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

$$\begin{aligned} \Phi _E: D^b(X_1) \rightarrow D^b(X_2) \end{aligned}$$
(1)

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

$$\begin{aligned} G = L i^* \circ i_* \end{aligned}$$

We shall see that in general G is not of the Fourier–Mukai type.

FormalPara Theorem

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. (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. (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.

FormalPara Proof

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

$$\begin{aligned} G(M)\buildrel {\sim }\over {\longrightarrow }Rp_{2 *}(E{\mathop {\otimes }\limits ^{L}} p^*_1 M). \end{aligned}$$
(2)

By the projection formula (Hartshorne 1966, Chapter II, Prop. 5.6) we have that

$$\begin{aligned} i_* \circ L i^* \circ i_*(M)&\buildrel {\sim }\over {\longrightarrow }i_*(M) {\mathop {\otimes }\limits ^{L}} i_*({{\mathcal {O}}}_X) \buildrel {\sim }\over {\longrightarrow }i_*(M) \otimes ({{\mathcal {O}}}_Y {\mathop {\longrightarrow }\limits ^{p}} {{\mathcal {O}}}_Y)\\&\buildrel {\sim }\over {\longrightarrow }i_*(M) \oplus i_*(M)[1] \end{aligned}$$

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 \( D^b(X\times X)\)

$$\begin{aligned} {{\mathcal {O}}}_{\Delta _X}[1] {\mathop {\longrightarrow }\limits ^{\alpha }} E {\mathop {\longrightarrow }\limits ^{}} {{\mathcal {O}}}_{\Delta _X} {\mathop {\longrightarrow }\limits ^{\beta }} {{\mathcal {O}}}_{\Delta _X}[2] \end{aligned}$$
(3)

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

$$\begin{aligned} M[1] {\mathop {\longrightarrow }\limits ^{\alpha _M}} G(M){\mathop {\longrightarrow }\limits ^{}} M {\mathop {\longrightarrow }\limits ^{\beta _M}} M[2] \end{aligned}$$
(4)

Our main tool is the following result.

FormalPara Lemma

For a coherent sheaf M on X the following conditions are equivalent.

  1. (1)

    \(\beta _M=0\).

  2. (2)

    \(G(M)\buildrel {\sim }\over {\longrightarrow }M\oplus M[1]\).

  3. (3)

    There exists a morphism \(\lambda : G(M) \rightarrow M[1]\) such that \(\lambda \circ \alpha _M\) is an isomorphism.

  4. (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 \(i^*{{\tilde{M}}} \simeq M\).

FormalPara Proof

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:

$$\begin{aligned} 0{\mathop {\longrightarrow }\limits ^{}} i_*M {\mathop {\longrightarrow }\limits ^{v}} {{\tilde{M}}} {\mathop {\longrightarrow }\limits ^{u}} i_*M {\mathop {\longrightarrow }\limits ^{}} 0. \end{aligned}$$
(5)

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

$$\begin{aligned} Li^*i_* M \rightarrow Li^* {{\tilde{M}}} \rightarrow Li^*i_*M \rightarrow Li^*i_* M[1]. \end{aligned}$$

This, in turn, yields a long exact sequence of the cohomology sheaves

$$\begin{aligned} 0= L_2 i^*i_* M \rightarrow L_1 i^*i_* M {\mathop {\longrightarrow }\limits ^{L_1i^*(v)}} L_1i^* {{\tilde{M}}} \rightarrow M{\mathop {\longrightarrow }\limits ^{\lambda \circ \alpha _M[-1]}} M\rightarrow i^* {{\tilde{M}}} {\mathop {\longrightarrow }\limits ^{L_1i^*(u)}} M \rightarrow 0. \end{aligned}$$

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

$$\begin{aligned} Ext^2_{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p} X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X}) \rightarrow H^0(X, {{\mathcal {E}}}xt^2 _{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p} X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X})) \buildrel {\sim }\over {\longrightarrow }H^0(X, \wedge ^2 T_X). \end{aligned}$$

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

$$\begin{aligned} v: H^2(X, {{\mathcal {O}}}_X) \buildrel {\sim }\over {\longrightarrow }H^2 (X, {{\mathcal {E}}}xt^0 _{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p} X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X}) ) \rightarrow Ext^2_{{{\mathcal {O}}}_{X\times _{{{\mathbb {F}}}_p} X}}( {{\mathcal {O}}}_{\Delta _X}, {{\mathcal {O}}}_{\Delta _X}) .\nonumber \\ \end{aligned}$$
(6)

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. \(\square \)

FormalPara Remark

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

$$\begin{aligned} \text {Ho}(R{\underline{{{\,\mathrm{{End}}\,}}}}_{{{\mathbb {F}}}_p}(L_{parf}(X) )) \rightarrow {{\,\mathrm{{End}}\,}}(D^b(X)) \end{aligned}$$
(7)

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.

$$\begin{aligned} \text {Ho}(R{\underline{{{\,\mathrm{{End}}\,}}}}_{{{\mathbb {F}}}_p}(L_{parf}(X) )) \rightarrow \text {Ho}(R{\underline{{{\,\mathrm{{End}}\,}}}}_{{{\mathbb {Z}}}_p}(L_{parf}(X) )) \rightarrow {{\,\mathrm{{End}}\,}}(D^b(X)). \end{aligned}$$
(8)

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 \( \text {Ho}(R{\underline{{{\,\mathrm{{End}}\,}}}}_{{{\mathbb {F}}}_p}(L_{parf}(X) ))\).