Abstract
In Rizzardo and Van den Bergh (An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves, 2014), constructed an example of a triangulated functor between the derived categories of coherent sheaves on smooth projective varieties over a field k of characteristic 0 which is not of the Fourier–Mukai type. The purpose of this note is to show that if \({{\,\mathrm{{char}}\,}}k =p\) then there are very simple examples of such functors. Namely, for a smooth projective Y over \({{\mathbb {Z}}}_p\) with the special fiber \(i: X\hookrightarrow Y\), we consider the functor \(L i^* \circ i_*: D^b(X) \rightarrow D^b(X)\) from the derived categories of coherent sheaves on X to itself. We show that if Y is a flag variety which is not isomorphic to \({{\mathbb {P}}}^1\) then \(L i^* \circ i_*\) is not of the Fourier–Mukai type. Note that by a theorem of Toën (Invent Math 167:615–667, 2007: Theorem 8.15) the latter assertion is equivalent to saying that \(L i^* \circ i_*\) does not admit a lifting to a \({{\mathbb {F}}}_p\)-linear DG quasi-functor \(D^b_{dg}(X) \rightarrow D^b_{dg}(X)\), where \(D^b_{dg}(X)\) is a (unique) DG enhancement of \(D^b(X)\). However, essentially by definition, \(L i^* \circ i_*\) lifts to a \({{\mathbb {Z}}}_p\)-linear DG quasi-functor.
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Given smooth proper schemes \(X_1, X_2\) over a field k and an object \(E\in D^b(X_1\times _k X_2)\) of the bounded derived category of coherent sheaves on \(X_1\times _k X_2\) define a triangulated functor
sending a bounded complex M of coherent sheaves on \(X_1\) to \( Rp_{2 *}(E{\mathop {\otimes }\limits ^{L}} p^*_1 M)\), where \(p_i: X_1\times _k X_2 \rightarrow X_i\) are the projections. Recall that a triangulated functor \(D^b(X_1) \rightarrow D^b(X_2)\) is said to be of the Fourier–Mukai type if it is isomorphic to \(\Phi _E\) for some E.
Let Y be a smooth projective scheme over \({{\,\mathrm{{Spec}}\,}}{{\mathbb {Z}}}_p\), and let X be its special fiber, \(i: X \hookrightarrow Y\) the closed embedding. Consider the triangulated functor \(G: D^b(X) \rightarrow D^b(X)\) given by the formula
We shall see that in general G is not of the Fourier–Mukai type.
Let Z a smooth projective scheme over \({{\,\mathrm{{Spec}}\,}}{{\mathbb {Z}}}_p\), \(Y= Z\times _{{{\mathbb {Z}}}_p} Z\), \(X= Y\times _{{{\mathbb {Z}}}_p} {{\,\mathrm{{Spec}}\,}}{{\mathbb {F}}}_p\) . Assume that
- (1)
The Frobenius morphism \(Fr: {\overline{Z}}\rightarrow {\overline{Z}}\), where \({\overline{Z}}= Z \times _{{{\mathbb {Z}}}_p} {{\,\mathrm{{Spec}}\,}}{{\mathbb {F}}}_p\), does not lift modulo \(p^2\).
- (2)
\(H^1(X, T_X)=0\), where \(T_X\) is the tangent sheaf on X.
Then \(G = L i^* \circ i_*: D^b(X) \rightarrow D^b(X)\) is not of the Fourier–Mukai type.
For example, let \(GL_n\) be the general linear group over \({{\,\mathrm{{Spec}}\,}}{{\mathbb {Z}}}_p\), \(B\subset GL_n\) a Borel subgroup. Then, by Theorem 6 from Buch et al. (1997), for any \(n>2\), the flag variety \(Z= GL_n/B\) satisfies the first assumption of the Theorem i.e., the Frobenius \(Fr: {\overline{Z}}\rightarrow {\overline{Z}}\) does not lift on \(Z \times _{{{\mathbb {Z}}}_p} {{\,\mathrm{{Spec}}\,}}{{\mathbb {Z}}}/p^2 {{\mathbb {Z}}}\). By Kumar et al. (1999), Theorem 2, we have that \(H^1({\overline{Z}}, T_{{\overline{Z}}})= H^1({\overline{Z}}, {{\mathcal {O}}}_{{\overline{Z}}})=0\). It follows that \(H^1(X, T_X)=0\). Hence, by the Theorem, for \(n>2\), \(G: D^b(X) \rightarrow D^b(X)\) is not of the Fourier–Mukai type.
Assume the contrary and let \(E\in D^b(X\times _{{{\mathbb {F}}}_p} X)\) be a Fourier–Mukai kernel of G. By definition, for every \(M\in D^b(X)\) we have a functorial isomorphism
By the projection formula (Hartshorne 1966, Chapter II, Prop. 5.6) we have that
In particular, if M is a coherent sheaf then \({\underline{H}}^i(G(M)) \simeq M\) for \(i=0, -1\) and \({\underline{H}}^i(G(M))=0\) otherwise. Applying this observation and formula (2) to skyscraper sheaves, \(M= \delta _x\), \(x\in X({{\overline{{{\mathbb {F}}}}}}_p)\), we conclude that the coherent sheaves \({\underline{H}}^i(E)\) are set theoretically supported on the diagonal \(\Delta _X \subset X\times _{{{\mathbb {F}}}_p}X\). Applying the same formulas to \(M= {{\mathcal {O}}}_X\) we see that \( p_{2 *}({\underline{H}}^i(E))={{\mathcal {O}}}_X\) for \(i=0, -1\) and \( p_{2 *}({\underline{H}}^i(E))=0\) otherwise. In fact, every coherent sheaf F on \(X\times _{{{\mathbb {F}}}_p} X\) which is set theoretically supported on the diagonal and such that \(p_{2 *} F = {{\mathcal {O}}}_X\) is isomorphic to \({{\mathcal {O}}}_{\Delta _X}\). It follows that \({\underline{H}}^0(E)= {\underline{H}}^{-1}(E)= {{\mathcal {O}}}_{\Delta _X}\). In the other words, E fits into an exact triangle in \( D^b(X\times X)\)
for some \(\beta \in Ext^2_{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p} X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X}).\) We wish to show that the second assumption in the Theorem implies that \(\beta = 0\), while the first one implies that \(\beta \ne 0\). For every \(M \in D^b(X)\), (3) gives rise to an exact triangle
Our main tool is the following result.
FormalPara LemmaFor a coherent sheaf M on X the following conditions are equivalent.
- (1)
\(\beta _M=0\).
- (2)
\(G(M)\buildrel {\sim }\over {\longrightarrow }M\oplus M[1]\).
- (3)
There exists a morphism \(\lambda : G(M) \rightarrow M[1]\) such that \(\lambda \circ \alpha _M\) is an isomorphism.
- (4)
M admits a lift modulo \(p^2\)i.e., there is a coherent sheaf \({{\tilde{M}}}\) on Y flat over \({{\mathbb {Z}}}/p^2 {{\mathbb {Z}}}\) such that \(i^*{{\tilde{M}}} \simeq M\).
The equivalence of (1), (2) and (3) is immediate. Let us check that (3) is equivalent to (4). By adjunction a morphism \(\lambda : G(M) \rightarrow M[1]\) gives rise to a morphism \(\gamma : i_*M \rightarrow i_* M[1]\). Note that \({{\tilde{M}}} : =( {{\,\mathrm{{cone}}\,}}\gamma ) [-1]\) is a coherent sheaf on Y which is an extension of \(i_*M\) by itself:
It suffices to prove that \(\lambda \circ \alpha _M: M[1] \rightarrow M[1]\) is an isomorphism if and only if \({{\tilde{M}}}\) is flat over \({{\mathbb {Z}}}/p^2 {{\mathbb {Z}}}\).
The exact sequence (5) gives rise to an exact triangle
This, in turn, yields a long exact sequence of the cohomology sheaves
It follows that the morphism \(\lambda \circ \alpha _M\) is an isomorphism if and only if the morphisms v and u from exact sequence (5) induce isomorphisms \( i_*M \buildrel {\sim }\over {\longrightarrow }{{\,\mathrm{{Ker}}\,}}({{\tilde{M}}} {\mathop {\longrightarrow }\limits ^{p}} {{\tilde{M}}})\), \({{\,\mathrm{{Coker}}\,}}({{\tilde{M}}} {\mathop {\longrightarrow }\limits ^{p}} {{\tilde{M}}}) \buildrel {\sim }\over {\longrightarrow }i_*M\). The latter condition is equivalent to the flatness of \({{\tilde{M}}}\) over \({{\mathbb {Z}}}/p^2 {{\mathbb {Z}}}\). \(\square \)
We have a spectral sequence converging to \(Ext^*_{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p}X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X}) \) whose second page is \( H^*(X, {{\mathcal {E}}}xt^* _{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p} X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X}))\). In particular, we have a homomorphism
Let us check that the image \(\mu \) of \(\beta \) under this map is 0. To do this we apply the Lemma to skyscraper sheaves \(\delta _x\), where x runs over closed points of X. On the one hand, the evaluation of the bivector field \(\mu \) at x is equal to the class of \(\beta _{\delta _x}\) in \(Ext^2_{{{\mathcal {O}}}_{X}}( \delta _x, \delta _x) \buildrel {\sim }\over {\longrightarrow }\wedge ^2 T_{x, X}.\) On the other hand, by the Lemma, \(\beta _{\delta _x}=0\) since \(\delta _x\) is liftable modulo \(p^2\). Next, the assumption that \(H^1(X, T_X)=0\) implies that \(\beta \) lies in the image of the map
The map (6) has a left inverse \(u: Ext^2_{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p} X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X}) \rightarrow H^2(X, {{\mathcal {O}}}_X)\) which takes \(\beta \) to \(\beta _{{{\mathcal {O}}}_X}\). But, by the Lemma, the later class is equal to 0 since \({{\mathcal {O}}}_X\) is liftable modulo \(p^2\). It follows that \(\beta \) is 0.
On the other hand, let \(\Gamma \subset X= {\overline{Z}}\times _{{{\mathbb {F}}}_p} {\overline{Z}}\) be the graph of the Frobenius morphism \(Fr: {\overline{Z}}\rightarrow {\overline{Z}}\) and \({{\mathcal {O}}}_{\Gamma }\) the structure sheaf of \(\Gamma \) viewed as a coherent sheaf on X. Then, by our first assumption, the sheaf \({{\mathcal {O}}}_{\Gamma }\) is not liftable modulo \(p^2\). Hence, by the Lemma, \(\beta _{{{\mathcal {O}}}_{\Gamma }}\) is not 0. This contradiction completes the proof. \(\square \)
Let X be a smooth proper scheme over \({{\mathbb {F}}}_p\). The bounded derived category \(D^b(X)\) of coherent sheaves on X has a natural dg enhencement \(L_{parf}(X)\) which is a dg category over \({{\mathbb {F}}}_p\) whose homotopy category \(\text {Ho}(L_{parf}(X))\) is equivalent to \(D^b(X)\) (see, for example, Toën 2007, Sect. 8.3). One has a functor
from the homotopy category of \({{\mathbb {F}}}_p\)-linear dg quasi-endofunctors of \(L_{parf}(X)\) to the category of triangulated endofunctors of \(D^b(X)\). According to (Toën 2007, Theorem 8.15) the dg category \(R{\underline{{{\,\mathrm{{Hom}}\,}}}}_{{{\mathbb {F}}}_p}(L_{parf}(X), L_{parf}(X))\) is homotopy equivalent to the dg category \(L_{parf}(X\times _{{{\mathbb {F}}}_p} X)\), so that the essential image of (7) consists of triangulated endofunctors of the Fourier–Mukai type. On the other hand, any dg category over \({{\mathbb {F}}}_p\) can be viewed as a dg category over \({{\mathbb {Z}}}_p\). In particular, one can consider the dg category \(R{\underline{{{\,\mathrm{{Hom}}\,}}}}_{{{\mathbb {Z}}}_p}(L_{parf}(X), L_{parf}(X))\) of \({{\mathbb {Z}}}_p\)-linear dg quasi-endofunctors of \(L_{parf}(X)\). Functor (7) factors as follows.
Given a flat lifting Y of X over \({{\mathbb {Z}}}_p\) one can view the functor \(Li^* i_*\) as an object of the category \( \text {Ho}(R{\underline{{{\,\mathrm{{End}}\,}}}}_{{{\mathbb {Z}}}_p}(L_{parf}(X) )) \). The construction from this paper is inspired by the simple observation that for any X and Y (for example, one can take \(X={{\,\mathrm{{Spec}}\,}}{{\mathbb {F}}}_p\), \(Y={{\,\mathrm{{Spec}}\,}}{{\mathbb {Z}}}_p\)) the \({{\mathbb {Z}}}_p\)-linear dg quasi-functor \(Li^* i_*\) is not in the image of \( \text {Ho}(R{\underline{{{\,\mathrm{{End}}\,}}}}_{{{\mathbb {F}}}_p}(L_{parf}(X) ))\).
References
Buch, A., Thomsen, J.F., Lauritzen, N., Mehta, V.: The Frobenius morphism on a toric variety. Tohoku Math. J. (2) 49(3), 355–366 (1997)
Hartshorne, R.: Resudues and duality. LNM 20, (1966)
Kumar, S., Lauritzen, N., Thomsen, J.F.: Frobenius splitting of cotangent bundles of flag varieties. Invent. Math. 136, 603–62 (1999)
Rizzardo, A., Van den Bergh, M.: An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves (2014). arXiv:1410.4039
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Acknowledgements
I would like to thank Alberto Canonaco and Paolo Stellari: their interest prompted writing this note. Also, I am grateful to Alexander Samokhin for stimulating discussions and references. I would like to thanks the referee for his comments which helped to improve the exposition. The author was partially supported by the Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. No. 14.641.31.0001.
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Dedicated to Rafail Kalmanovich Gordin on the occasion of his 70th birthday.
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Vologodsky, V. Triangulated Endofunctors of the Derived Category of Coherent Sheaves Which do not Admit DG Liftings. Arnold Math J. 5, 387–391 (2019). https://doi.org/10.1007/s40598-019-00127-6
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DOI: https://doi.org/10.1007/s40598-019-00127-6