i(K)$ is nonzero. The same argument with $\mathcal{E}$ and $\mathcal{G}$ support of dimension $0$ shows that $i \leq 0$. Hence we conclude that $K$ is given by a single coherent $\mathcal{O}_{X \times X}$-module $\mathcal{K}$ sitting in degree $0$.
Since $R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})$ is $\mathcal{F}$, by taking $\mathcal{F}$ supported at closed points we see that the support of $\mathcal{K}$ is finite over $X$ via $\text{pr}_2$. Since $R\text{pr}_{2, *}(\mathcal{K}) \cong \mathcal{O}_ X$ we conclude by Functors and Morphisms, Lemma 56.7.6 that $\mathcal{K} = s_*\mathcal{O}_ X$ for some section $s : X \to X \times X$ of the second projection. Then $\Phi _ K(M) = f^*M$ where $f = \text{pr}_1 \circ s$ and this can happen only if $s$ is the diagonal morphism as desired. $\square$
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