1.1 Frobenius-constant quantizations

For the duration of this paper, let $k$ be a perfect field of characteristic $p>2$. Given a scheme $X$ over $k$ we denote by $X'$ the Frobenius twist of $X$ and by $F: X\to X'$ the $k$-linear Frobenius morphism. Since $F$ is a homeomorphism on the underlying topological spaces, we shall identify the categories of sheaves on $X$ and $X'$.

Let $X$ be a smooth variety over $k$ equipped with a symplectic $2$-form $\omega$. Recall, that a quantization $(X, {\mathcal {O}}_h)$ of $(X, \omega )$ is a sheaf ${\mathcal {O}}_h$ on the Zariski site of $X$ of flat $k[[h]]$-algebras complete with respect to the $h$-adic topology together with an isomorphism of $k$-algebras

\[ {\mathcal{O}}_h/h \buildrel{\sim}\over{\longrightarrow} {\mathcal{O}}_X \]

such that, for any two local sections $\tilde {f}$, $\tilde {g}$ of ${\mathcal {O}}_h$, one has

\[ \{f,g\}\equiv \frac{\tilde{f} \tilde{g} - \tilde{g} \tilde{f}}{h} \mod h. \]

Here $f$ and $g$ stand for the images in ${\mathcal {O}}_X$ of $\tilde {f}$ and $\tilde {g}$ respectively and $\{, \}$ for the Poisson bracket ${\mathcal {O}}_X$ induced by the symplectic structure. Note that if $X$ is affine then giving a quantization $(X, {\mathcal {O}}_h)$ of $(X, \omega )$ is equivalent to giving a quantization ${\mathcal {O}}_h(X)$ of the Poisson algebra ${\mathcal {O}}_X(X)$ (see e.g. [Reference Bezrukavnikov and KaledinBK04a, Remark 1.6]).

A feature special to characteristic $p$ is that the Poisson algebra ${\mathcal {O}}_X$ of a symplectic variety has a large center consisting of $p$th powers of functions. We are going to identify it with the sheaf ${\mathcal {O}}_{X'}$ using the Frobenius morphism

\[ F^*: {\mathcal{O}}_{X'} \buildrel{\sim}\over{\longrightarrow} {\mathcal{O}}_{X}^p \subset {\mathcal{O}}_{X}. \]

Given a quantization $(X, {\mathcal {O}}_h)$ of $(X, \omega )$ we have $k$-linear homomorphisms

(1.1)\begin{equation} {\mathcal{Z}}_h \twoheadrightarrow {\mathcal{Z}}_h/h \hookrightarrow {\mathcal{O}}_{X'}. \end{equation}

from the center ${\mathcal {Z}}_h$ of the quantization ${\mathcal {O}}_h$ to the Poisson center. Following [Reference Bezrukavnikov and KaledinBK08], a quantization is called central if the composition (1.1) is surjective. A Frobenius-constant quantization of $(X, \omega )$ is a pair consisting of a quantization $(X, {\mathcal {O}}_h)$ of the symplectic variety $X$ together with a $k[[h]]$-algebra isomorphism

(1.2)\begin{equation} s: {\mathcal{O}}_{X'}[[h]] \buildrel{\sim}\over{\longrightarrow} {\mathcal{Z}}_h \end{equation}

such that, for any local section $f^p \in {\mathcal {O}}_{X}^p = {\mathcal {O}}_{X'} \subset {\mathcal {O}}_{X'}[[h]]$ and a lift $\tilde {f}\in {\mathcal {O}}_h$ of $f\in {\mathcal {O}}_X$, one has thatFootnote ¹

\[ s(f^p) = \tilde{f}^p \mod h^{p-1}. \]

It is clear that a quantization that admits a Frobenius-constant structure is central.

A Frobenius-constant structure on $(X, {\mathcal {O}}_h)$ makes ${\mathcal {O}}_h$ into a sheaf of algebras over ${\mathcal {O}}_{X'}[[h]]$. It was shown in [Reference Bezrukavnikov and KaledinBK08] that ${\mathcal {O}}_h$ is locally free of rank $p^{\dim X}$ as an ${\mathcal {O}}_{X'}[[h]]$-module for the Zariski topology on $X'$.

Frobenius-constant quantizations of symplectic varieties have been first introduced by Bezrukavnikov and Kaledin as a tool for proving the categorical McKay correspondence for symplectic resolutions of singularities (see [Reference Bezrukavnikov and KaledinBK04b]). Most of the foundational results have been obtained in [Reference Bezrukavnikov and KaledinBK08]. The technique introduced in [Reference Bezrukavnikov and KaledinBK04b] has found some other applications in geometric representations theory (see e.g. [Reference Bezrukavnikov and FinkelbergBF14, Reference Bezrukavnikov and LosevBL21]). A key to all these applications is the Azumaya property of the algebra ${\mathcal {O}}_h(h^{-1})$ obtained from ${\mathcal {O}}_h$ by inverting $h$: it was shown in [Reference Bezrukavnikov and KaledinBK08] that for any Frobenius-constant quantization on $(X, {\mathcal {O}}_h)$, the algebra ${\mathcal {O}}_h(h^{-1})$ is isomorphic, that is locally, for the fppf topology on $X'$, to a matrix algebra over ${\mathcal {O}}_{X'}((h))$. Since the algebra ${\mathcal {O}}_h$ has no zero divisors ${\mathcal {O}}_h(h^{-1})$ does not split even locally for the étale topology on $X'$ (unless $\dim X =0$). In [Reference Bezrukavnikov and KaledinBK08, Proposition 1.24] a formula for the class of this Azumaya algebra in an appropriate Brauer group was proposed. However, it has been observed in [Reference MolokovMol17] that the formula in [Reference Bezrukavnikov and KaledinBK08] is not correct as stated. The immediate goal of this paper is to correct it. The technique introduced along its proof (in particular, the Basic Lemma from § 1.8) plays an essential role in a sequel paper joint with Kubrak and Travkin [Reference Bogdanova, Kubrak, Travkin and VologodskyBKTV22], where we prove that the category of quasi-coherent sheaves on any restricted symplectic variety admits a canonical Frobenius-constant quantization.

1.2 Differential operators as a Frobenius-constant quantization

A basic example of a Frobenius-constant quantization is as follows. Let $Y$ be a smooth variety over $k$, $X:= {\mathbb {T}}^*_Y$ the cotangent bundle to $Y$ equipped with the canonical symplectic structure $\omega$. Denote by $D_{Y}$ the sheaf of differential operators on $Y$. This comes with a filtration given by the order of a differential operator. Applying the Rees construction to the filtered algebra $D_{Y}$ we obtain a sheaf of algebras $D_{Y, h}$ flat over $k[h]$ whose fiber over $h=1$ is $D_Y$ and whose fiber over $h=0$ is the symmetric algebra $S^\cdot T_Y$. Explicitly, $D_{Y, h}$ is the subalgebra of $D_{Y}[h]$ generated by $h$, ${\mathcal {O}}_Y$, and $hT_Y$. The $p$-curvature homomorphism

\[ S^{\cdot} T_{Y'} \to D_{Y, h} \]

sending a function $f\in {\mathcal {O}}_Y$ to $f^p$ and a vector field $\theta \in T_Y$ to $(h \theta )^p - h^{p-1}(h\theta ^{[p]})$ induces an isomorphism between the algebra $S^{\cdot } T_{Y'}[h]$ and the center of $D_{Y, h}$. In particular, $D_{Y, h}$ can be viewed as a quasi-coherent sheaf on ${\mathbb {T}}_{Y'}^*$. The canonical Frobenius-constant quantization of $({\mathbb {T}}^*_Y, \omega )$ is obtained from $D_{Y, h}$ by $h$-completion. We shall denote this canonical Frobenius-constant quantization of $({\mathbb {T}}^*_Y, \omega )$ by $({\mathbb {T}}^*_Y, {\mathcal {D}}_{Y, h})$.

1.3 Restricted Poisson structures

There is a local obstruction to the existence of a central quantization of a symplectic variety $(X, \omega )$. It was observed in [Reference Bezrukavnikov and KaledinBK08] that if $f^p \in {\mathcal {O}}_{X'} \buildrel {\sim }\over {\longrightarrow } {\mathcal {O}}_{X}^p$ is in the image (1.1), then the restricted power $H_f^{[p]}$ of the Hamiltonian vector field $H_f$ is again Hamiltonian: $H_f^{[p]}= H_{f^{[p]}}$ for some $f^{[p]}\in {\mathcal {O}}_{X}$. For example, it follows that the torus $({\mathbb {G}}_m \times {\mathbb {G}}_m, \omega = {{d}x}/{x}\wedge {{d}y}/{y})$ does not admit central quantizations.

A Frobenius-constant structure on $(X, {\mathcal {O}}_h)$ provides a canonical Hamiltonian for $H_f^{[p]}$. In fact, given a Frobenius-constant quantization $(X, {\mathcal {O}}_h, s)$ the formula

(1.3)\begin{equation} f^{[p]}= \frac{1}{h^{p-1}} (\tilde{f}^p - s(f)) \mod h \end{equation}

defines a restricted structure on the Poisson algebra ${\mathcal {O}}_X$, that is, the structure of a restricted Lie algebra on ${\mathcal {O}}_X$ such that $(f^2)^{[p]}= 2 f^{[p]} f^p$ and $H_f^{[p]}= H_{f^{[p]}}$.Footnote ²

It was shown in [Reference Bezrukavnikov and KaledinBK08] that, for every symplectic variety $(X, \omega )$, giving a restricted Poisson structure on ${\mathcal {O}}_X$ is equivalent to giving a class

\[ [\eta] \in H^0_{\rm Zar}(X, \operatorname{coker}({\mathcal{O}}_X \stackrel{d}{\longrightarrow} \Omega^1_X)) \]

such that

\[ d([\eta])=\omega. \]

In one direction, if $\eta \in \Omega ^1_X$, $d\eta =\omega$, then the formula

\[ f^{[p]}= L^{p-1}_{H_f} \iota_{H_f} \eta - \iota_{H_f^{[p]}} \eta \]

defines a restricted structure on ${\mathcal {O}}_X$. In particular, if $(X, \omega )$ admits a restricted structure, then $\omega$ is exact locally for Zariski topology on $X$.

1.4 Classification of Frobenius-constant quantizations

Fix a symplectic variety $X$ with a restricted Poisson structure $[\eta ]$. Denote by $Q(X, [\eta ])$ the set of isomorphism classes of Frobenius-constant quantizations $(X, {\mathcal {O}}_h, s)$ compatible with $[\eta ]$. In [Reference Bezrukavnikov and KaledinBK08], Bezrukavnikov and Kaledin constructed a map of sets

(1.4)\begin{equation} \rho: Q(X, [\eta]) \to H^1_{\rm et}(X', {\mathcal{O}}_{X'}^*/{\mathcal{O}}_{X'}^{* p} ) \end{equation}

and showed that if $H^1_{\rm Zar}(X', {\mathcal {O}}_{X'}/{\mathcal {O}}_{X'}^p )=0$, then $\rho$ is injective, and if $H^2_{\rm Zar}(X', {\mathcal {O}}_{X'}/{\mathcal {O}}_{X'}^p )=0$, then $\rho$ is surjective. Consequently, if both cohomology groups vanish the map $\rho$ is a bijection and there is a canonical Frobenius-constant quantization of $(X, [\eta ])$ corresponding to $0\in H^1_{\rm et}(X', {\mathcal {O}}_{X'}^*/{\mathcal {O}}_{X'}^{* p} )$. This quantization $(X, {\mathcal {O}}_h, s)$ is uniquely characterized as the one that admits a ${\mathbb {Z}}/2$-equivariant structure: an isomorphism ${\mathcal {O}}_{X'}[[h]]$-algebras

(1.5)\begin{equation} \alpha: {\mathcal{O}}_{-h}^{\rm op}\buildrel{\sim}\over{\longrightarrow} {\mathcal{O}}_h \end{equation}

identical modulo $h$ and such that $\alpha \circ \alpha = \operatorname {Id}$. We review the construction of $\rho$ in § 2.

1.5 A central reduction of the algebra of differential operators

A class

\[ [\eta] \in H_{\rm Zar}^{0}\Big(X, \operatorname{coker}\Big(\mathcal{O}_{X} \stackrel{d}{\longrightarrow} \Omega_{X}^{1}\Big)\Big) \]

gives rise to a certain central reduction $\mathcal {D}_{X, [\eta ], h}$ of the algebra $\mathcal {D}_{X, h}$. We first construct this reduction locally and then glue. For any open subset $U$ together with a $1$-form $\eta \in \Omega ^1(U)$ representing $[\eta ]$, consider the graph $\Gamma _{\eta }: U' \rightarrow \mathbb {T}_{U'}^*$ of $\eta \otimes 1 \in \Omega ^1 (U')$. Let $\mathcal {D}_{U, h}$ be the quantization of $\mathbb {T}^*_{U}$ defined above regarded as a locally free sheaf of modules over $S^{\cdot }T_{U'}[[h]]$ on $U$. Set

(1.6)\begin{equation} \Gamma_{\eta_1}^{*} \mathcal{D}_{U, h} = \mathcal{D}_{U, h}/ I_{\Gamma_{\eta}} \mathcal{D}_{U, h}. \end{equation}

Here $I_{\Gamma _{\eta }} \subset S^{\cdot }T_{U'}$ is sheaf of ideals defined by the closed embedding $\Gamma _{\eta }$. Note that $\Gamma _{\eta _1}^{*} \mathcal {D}_{U, h}$ is a sheaf of algebras over $S^{\cdot }T_{U'}/I_{\Gamma _{\eta }} [[h]]\buildrel {\sim }\over {\longrightarrow } {\mathcal {O}}_{U'}[[h]]$.

Suppose we are given two forms $\eta _1$, $\eta _2$ on $U$ representing the class $[\eta ]$. Let us construct a canonical isomorphism between the algebras $\Gamma _{\eta _1}^{*} \mathcal {D}_{X, h}$ and $\Gamma _{\eta _2}^{*} \mathcal {D}_{X, h}$. Set $\mu = \eta _1 - \eta _2$. Define the automorphism $\phi _{\mu }$ of $\mathcal {D}_{X, h}$ by setting $\phi _{\mu }(f)=f$ and $\phi _{\mu }(h\theta )=h\theta + \iota _{\theta }\mu$, for any function $f$ and vector field $\theta$.Footnote ³ Let $t_{\mu }$ be the translation by $\mu$ on $S^{\cdot }T_{U'}$ i.e. an automorphism sending a vector field $\theta$ to $\theta + \iota _{\theta }\mu$. Then using the Katz formula [Reference KatzKat72, § 7.22] (and the exactness of $\mu$) the following diagram is commutative.

The desired isomorphism is given by the formula

\[ \Gamma_{\eta_1}^{*} \mathcal{D}_{U, h} =\mathcal{D}_{U, h} \otimes_{S^{\cdot}T_{U'}} S^{\cdot}T_{U'} \ \mathcal{I}_{\Gamma_{\eta_1}}\stackrel{\phi_{\mu}\otimes t_\mu}{\longrightarrow} \mathcal{D}_{U, h} \otimes_{S^{\cdot}T_{U'}} S^{\cdot}T_{U'} \ \mathcal{I}_{\Gamma_{\eta_2}}=\Gamma_{\eta_2}^{*} \mathcal{D}_{U, h}. \]

Given three $1$-forms $\eta _1$, $\eta _2$, and $\eta _3$ representing the class $[\eta ]$ one has

\[ (\phi_{\eta_1 - \eta_2} \otimes t_{\eta_1 - \eta_2}) \circ (\phi_{\eta_2 - \eta_3} \otimes t_{\eta_2 - \eta_3}) = \phi_{\eta_1 - \eta_3}\otimes t_{\eta_1 - \eta_3} . \]

The sheaf of algebras $\mathcal {D}_{X, [\eta ], h}$ is obtained by gluing $\Gamma _{\eta }^{*} \mathcal {D}_{U, h}$ along the above isomorphisms.

The sheaf ${\mathcal {D}}_{X, [\eta ], h}$ of ${\mathcal {O}}_{X'}[[h]]$-algebras is locally free as a ${\mathcal {O}}_{X'}[[h]]$-module of rank $p^{2\dim X}$. The commutative algebra ${\mathcal {D}}_{X, [\eta ], h}/h$ is isomorphic to the algebra of functions on the Frobenius neighborhood of the zero section $X\hookrightarrow {\mathbb {T}}^*_X$ with the Poisson structure given by the symplectic form $\omega _{{\rm can}} + \operatorname {pr} ^* \omega$ on ${\mathbb {T}}^*_X$. Here $\omega _{{\rm can}}$ is the canonical symplectic form on the cotangent bundle, and $\operatorname {pr}: {\mathbb {T}}^*_X \to X$ is the projection.

1.6 Main result

Denote by $\operatorname {Br}(X'[[h]])$ the Brauer group of the formal scheme $(X', {\mathcal {O}}_{X'}[[h]])$ obtained from $X' \times \operatorname {Spec} k[h]$ by completion along the closed subscheme cut by the equation $h=0$. We have homomorphisms:

(1.7)\begin{equation} \delta \colon H^1_{\rm et}(X', {\mathcal{O}}_{X'}^*/{\mathcal{O}}_{X'}^{* p} ) \to H^2_{\rm et}(X', {\mathcal{O}}_{X'}^* ) \cong \operatorname{Br}(X') \hookrightarrow \operatorname{Br}(X'[[h]]). \end{equation}

The first map in (1.7) is the boundary morphism associated to the short exact sequence of sheaves for the étale topology

\[ 0 \to {\mathcal{O}}_{X'}^* \stackrel{p}{\longrightarrow} {\mathcal{O}}_{X'}^* \stackrel{}{\longrightarrow} {\mathcal{O}}_{X'}^*/{\mathcal{O}}_{X'}^{* p} \to 0. \]

The right arrow in (1.7) is the pullback homomorphism which is a split injection because its composition with the restriction homomorphism

\[ i^*\colon \operatorname{Br}(X'[[h]]) \to \operatorname{Br}(X') \]

is the identity. Given a class $\gamma \in H^1_{\rm et}(X', {\mathcal {O}}_{X'}^*/{\mathcal {O}}_{X'}^{* p} )$ we denote by $\delta (\gamma ) \in \operatorname {Br}(X'[[h]])$ the image of $\gamma$ under the composition (1.7). Finally, we can state the main result of this paper.

For example, let $(X, {\mathcal {O}}_h, s)$ be a Frobenius-constant quantization that admits ${\mathbb {Z}}/2{\mathbb {Z}}$-equivariant structure (1.5). Assume that $H^2(X, {\mathcal {O}}_X) =0$. Then by Theorem 1 ${\mathcal {O}}^\sharp _h$ is a split Azumaya algebra, that is there exists a locally free ${\mathcal {O}}_{X'}[[h]]$-module $E$ of finite rank and an isomorphism of ${\mathcal {O}}_{X'}[[h]]$-algebras

\[ {\mathcal{O}}^\sharp_h \buildrel{\sim}\over{\longrightarrow} \operatorname{End}_{{\mathcal{O}}_{X'}[[h]]}(E). \]

Using (1.8) and the Azumaya property of ${\mathcal {O}}_h (h^{-1})$ and ${\mathcal {D}}_{X, [\eta ], h}(h^{-1})$ we observe an equivalence of categories

(1.10)\begin{equation} \operatorname{Mod}({\mathcal{D}}_{X, [\eta], h}(h^{-1}))\buildrel{\sim}\over{\longrightarrow} \operatorname{Mod}({\mathcal{O}}_h (h^{-1})) \end{equation}

between the category of ${\mathcal {O}}_h (h^{-1})$-modules and the category of ${\mathcal {D}}_{X, [\eta ], h}(h^{-1})$-modules. The functor from left to right carries a ${\mathcal {D}}_{X, [\eta ], h}(h^{-1})$-module $M$ to $E\otimes _{{\mathcal {D}}_{X, [\eta ], h}} M$; the quasi-inverse functor takes an ${\mathcal {O}}_h (h^{-1})$-module $N$ to ${{\mathcal {H}}}om_{{\mathcal {O}}_h }(E, N)$.

Also note that if $H^1_{\rm Zar}(X', {\mathcal {O}}_{X'}/{\mathcal {O}}_{X'}^p )=0$, then the map

\[ H^0(X, \Omega^1_X) \to H^0_{\rm Zar}(X, \operatorname{coker}({\mathcal{O}}_X \stackrel{d}{\longrightarrow} \Omega^1_X)) \]

is surjective and, thus, any restricted Poisson structure arises from a global $1$-form $\eta$. In this case objects of $\operatorname {Mod}({\mathcal {D}}_{X, [\eta ], h}(h^{-1}))$ can be viewed as ${\mathcal {D}}_{X, h}(h^{-1})$-modules whose $p$-curvature equals $\eta$.

1.7 ${\mathbb {G}}_m$-equivariant quantizations

Let $(X, \omega )$ be a symplectic variety equipped with an action

(1.11)\begin{equation} \lambda \colon {\mathbb{G}}_m \times X \to X \end{equation}

of the multiplicative group such that $\omega$ has a positive weight $m$ with respect to this action. Moreover, we shall assume that $m$ invertible in $k$. Denote by $\theta$ the Euler vector field on $X$ corresponding to the ${\mathbb {G}}_m$-action. Then the formula $\eta = ({1}/{m}) \iota _{\theta } \omega$ defines a restricted structure on $X$. Define a ${\mathbb {G}}_m$-action on $X'$ twisting (1.11) by the $p$th power map ${\mathbb {G}}_m \stackrel {F}{\longrightarrow } {\mathbb {G}}_m$. Also let ${\mathbb {G}}_m$ act on $X'[h]:= X' \times \operatorname {Spec} k[h]$ as above on the first factor and by $z * h = z^m h$ on the second one.

A ${\mathbb {G}}_m$-equivariant Frobenius-constant quantization of $X$ is a ${\mathbb {G}}_m$-equivariant sheaf $O_h$ of associative ${\mathcal {O}}_{X'[h]}$-algebras on $X'[h]$, locally free as an ${\mathcal {O}}_{X'[h]}$-module, such that the restriction ${\mathcal {O}}_h$ of $O_h$ to the formal completion of $X'[h]$ along the divisor $h=0$ is a Frobenius-constant quantization of $X$ compatible with the restricted structure $[\eta ]$. Examples of ${\mathbb {G}}_m$-equivariant quantizations arise in geometric representation theory (see e.g. [Reference Bezrukavnikov and KaledinBK04b, Reference Bezrukavnikov and FinkelbergBF14, Reference Bezrukavnikov and LosevBL21, Reference Kubrak and TravkinKT19]).

Assume that morphism (1.11) extends to a morphism

(1.12)\begin{equation} \tilde{\lambda}: {\mathbb{A}}^1 \times X \to X. \end{equation}

Then the restriction of $O_h$ to the open subscheme $X'[h, h^{-1}]\hookrightarrow X'[h]$ is an Azumaya algebra. As an application of Theorem 1 we prove in § 5 a conjecture of Kubrak and Travkin concerning the class of this algebra in the Brauer group. Namely, we show that, for every ${\mathbb {G}}_m$-equivariant Frobenius-constant quantization $O_h$, the following equality in $\operatorname {Br}(X')$ holds:

\[ [O_{h=1}] = \bigg[\frac{1}{m} \eta\bigg] + \tilde{\lambda}_0^*[\rho(O_h)]. \]

Here $[\eta ]$ denotes the image of $\eta$ under the canonical map $\Gamma (X', \Omega ^1_{X'}) \to \operatorname {Br}(X')$, $\rho (O_h)\in H^1_{\rm et}(X', {\mathcal {O}}_{X'}^*/{\mathcal {O}}_{X'}^{* p} )$ for the class associated to the formal quantization via (1.4), and $[\rho (O_h)]$ for its image in the Brauer group.

1.8 Plan of the proof

Using the language of formal geometry we reduce the theorem to a group-theoretic statement. We shall start by explaining the latter.

Let $(V, \omega _V)$ be a finite-dimensional symplectic vector space over $k$, and let $A_h$ be the algebra over $k[[h]]$ generated by the dual vector space $V^*$ subject to the relations

\[ f g - g f = \omega_V^{-1}(f, g) h, \quad f^p = 0 \]

for any $f, g \in V^*$. We refer to $A_h$ as the restricted Weyl algebra. This is a flat $k[[h]]$-algebra whose reduction modulo $h$ is the finite-dimensional commutative algebra of functions on the Frobenius neighborhood of the origin in the affine space $\operatorname {Spec}(S^{\cdot } V^*) := \bf {V}$. Explicitly, $A_0 := S^{\cdot } V^* / J_V$ where $J_V$ is the ideal generated by $f^p$ for all $f \in V^*$. The quantization $A_h$ of $A_0$ specifies a restricted Poisson structure $[\eta _V]\in \operatorname {coker}(A_0 \stackrel {d}{\longrightarrow } \Omega ^1_{A_0})$.Footnote ⁴

Denote by $G$ the group scheme $\underline {\operatorname {Aut}}(A_h)$ of $k[[h]]$-linear automorphisms of the algebra $A_h$, by $G^{\geq 1}$ the subgroup of automorphisms identical modulo $h$, and by $G_0$ the quotient of $G$ by $G^{\geq 1}$. As shown in [Reference Bezrukavnikov and KaledinBK08], $G_0$ is the group scheme of automorphisms of $A_0$ preserving the class $[\eta _V]$.

A pair $(W, W^*)$ of transversal Lagrangian subspaces of $V$ defines an isomorphism between the algebra $A_h(h^{-1})$ and the matrix algebra $\operatorname {End}_{k}(S^{\cdot } W^* / J_W)((h))$ which, in turn, gives an embedding $G \hookrightarrow L\operatorname {PGL}(p^n)$, where $L\operatorname {PGL}(p^n)$ is the loop group of $\operatorname {PGL}(p^n)$ (viewed as a sheaf for the fpqc topology). Then the extension

\[ 1 \to \mathbb{G}_m \to \operatorname{GL}(p^n) \to \operatorname{PGL}(p^n) \to 1 \]

gives rise toFootnote ⁵

(1.13)\begin{equation} 1 \to L{\mathbb{G}}_m \to \tilde{G} \to G \to 1. \end{equation}

In [Reference Bezrukavnikov and KaledinBK08] it is proved that $G^{\geq 1}$ is the subgroup of inner automorphisms. Hence, we have a subextension of (1.13)

(1.14)\begin{equation} 1 \to L^+{\mathbb{G}}_m \to \underline{A_h^*} \to G^{\geq 1} \to 1, \end{equation}

where $L^+{\mathbb {G}}_m$ is the positive loop group of $\mathbb {G}_m$. Then passing to the quotient we get a central extension by the affine grassmannianFootnote ⁶

(1.15)\begin{equation} 1 \to \operatorname{Gr}_{{\mathbb{G}}_m} \to \tilde{G}_0 \to G_0 \to 1. \end{equation}

Let $i: V\hookrightarrow V^{\flat }$ be a morphism of symplectic vector spaces such that the restriction to $V$ of the symplectic form on $V^{\flat }$ is $\omega _V$. Let $\widetilde {G}_0 \to G_0$ and $\widetilde {G}_0^{\flat } \to G_0^{\flat }$ be the corresponding extensions. We emphasize that $\widetilde {G}_0$ and $\widetilde {G}_0^{\flat }$ depend on a choice of Lagrangian pairs in $V$ and $V^{\flat }$.

Finally, denote by $G_0^{\sharp }\subset G_0^\flat$ the group subscheme that consists of automorphisms preserving the kernel of the homomorphism $i^*: A_0^{\flat } \to A_0$. We have a natural homomorphism $G_0^{\sharp } \to G_0$. In § 4 we prove the following assertion.

Theorem 1 The homomorphism $G_0^{\sharp } \to G_0$ lifts uniquely to a homomorphism of central extensions

\[ \widetilde{G}_0^\flat \times _{G_0^\flat} G_0^{\sharp} \to \widetilde{G}_0. \]

Our proof of the Basic Lemma, that occupies almost the half of the paper, is based on a new construction of (1.15) that makes this functoriality property obvious. Namely, consider two subgroups $\alpha \subset G_0 \supset G_0^0$, where $G_0^0$ is the subgroup of automorphisms preserving the origin in $\bf {V}$ (which by a result of Bezrukavnikov and Kaledin coincides with the reduced subgroup of $G_0$) and $\alpha =\operatorname {Spec} A_0 (\cong \alpha _p^{\dim V})$ is the finite group scheme of translations. Then the product map $\alpha \times G_0^0 \to G_0$ induces an isomorphism of the underlying schemes. Let $\widetilde {\alpha }$ be the restriction of the central extension (1.15) to $\alpha$. This is a version of the Heisenberg group. We show that the extension (1.15) splits uniquely over the reduced subgroup $G_0^0$.Footnote ⁷ Thus, we can view $G_0^0$ as a subgroup of $\widetilde {G_0}$, and the quotient $\widetilde {G_0}/G_0^0$ is identified (as an ind-scheme) with $\widetilde {\alpha }$. The left action of $\widetilde {G_0}$ on $\widetilde {G_0}/G_0^0$ defines an embedding of $\widetilde {G_0}$ into the group of automorphisms of $\widetilde {\alpha }$ viewed as a space with an action of $\operatorname {Gr}_{{\mathbb {G}}_m}$. We prove in Theorem 3 that the image of this embedding is precisely the group of automorphisms that preserve a unique $\operatorname {Sp}_{2 n} \ltimes \alpha _{p}^{2 n}$-invariant connection on $\operatorname {Gr}_{{\mathbb {G}}_m}$-torsor $\widetilde {\alpha }$.

To derive the Basic Lemma from the above, we classify all central extensions of $\alpha$ by $\operatorname {Gr}_{\mathbb {G}_m}$ in § 4.2. In particular, we show that extensions of $\alpha$ by $\operatorname {Gr}_{\mathbb {G}_m}$ that split over every $\alpha _{p}$ factor are classified by $\operatorname {Lie}(\operatorname {Gr}_{\mathbb {G}_m})$-valued skew-symmetric $2$-forms on $\operatorname {Lie}(\alpha )$. Then it follows that the morphism $\alpha \rightarrow \alpha ^{\flat }$ induced by $i$ lifts uniquely to a morphism of extensions $\widetilde {\alpha } \rightarrow \widetilde {\alpha }^{\flat }$ respecting the connections. Since $\widetilde G_0^\flat \times _{G_0^\flat } G_0^{\sharp }$ is the group of automorphisms of $\widetilde {\alpha }^{\flat }$ that preserve the connection and the subspace $\widetilde {\alpha } \hookrightarrow \widetilde {\alpha }^{\flat }$, by restriction we get the desired lifting $\widetilde {G}_0^\flat \times _{G_0^\flat } G_0^{\sharp } \to \tilde {G}_0.$

Let us explain how the Basic Lemma implies the theorem. The Bezrukavnikov–Kaledin construction of Frobenius-constant quantizations is based on a characteristic-$p$ version of the Gelfand–Kazhdan formal geometry. Namely, it is shown in [Reference Bezrukavnikov and KaledinBK04b] that any Frobenius-constant quantization is locally for the fpqc topology on $X^{\prime }$ isomorphic to the constant quantization $\mathcal {O}_{X^{\prime }}[[h]] \otimes _{k[[h]]} A_h$ for a fixed finite-dimensional space $V$ of dimension $2n = \dim X$. It follows that a Frobenius-constant quantization $(X, \mathcal {O}_h, s)$ gives rise to a torsor $\mathcal {M}_{X, \mathcal {O}_h, s}$ over $G$. Conversely, the algebra $\mathcal {O}_h$ is the twist of $\mathcal {O}_{X^{\prime }}[[h]] \otimes _{k[[h]]} A_h$ by the torsor ${\mathcal {M}}_{X, \mathcal {O}_h, s}$, i.e.

\[ {\mathcal{M}}_{X, \mathcal{O}_h, s} {\times} ^{G} (\mathcal{O}_{X^{\prime}}[[h]] \otimes_{k[[h]]} A_h) \buildrel{\sim}\over{\longrightarrow} \mathcal{O}_h. \]

The reduction of differential operators ${\mathcal {D}}_{X, [\eta ], h}$ also can be constructed using formal geometry. Namely, choosing a homogeneous form $\eta _V$ in the class $[\eta _V]$ on $V$ consider its graph

\[ {\bf{V}} \hookrightarrow \mathbb{T}^*_{\bf{V}}, \]

and let $i : V \to V\oplus V^*= V^{\flat }$ be the corresponding linear map of vector spaces. Let $G_0^{\sharp, f}$ be the subgroup of $G_0^{\sharp } \subset G_0^{\flat }$ of automorphisms $g$ of $\alpha ^{\flat }$ respecting the fibers of the projection $\pi : \alpha ^{\flat } \to \alpha$, that is fitting in the following diagram.

Then the restriction of the natural map $G_0^{\sharp } \to G_0$ to $G_0^{\sharp, f}$ is an isomorphism. This yields a homomorphism $\psi _0 : G_0 \hookrightarrow G_0^{\sharp } \subset G_0^{\flat }$. In § 3.1 we construct a lifting $\psi : G \to G^{\flat }$ of $\psi _0$ that makes $\mathcal {D}_{X, [\eta ], h}$ a twist of $\mathcal {O}_{X^{\prime }}[[h]] \otimes _{k[[h]]} A_h^{\flat }$ by ${\mathcal {M}}_{X, \mathcal {O}_h, s}$.

Then we consider the following diagram.

Here the loop group $L\operatorname {PGL}$ is the group ind-scheme of projective automorphisms of a certain vector space $U$ over $k((h))$. Suppose we can construct a $G$-invariant $k[[h]]$-lattice $\Lambda$ in $U$. Then

\[ {\mathcal{O}}^\sharp_h: = {\mathcal{M}}_{X, {\mathcal{O}}_h, s} \times ^G (\mathcal{O}_{X^{\prime}}[[h]] \otimes_{k[[h]]} \operatorname{End}_{k[[h]]}(\Lambda)) \]

does the job for the first part of the theorem. By a general result proven in the Appendix (Proposition A.5) the existence of an invariant lattice is equivalent to the existence of the dashed arrow making the diagram above commutative. This is where we use the Basic Lemma. It follows from the latter that

\[ \widetilde{G}_0^\flat \times _{G_0^\flat} G_0^{\sharp, f} \cong \widetilde{G}_0. \]

We infer (Proposition 3.3) that $\psi _0$ lifts to $\widetilde {\psi }_0 : \widetilde {G}_0 \to \widetilde {G}_0^{\flat }$, which implies the existence of the dashed arrow.

The proof of the second part of the theorem amounts to unveiling the Bezrukavnikov–Kaledin construction of the map $\rho$.

1.9 Further directions

In this subsection we briefly discuss some applications of Theorem 1 obtained in a sequel paper joint with Dmitry Kubrak and Roman Travkin [Reference Bogdanova, Kubrak, Travkin and VologodskyBKTV22].

According to the Bezrukavnikov–Kaledin theorem from § 1.4 every smooth affine restricted symplectic variety admits a unique up to a non-canonical isomorphism Frobenius-constant quantization $O_h$ with $\rho [O_h]=0$. The formation of $O_h$ is not functorial in $(X, [\eta ])$. However, we show in [Reference Bogdanova, Kubrak, Travkin and VologodskyBKTV22] that the assignment $(X, [\eta ]) \mapsto \operatorname {Mod}(O_h(X))$ extends to a contravariant functor from the category of smooth affine restricted symplectic varieties and open embeddings to the category of abelian categories. Applying the right Kan extension this yields a functorial quantization $\operatorname {QCoh}_h$ of the category of quasi-coherent sheaves of any smooth restricted symplectic variety. Moreover, using Remark 1.1 and equivalence (1.10) we extend the range of quantum parameter $h$ from being a formal variable to a genuine coordinate on ${\mathbb {P}}^1$. The construction of $\operatorname {QCoh}_h$ uses in an essential way Corollary 3.5 of the Basic Lemma.

Let $Y\hookrightarrow X$ be a smooth Lagrangian subvariety such that $[\eta ]_{|Y} = 0$ in $H_{\rm Zar}^{0}\big (Y, \Omega _{Y}^{1}/d{\mathcal {O}}_Y\big )$. Using results from [Reference MundingerMun22] we show in § 6.1 of [Reference Bogdanova, Kubrak, Travkin and VologodskyBKTV22] that every such $Y$ determines a canonical object in $\operatorname {QCoh}_h$, which is a quantization of the line bundle $(\Omega _{Y}^n)^{ ({1-p})/{2}}$ viewed as a quasi-coherent sheaf on $X$.

1.10 Plan of the paper

In § 2 we review the Bezrukavnikov–Kaledin construction of Frobenius-constant quantizations which is based on a characteristic $p$ version of the Gelfand–Kazhdan formal geometry. In § 3 we recast the construction of ${\mathcal {D}}_{X, h}$ using the language of formal geometry and reduce Theorem 1 to a certain statement on central extensions of the group of automorphisms of the restricted Weyl algebra. In § 4 we prove this statement. In § 5 we study $\mathbb {G}_m$-equivariant quantizations and prove a conjecture of Kubrak and Travkin. Finally, in the Appendix we prove some results (probably known to experts) on loop groups that are used in the main body of the paper.