57.12 Special functors
In this section we prove some results on functors of a special type that we will use later in this chapter.
Definition 57.12.1. 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 57.10.1 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 57.12.2. 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 57.8.5. Observe that F and F' satisfy the first and second condition of Lemma 57.10.2 and hence are siblings.
\square
Lemma 57.12.4. 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 57.7.1. 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 57.11.3 and we win.
\square
Lemma 57.12.5.reference 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 57.11.6 the functor F is fully faithful. By Lemma 57.12.4 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 57.11.4 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 57.10.2 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 57.12.6. 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 57.12.4. Consider the functor H = F^{-1} \circ G which makes sense as F is fully faithful. By Lemma 57.12.5 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 57.10.3 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 57.8.3 we conclude.
\square
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