\chapter{DG Affinity of DQ-modules}
\selectlanguage{english}
\section{Introduction}

Many classical results of complex algebraic or analytic geometry have a counterpart in the framework of Deformation Quantization modules (see \cite{KS3}). Let us mention a few of them: Serre duality, convolution of coherent kernels, the construction of Hochschild classes for coherent DQ-modules in \cite{KS3}, a GAGA type theorem in \cite{Gaga_DQ} and Fourrier-Mukai transforms in \cite{Mukai_Pantev}, etc.

In this chapter, extracted from \cite{dgaff}, we give a non-commutative analogue of a famous result of Bondal-Van den Bergh asserting the dg affinity of quasi-compact quasi-separated schemes (see \cite[Corollary 3.1.8]{BVdB}). In the framework of formal deformation algebroid stacks, the notion of quasi-coherent object is no more suited for this purpose. Thus, we introduce the notion  of cohomologically complete and graded quasi-coherent objects ($\qcc$ for short). The $\qcc$ objects of the derived category $\Der(\cA_X)$, where $\cA_X$ is a formal deformation algebroid stack, form a full triangulated subcategory of $\Der(\cA_X)$ denoted $\Der_{\qcc}(\cA_X)$. This category can be thought of as the deformation of $\Der_{\qcoh}(\cO_X)$ while deforming $\cO_X$  into $\cA_X$ (see Theorem \ref{deforq}). We prove that the image of a compact generator of $\Der_{\qcoh}(\cO_X)$ is a compact generator of $\Der_{\qcc}(\cA_X)$. The existence of a compact generator in $\Der_{\qcoh}(\cO_X)$ is granted by a result of Bondal-Van den Bergh (see loc. cit.). Hence, the category $\Der_{\qcc}(\cA_X)$ is dg affine.

The study of generators in derived categories of geometric origin has been initiated by Beilinson in \cite{Beil_gen}. The results of \cite{BVdB}  have been refined by Rouquier in \cite{Rou} where he introduced a notion of dimension for triangulated categories. Recently, in \cite{Toen_generateur} To�n generalized the results of Bondal and Van den Bergh and reinterpreted them in the framework of homotopical algebraic geometry.\\

This chapter is organised as follows. In the first part, we recall some classical material concerning generators in a triangulated category. We review, following \cite{KS3}, the notion of cohomological completeness and its link with the functor of $\hbar$-graduation. We finally state some results specific to deformation algebroid stacks on smooth algebraic varieties.

In the second part of the chapter, we introduce the triangulated category of $\qcc$ objects, that is to say objects of $\Der(\cA_X)$ that are cohomologically complete and whose associated graded module is quasi-coherent. We prove that the category $\Der_{\qcc}(\cA_X)$ admits arbitrary coproducts. The coproduct is given by the cohomological completion of the usual direct sum (Proposition \ref{cocoqcc}). Then we prove that $\Der_{\qcc}(\cA_X)$ is compactly generated (see Proposition \ref{genqcc} and Lemma \ref{impli}). Relying on a theorem of Ravenel and Neeman (see \cite{Rav} and \cite{Nee_comp}) we describe completely the compact objects of $\Der_{\qcc}(\cA_X)$ (see Theorem \ref{qccompact}). They are objects of 
$\Der^{b}_{\coh}(\cA_X)$ satisfying certain torsion conditions. Finally, we conclude this section by proving that $\Der_{\qcc}(\cA_X)$ is equivalent as a triangulated category to the derived category of a suitable dg algebra with bounded cohomology (see Theorem \ref{affinity}).

In the last section, we study $\qcc$ sheaves on an affine variety and prove that the equivalence of triangulated categories between $\Der^+_{\qcoh}(\cO_X)$ and $\Der^+(\cO_X(X))$ lifts to an equivalence between $\Der^+_{\qcc}(\cA_X)$ and the triangulated category $\Der^+_{cc}(\cA_X(X))$ of cohomologically complete $\cA_X(X)$-modules (see Theorem \ref{deforq}).
 
\section{Review}

\subsection{Generators and compactness in triangulated categories: a review}\label{envsec}

We start with some classical definitions. See \cite{Nee_book}, \cite{BVdB}.



Recall that if $\mathcal{T}$ is a triangulated category, then a triangulated subcategory $\mathcal{B}$ of $\mathcal{T}$ is called thick if it is closed under isomorphisms and direct summands.

\begin{defi}\label{epais}
Let $\mathcal{S}$ be a set of objects of $\mathcal{T}$. The smallest thick triangulated subcategory of $\mathcal{T}$ containing $\mathcal{S}$  is called the thick envelope of $\mathcal{S}$ and is denoted $\langle \mathcal{S} \rangle$\index{1env@$\langle \mathcal{S} \rangle$}.
One says that $\mathcal{S}$ classically generates $\mathcal{T}$ if $\langle \mathcal{S} \rangle$ is equal to $\mathcal{T}$.
\end{defi}

\begin{defi}
Let $\mathcal{T}$ be a triangulated category. Let $\mathfrak{G}=(G_i)_{i \in I}$ be a set of objects of $\mathcal{T}$. One says that $\mathfrak{G}$ generates $\mathcal{T}$ if for every $F \in \mathcal{T}$ such that $\Hom_{\mathcal{T}}(G_i[n],F)=0$ for every $G_i \in \mathfrak{G}$ and $n \in \Z$, we have $F \simeq 0$.
\end{defi}


\begin{defi}
Assume that  $\mathcal{T}$ is cocomplete. 

\begin{enumerate} [(a)]
\item An object $L$ in $\mathcal{T}$ is compact if $\Hom_{\mathcal{T}}(L, \cdot)$ commutes with direct sums. We denote by $\mathcal{T}^{c}$ the full subcategory of $\mathcal{T}$ consisting of compact objects.

\item The category $\mathcal{T}$ is compactly generated if it is generated by a set of compact objects.
\end{enumerate}
\end{defi}

The following result was proved independently by Ravenel and Neeman, see  \cite{Nee_comp} and \cite{Rav}.

\begin{thm} \label{Rav_Nee}
Assume that $\mathcal{T}$ is compactly generated. Then a set of objects $\mathcal{S} \subset \mathcal{T}^{c}$ classically generates $\mathcal{T}^{c}$ if and only if it generates $\mathcal{T}$.
\end{thm}

We give an inductive description of the thick envelope of a subset of a triangulated category. For that purpose, we introduce a multiplication on the set of full subcategories of a triangulated category. We follow closely the exposition of \cite{BVdB}. 

\begin{defi}
Let $\mathcal{T}$ be a triangulated category. Let $\mathcal{C}$ and $\mathcal{D}$ be full subcategories of $\mathcal{T}$. One denotes by $\mathcal{C} \star \mathcal{D}$\index{1star@$\star$} the strictly full subcategory of $\mathcal{T}$ whose objects $E$ occur in a triangle of the form 
\begin{equation*}
C \to E \to D \to C \lbrack1 \rbrack
\end{equation*}
where $C \in \mathcal{C}$ and $B \in \mathcal{D}$.
\end{defi}

\begin{prop} \label{asso}
The operation $\star$ is associative.
\end{prop}

Let $\mathcal{S}$ be a set of objects of $\mathcal{T}$. We denote by $\add(\mathcal{S})$\index{add@$\add$} the smallest full subcategory in $\mathcal{T}$ which contains $\mathcal{S}$ and is closed under taking finite direct sums and shifts.\\
We denote by $\smd(\mathcal{S})$\index{smd@$\smd$} the smallest full subcategory which contains $\mathcal{S}$ and is closed under taking direct summands.

\begin{lemme} \label{rec}
If $\mathcal{C}$ and $\mathcal{D}$ are closed under finite direct sums, then
$\smd(\smd(\mathcal{C}) \star \mathcal{D})=\smd(\mathcal{C} \star \mathcal{D})$.
\end{lemme}%
\begin{center}%
\begin{flushleft}
Set
\end{flushleft}
\begin{equation*}
\; \langle \mathcal{S} \rangle_1=\smd(\add(\mathcal{S})),
\end{equation*}
\begin{equation*}
\langle \mathcal{S} \rangle_k=\smd(\underbrace{\langle \mathcal{S} \rangle_1 \star \ldots \star \langle \mathcal{S} \rangle_1}_{k \; factors}),
\end{equation*}
\begin{equation*}
\langle \mathcal{S} \rangle = \bigcup_k \langle \mathcal{S} \rangle_k.
\end{equation*}
\end{center}

Then $\langle \mathcal{S} \rangle$ is the thick envelope of $\mathcal{S}$ (see Definition \ref{epais}).

\subsection{Recollections on algebraic categories}
In this section, we recall some classical facts on algebraic categories, \cite{Keller_deriving}, \cite{Keller_construction}, \cite{Keller_dg}. In this section $R$ is a commutative unital ring.

\begin{defi}
A Frobenius category $\mathcal{E}$ is an exact category (in the sense of Quillen \cite{Quill}) with enough projective and injective objects such that an object is projective if and only if it is injective.
\end{defi}

Let $\sigma$ and $\sigma^{'}$ in $\mathcal{E}$. We denote by $\mathcal{N}(\sigma, \sigma^{'})$ the subgroup of $\Hom_{\mathcal{E}}(\sigma, \sigma^{'})$ formed by the maps that can be factorized through an injective object.
We denote by $\underline{\mathcal{E}}$ the category with the same objects as $\mathcal{E}$ and whose morphisms spaces are the quotients $\Hom_{\mathcal{E}}(\sigma, \sigma^{'})/\mathcal{N}(\sigma, \sigma^{'})$. The category  $\underline{\mathcal{E}}$\index{ecst@$\underline{\mathcal{E}}$} is called the stable category of $\mathcal{E}$. A classical result states that $\underline{\mathcal{E}}$ is a triangulated category.

\begin{defi}
One says that an $R$-linear triangulated category is algebraic if it is equivalent as a triangulated category to the stable category of an $R$-linear Frobenius category.
\end{defi}

\begin{prop}\label{trialg}
A triangulated subcategory of an algebraic triangulated category is algebraic.
\end{prop}

\begin{prop}\label{deralg}
The derived category of an Abelian category is algebraic.
\end{prop}

We have the following theorem from \cite{Keller_construction} which is a consequence of 
\cite[Theorem 4.3] {Keller_deriving}. If $\Lambda$ is a dg category, we denote by $\Der(\Lambda)$ its derived category in the sense of \cite{Keller_deriving} (note that  $\Der(\Lambda)$ is not a dg category).

\begin{thm}\label{equidg}
Let $\mathcal{E}$ be a Frobenius category and set $\mathcal{T}=\underline{\mathcal{E}}$. Assume that $\mathcal{T}$ is cocomplete and has a compact generator $G$. Then, there is a dg algebra $\Lambda$ and an equivalence of triangulated categories $F: \Der(\Lambda) \to \mathcal{T}$ with $F(\Lambda) \stackrel{\sim}{\to}G$. In particular, we have
\begin{equation*}
\Hn^n(\Lambda) \stackrel{\sim}{\to}\Hom_{\Der(\Lambda)}(\Lambda,\Lambda [n]) \stackrel{\sim}{\to} \Hom_{\mathcal{T}}(G,G[n]),\; n \in \Z. 
\end{equation*}
\end{thm}


\subsection{The case of $\Der_{\qcoh}(\cO_X)$}

 Let $(X, \cO_X)$ be a scheme. We denote by $Qcoh(X)$\index{qcoh@$Qcoh(X)$} the category of quasi-coherent $\cO_X$-modules. Its derived category is denoted by $\Der(Qcoh(X))$. We write $\Der_{\qcoh}(\cO_X)$\index{deriveqcoh@$\Der_{\qcoh}(\cO_X)$} for the full triangulated subcategory of $\Der(\cO_X)$ consisting of complexes with quasi-coherent cohomology. 

\begin{thm}[\cite{Nee}] \label{eqqc}
If $X$ is a quasi-compact and separated scheme then the canonical functor $\Der(Qcoh(X)) \to \Der_{\qcoh}(\cO_X)$ is an equivalence.
\end{thm}

\begin{defi}
  Let $(X, \cO_X)$ be a scheme. A perfect complex on $X$ is a complex of $\cO_X$-modules which is locally quasi-isomorphic to a bounded complex of locally free $\cO_X$-modules of finite type. We denote by $\Dper(\cO_X) \subset \Der_{\qcoh}(\cO_X)$ the category of perfect complexes.
\end{defi}

In this chapter, we are interested in complex smooth algebraic varieties. We give a few properties of perfect complexes in this setting.  Since $X$ is an algebraic variety, X is a Noetherian topological space. Thus, a perfect complex in $\Der_{\qcoh}(\cO_X)$ is in $\Der^{b}_{\qcoh}(\cO_X)$. Since $\cO_X$ is Noetherian it follows that $\Dper(\cO_X) \subset \Der^{b}_{\coh}(\cO_X)$. Finally since $X$ is smooth, we have $\Der^{b}_{\coh}(\cO_X) \subset \Dper(\cO_X)$. Thus, on a smooth algebraic variety, $\Dper(\cO_X)=\Der^{b}_{\coh}(\cO_X)$.\\

Recall the following theorem from \cite{BVdB}.

\begin{thm}\label{BV}
Assume that $X$ is a quasi-compact and quasi-separated scheme. Then,
\begin{enumerate}[(i)]
\item the compact  objects in $\Der_{\qcoh}(\cO_X)$ are the perfect complexes,
\item $\Der_{\qcoh}(\cO_X)$ is generated by a single perfect complex.
\end{enumerate}
\end{thm}

As a corollary Bondal and Van den Bergh obtain

\begin{thm}\label{dgaff}
Assume that $X$ is a quasi-compact quasi-separated scheme. Then $\Der_{\qcoh}(\cO_X)$ is equivalent to $\Der(\Lambda_0)$ for a suitable dg algebra $\Lambda_0$ with bounded cohomology.
\end{thm}

\subsection{$\hbar$-graduation}\label{sectiongr}

\subsubsection{The case of ringed space} \label{sheafcase}
In this section, X is a topological space and $\mathcal{R}$ is a $\Z [ \hbar]_X$-algebra on $X$ without $\hbar$-torsion. Throughout this text we assume that $\hbar$ is central in $\mathcal{R}$. We set $\cR_0= \cR / \hbar \cR$.
We refer the reader to \cite{KS3} for more details.
\begin{defi}
We denote by $\gr_\hbar: \Der(\cR) \to \Der(\cR_0)$\index{grh@$\gr_\hbar$} the left derived functor of the right exact functor $\Mod(\cR) \to \Mod(\cR_0)$ given by $\mathcal{M} \mapsto \mathcal{M} / \hbar \mathcal{M}$. For $\mathcal{M} \in \Der(\cR)$ we call $\gr_\hbar(\mathcal{M})$ the graded module associated to $\mathcal{M}$. We have
\begin{equation*}
\gr_{\hbar} \mathcal{M} \simeq \cR_0 \Lte_\cR \mathcal{M}.
\end{equation*}
\end{defi}

\begin{prop}\label{grhtens}
(i) Let $\cK_1 \in \Der(\op{\cR})$ and $\cK_2 \in \Der(\cR)$. Then,
\begin{equation*}
\gr_\hbar(\cK_1 \Lte_\cR \cK_2) \simeq \gr_\hbar(\cK_1) \Lte_{\cR_0} \gr_\hbar(\cK_2).
\end{equation*}
(ii) Let $\cK_i \in \Der(\cR) \; (i=1,2)$. Then
\begin{equation*}
\gr_\hbar(\fRHom_\cR(\cK_1,\cK_2))\simeq \fRHom_{\cR_0}(\gr_\hbar\cK_1,\gr_\hbar\cK_2).
\end{equation*}
\end{prop}

\begin{prop}
Let $X$ and $Y$ be two topological spaces and $f:X \to Y$ a continuous map. The functor $\gr_\hbar : \Der(\Z[\hbar]_Z) \to \Der(\Z[\hbar]_Z)$, $Z=X, \, Y$ commutes with the operations of direct images $\dR f_\ast$ and of inverse images. 
\end{prop}

\subsubsection{The case of algebroid stacks}

We write $\C^{\hbar}$\index{Ch@$\C^{\hbar}$} for the ring $\C[[\hbar]]$. In this section $\cA_X$ denotes a $\C^{\hbar}$-algebroid stack without $\hbar$-torsion. As in the previous subsection we refer the reader to \cite{KS3}.

\begin{defi}
Let $\cA_X$ be a $\C^{\hbar}$-algebroid stack without $\hbar$-torsion on a topological space X. One denotes by $\gr_\hbar (\cA_X)$ the $\C$-algebroid associated with the prestack $\mathfrak{S}$ given by:%
\begin{flushleft}
$\Ob(\mathfrak{S}(U))=\Ob(\cA(U))$ for an open set $U$ of $X$,\\
$\Hom_{\mathfrak{S}(U)}(\sigma,\sigma^{'})=\Hom_{\cA}(\sigma,\sigma^{'})/ \hbar \Hom_{\cA}(\sigma,\sigma^{'})$ for $\sigma, \sigma^{'} \in \cA(U)$.
\end{flushleft}
\end{defi}

There is a natural functor $\cA_X \to \gr_\hbar(\cA_X)$ of $\C$-algebroid stacks. This functor induces a functor

\begin{equation*}
\iota_g:\Mod(\gr_\hbar \cA_X) \to \Mod(\cA_X).
\end{equation*}

The functor $\iota_g$ admits a left adjoint functor $\mathcal{M} \to \C \otimes_{\C^{\hbar}} \mathcal{M}$. The functor $\iota_g$ is exact and it induces a functor

\begin{equation*}
\iota_g:\Der(\gr_\hbar \cA_X) \to \Der(\cA_X).
\end{equation*}\index{iota@$\iota_g$}

One extends the definition of $\gr_\hbar$  by
\begin{equation*}
\gr_\hbar(\mathcal{M}) \simeq \gr_\hbar(\cA_X) \Lte_{\cA_X} \mathcal{M} \simeq \C \Lte_{\C^{\hbar}} \mathcal{M}.
\end{equation*}

The propositions of the preceding subsection concerning sheaves extend to the case of algebroid stacks. Finally we have the following important proposition.

\begin{prop}\label{double_adj}
The functor $\gr_\hbar$ and $\iota_g$ define pairs of adjoint functors $(\gr_\hbar,\iota_g)$ and $(\iota_g,\gr_\hbar[-1])$.
\end{prop}

\begin{proof}
We refer the reader to \cite[Proposition 2.3.6]{KS3}. 
\end{proof}

\subsection{Cohomologically Complete Modules}
In this subsection, we briefly recall some facts about cohomologically complete modules. We closely follow \cite{KS3} and refer the reader to it for an in depth treatment of the notion of cohomological completeness. 

In this section, $X$ is a topological space and $\mathcal{R}$ is a $\Z [\hbar]_X$-algebra without $\hbar$-torsion.
 We set $\mathcal{R}^{loc}:=\Z[\hbar, \hbar^{-1}]\otimes_{\Z[\hbar]} \mathcal{R}$\index{rloc@$\mathcal{R}^{loc}$}. The triangulated category $\Der(\mathcal{R}^{loc})$ is equivalent to the full subcategory of $\Der(\mathcal{R})$ consisting of objects $\mathcal{M}$ satisfying $\cM \stackrel{\sim}{\to} \mathcal{R}^{loc} \Lte_{\mathcal{R}} \cM$ or equivalently $\gr_\hbar \cM=0$. 


The right orthogonal category $\Der(\mathcal{R}^{loc})^{\bot r}$ to the full subcategory $\Der(\mathcal{R}^{loc})$ of $\Der(\mathcal{R})$ is the full triangulated subcategory consisting of objects $\mathcal{M} \in \Der(\mathcal{R})$ satisfying $\Hom_{\Der(\mathcal{R})}(\mathcal{N},\mathcal{M}) \simeq 0$ for any $\mathcal{N} \in \Der(\mathcal{R}^{loc})$.


\begin{defi}
An object $\mathcal{M} \in  \Der(\mathcal{R})$ is cohomologically complete if it belongs to $\Der(\mathcal{R}^{loc})^{\bot r}$. We write $\Der_{\cc}(\cR)$\index{derivecc@$\Der_{\cc}(\cR)$} for  $\Der(\mathcal{R}^{loc})^{\bot r}$.
\end{defi}

 Propositions \ref{ccfact}, \ref{ccgr} are proved in \cite{KS3}.

\begin{prop}[{\cite[Prop. 1.5.6]{KS3}}] \label{ccfact}
\begin{enumerate}[(i)]
\item For $\mathcal{M} \in \Der(\mathcal{R})$, the following conditions are equivalent:\\

\begin{enumerate}[(a)]
\item $\mathcal{M}$ is cohomologically complete,\\


\item $\fRHom_{\mathcal{R}}(\mathcal{R}^{loc}, \mathcal{M}) \simeq \fRHom_{\Z[\hbar]}(\Z[\hbar, \hbar^{-1}], \mathcal{M}) \simeq 0,$\\


\item For any $x \in X$, $j=0,1$ and any $i \in \Z$, 
\begin{equation*}
\varinjlim_{x \in U} \Ext^j_{\mathcal{R}}(\mathcal{R}^{loc}, \Hn^i(U,\mathcal{M})) \simeq 0. 
\end{equation*}
Here, $U$ ranges over an open neighborhood system of $x$.\\
\end{enumerate}
\item $\fRHom_{\mathcal{R}}(\mathcal{R}^{loc}/ \mathcal{R}, \mathcal{M})$ is cohomologically complete for any $\mathcal{M} \in \Der(\mathcal{R})$.\\
\item For any $\mathcal{M} \in \Der(\mathcal{R})$, there exists a distinguished triangle
\begin{equation*}
\mathcal{M}^{'} \to \mathcal{M} \to \mathcal{M}^{''} \stackrel{+1}{\to}
\end{equation*}
with $\mathcal{M}^{'} \in \Der(\mathcal{R}^{loc})$ and $\mathcal{M}^{''} \in \Der_{\cc}(\cR)$.

\item Conversely, if
\begin{equation*}
\mathcal{M}^{'} \to \mathcal{M} \to \mathcal{M}^{''} \stackrel{+1}{\to}
\end{equation*}
is a distinguished triangle with $\mathcal{M}^{'} \in \Der(\mathcal{R}^{loc})$ and $\mathcal{M}^{''} \in \Der_{\cc}(\cR)$, then $\mathcal{M}^{'} \simeq \fRHom_{\mathcal{R}}(\mathcal{R}^{loc}, \mathcal{M})$ and $\mathcal{M}^{''} \simeq \fRHom_{\mathcal{R}}(\mathcal{R}^{loc}/ \mathcal{R}[-1], \mathcal{M})$.
\end{enumerate}
\end{prop}

\begin{lemme}[{\cite[Prop. 1.5.10]{KS3}}]\label{Homcc}
Assume that $\mathcal{M} \in \Der(\mathcal{R})$ is cohomologically complete. Then $\fRHom_{\cR}(\mathcal{N},\mathcal{M}) \in \Der(\Z_X[\hbar])$ is cohomologically complete for any $\mathcal{N} \in \Der(\cR)$.
\end{lemme}

\begin{prop}[{\cite[Cor. 1.5.9]{KS3}}]\label{ccgr}
Let $\mathcal{M} \in \Der(\mathcal{R})$ be a cohomologically complete object. If $\gr_{\hbar} \mathcal{M} \simeq 0$, then $\mathcal{M} \simeq 0$.
\end{prop}

\begin{cor}\label{isogr}
Let $f:\mathcal{M} \to \mathcal{N}$ be a morphism of $\Der_{\cc}(\cR)$. If $\gr_{\hbar}(f)$ is an isomorphism then $f$ is an isomorphism. 
\end{cor}

%\begin{prop}[{\cite[Prop. 1.5.12]{KS3}}]\label{ccdirect}
%Let $f:X \to Y$ be a continuous map, and $\mathcal{M} \in \Der(\Z[\hbar]_X)$. If $\mathcal{M}$ is cohomologically complete, then so is $\dR \! f_\ast \mathcal{M}$.
%\end{prop}

The following result is implicit in \cite{KS3}. We make it explicit since we use it frequently.

\begin{prop} \label{torsion comp}
Let $\mathcal{M} \in \Der(\cR)$ be such that there locally exists $n \in \N$ with $\hbar^n \mathcal{M}\simeq 0$. Then $\mathcal{M}$ is cohomologically complete.
\end{prop}

\begin{proof}
 The action of $\hbar$ on $\cR^{loc}$ is an isomorphism. Thus the morphism
\begin{equation*}
\hbar \circ: \fRHom(\cR^{loc},\mathcal{M}) \to \fRHom(\cR^{loc},\mathcal{M})
\end{equation*}
is an isomorphism.
The morphism
\begin{equation*}
\circ \hbar: \fRHom(\cR^{loc},\mathcal{M}) \to \fRHom(\cR^{loc},\mathcal{M})
\end{equation*}
is locally nilpotent.
Since $\hbar$ is central in $\mathcal{R}$, $\hbar \circ= \circ \hbar$. Hence, $ \fRHom(\cR^{loc},\mathcal{M})=0$.
\end{proof}
%Note that the result still holds when $\cR$ is an algebroid.

\subsection{Modules over formal deformations after \cite{KS3}}

In this subsection, we recall some facts about formal deformation of ringed spaces. We refer the reader to \cite{KS3} for DQ-modules. We refer to \cite{Stacks_Gerbes}, \cite{categories_and_sheaves} for stacks and algebroid stacks. We denote by $\C^{\hbar}$ the ring $\C[[\hbar]]$.\\

\begin{defi}[\cite{KS3}]
Let $(X, \cO_X)$ be a commutative ringed space. Assume that $\cO_X$ is a Noetherian sheaf of $\C$-algebras. A formal deformation algebra $\cA_X$ of $\cO_X$ is a sheaf of $\C^{\hbar}$-algebras such that
\begin{enumerate}[(i)]
\item $\hbar$ is central in $\cA_X$
\item $\cA_X$ has no $\hbar$-torsion
\item $\cA_X$ is $\hbar$-complete
\item $\cA_X / \hbar \cA_X \simeq \cO_X$ as sheaves of $\mathbb{C}$-algebras.
\item There exists a base $\mathfrak{B}$ of open subsets of $X$ such that for any $U \in \mathfrak{B}$ and any coherent $\cO{_X}|_U$-module $\mathcal{F}$, we have $\Hn^n(U,F)=0$ for any $n>0$.
\end{enumerate}
\end{defi}

\begin{Rem}
Clearly, on a complex algebraic variety, condition (v) of the preceding definition is satisfied.
\end{Rem}

\begin{defi}
A formal deformation algebroid stack $\cA_X$  on $X$ is a $\C^{\hbar}$-algebroid such that for each open set $U \subset X$ and each $\sigma \in \cA_X(U)$, the $\C^{\hbar}$-algebra $\mathcal{E}nd_{\cA_X}(\sigma)$ is a  formal deformation algebra on U.
\end{defi}

\begin{Rem}
Note that formal deformation algebroid stack are called twisted deformations in \cite{Yeku1}.
\end{Rem}

Let $\cA_X$ be a formal deformation algebroid on $X$. We denote by $\sMod(\C^{\hbar}_X)$ the $\C^\hbar$-linear stack of sheaves of $\C^\hbar$-modules over $X$ and by $\Mod(\cA_X)$ the category of functors $\Fct(\cA_X, \sMod(\C^{\hbar}_X))$. The category $\Mod(\cA_X)$ is a Grothendieck category.
For a module $\mathcal{M}$ over an  algebroid $\cA_X$ the local notions of being coherent, locally free etc. still make sense. We denote by $\Der(\cA_X)$ the derived category of $\Mod(\cA_X)$, by $\Der^{b}(\cA_X)$ its bounded derived category and by $\Der^{b}_{\coh}(\cA_X)$\index{derivebcoh@$\Der^{b}_{\coh}(\cA_X)$} the full triangulated subcategory of $\Der^{b}(\cA_X)$ consisting of objects with coherent cohomologies.

\begin{defi}
We say that an algebroid is trivial if it is equivalent to the algebroid stack associated to a sheaf of rings.
\end{defi}

From now on, we assume that $X$ is a smooth algebraic variety  endowed with the Zarisky topology. There are the following results (see Remark 2.1.17 of \cite{KS3} due to Prof. Joseph Oesterl�).

\begin{prop}
On a smooth algebraic variety $X$, the group $\Hn^2(X,\cO_X^{\times})$ is zero.
\end{prop}

\begin{cor}\label{grgentil}
On a smooth algebraic variety, invertible $\cO_X$-algebroid stacks are trivial.
\end{cor}
See \cite[Definition 2.1.14 (ii)]{KS3} or the appendix for a definition of invertible algebroid stack.

By the definition of the functor $\gr_\hbar$, it is clear that $\gr_\hbar \cA_X$ is an $\cO_X$ invertible algebroid and by Corollary \ref{grgentil} it follows that

\begin{equation}
\gr_\hbar \cA_X \simeq \cO_X. 
\end{equation}

\begin{prop}
The functor $\gr_\hbar$ induces a functor
\begin{equation*}
\gr_\hbar: \Der(\cA_X) \to \Der(\cO_X).
\end{equation*}
which sends bounded objects to bounded objects and objects with coherent cohomology to objects with coherents cohomology.
\end{prop}
We have the following results from \cite{KS3}.

\begin{prop}\label{resloclibre}
Let $d \in \N$. Assume that any coherent $\cO_X$-module locally admits a resolution of length $\leq d$ by free $\cO_X$-modules  of finite rank. Let $\mathcal{M}^{\bullet}$ be a complex of $\cA_X$-modules concentrated in degrees $[a,b]$ and assume that $\Hn^i(\mathcal{M})$ is coherent for all $i$. Then, in a neighborhood of each $x \in X$, there exists a quasi-isomorphism $\mathcal{L}^{\bullet} \to \mathcal{M}^{\bullet}$ where $\mathcal{L}^{\bullet}$ is a complex of free $\cA_X$-modules of finite rank concentrated in degrees $[a-d-1,b]$.
\end{prop}

We have the following sufficient condition which is a corollary of more general results that ensure that under certain conditions, an algebroid stack of formal deformations is trivial (see \cite{Kos}, \cite{Defor_Tsy}, \cite{CalHal}, \cite{Yeku1}). 
%I thank Amnon Yekutieli for bringing to my attention the difference between twisted associative deformations and formal deformation algebroids.  

\begin{prop}\label{deforsheaf}
Let $X$ be a smooth algebraic variety endowed with a twisted deformation algebroid $\cA_X$ in the sense of \cite{Yeku1}. If $\Hn^{1}(X,\cO_X)=\Hn^{2}(X,\cO_X)=0$, then $\cA_X$ is equivalent to the algebroid stack associated to a  formal deformation algebra of $\cO_X$.
\end{prop}

\section{QCC Modules}

From now on, we assume that $X$ is a smooth algebraic variety  endowed with the Zarisky topology and equipped with a formal deformation algebroid $\cA_X$.

\subsection{Graded quasi-coherent modules and quasi-coherent $\cO_X$-modules}

We introduce the category of graded quasi-coherent modules.

\begin{defi}
Let $\mathcal{M} \in \Der(\cA_X)$. We say that $\mathcal{M}$ is graded quasi-coherent if $\gr_\hbar(\mathcal{M}) \in \Der_{\qcoh}(\cO_X)$. We denote by $\Der_{\gqcoh}(\cA_X)$\index{derivegcoh@$\Der_{\gqcoh}(\cA_X)$} the full subcategory of $\Der(\cA_X)$ formed by graded quasi-coherent modules.
\end{defi}


\begin{prop}
The category $\Der_{\gqcoh}(\cA_X)$ is a triangulated subcategory of $\Der(\cA_X)$.
\end{prop}

\begin{proof}
Obvious.
\end{proof}

\subsection{QCC objects}
In this subsection, we introduce the category of $\qcc$-modules.
\begin{defi}
An object $\mathcal{M} \in \Der(\cA_X)$ is $\qcc$ if it is graded quasi-coherent and cohomologically complete. The full subcategory of $\Der(\cA_X)$ formed by $\qcc$-modules is denoted by $\Der_{\qcc}(\cA_X)$\index{deriveqcc@$\Der_{\qcc}(\cA_X)$}.
\end{defi}
Since, $\Der_{\qcc}(\cA_X) = \Der_{\gqcoh}(\cA_X) \cap \Der_{\cc}(\cA_X)$, we have
\begin{prop}\label{qcc_tri}
The category $\Der_{\qcc}(\cA_X)$ is a $\C^{\hbar}$-linear triangulated subcategory of $\Der(\cA_X)$.
\end{prop}

\begin{prop}
If $\mathcal{M} \in \Der^{+}_{\qcc}(\cA_X)$ is such that $\gr_\hbar \mathcal{M} \in \Der^b_{\coh}(\cO_X)$, then $\mathcal{M} \in \Der^+_{\coh}(\cA_X)$.
\end{prop}

\begin{proof}
It is a direct consequence of  \cite[Theorem 1.6.4]{KS3}.
\end{proof}

\begin{prop}
If $\mathcal{M} \in \Der^{b}_{\coh}(\cA_X)$, then $\mathcal{M} \in \Der^b_{\qcc}(\cA_X)$.
\end{prop}

\begin{proof}
It is a direct consequence of  \cite[Theorem 1.6.1]{KS3}.
\end{proof}

We now prove that $\Der_{\qcc}(\cA_X)$ is cocomplete. For that, we first prove that $\Der_{cc}(\cA_X)$ is cocomplete.

\begin{defi}
We denote by $(\cdot)^{\cc}$\index{cc@$(\cdot)^{\cc}$} the functor 
\begin{equation*}
\fRHom_{\cA_X}((\cA_X^{loc}/\cA_X)[-1],\cdot): \Der(\cA_X) \to \Der(\cA_X). 
\end{equation*}
We call this functor the functor of cohomological completion.
\end{defi}

By Proposition \ref{ccfact}, the functor of cohomological completion takes its values in $\Der_{cc}(\cA_X)$.

The following exact sequence
\begin{equation}\label{suiteloc}
0 \to \cA_X \to \cA_X^{loc} \to \cA_X^{loc}/ \cA_X \to 0.
\end{equation}
induces a morphism
\begin{equation*}
\cA_X^{loc}/ \cA_X[-1] \to \cA_X.
\end{equation*}
This morphism yields a morphism of functors
\begin{equation} \label{morcc}
cc:\id \to (\cdot)^{cc}.
\end{equation}

\begin{prop}\label{cciso}
The morphism of functors 
\begin{equation*}
\gr_\hbar(cc):\gr_\hbar \circ \id \to\gr_\hbar \circ (\cdot)^{cc}
\end{equation*}
is an isomorphism.
\end{prop}

\begin{proof}
Let $\cM \in \Der(\cA_X)$. Applying Lemma \ref{grhtens} (ii), we get the following isomorphism
\begin{align*}
\gr_\hbar(\cM^{cc})&\simeq \gr_{\hbar}\fRHom_{\cA_X}((\cA_X^{loc}/ \cA_X)[-1],\mathcal{M})\\ 
& \simeq \fRHom_{\gr_\hbar \cA_X}(\gr_\hbar (\cA_X^{loc}/ \cA_X)[-1],\gr_\hbar\mathcal{M}).
\end{align*}

Applying the functor $\gr_\hbar$ to (\ref{suiteloc}) and noticing that $ \gr_\hbar \cA_X^{loc} \simeq 0$, we deduce that $\gr_\hbar (\cA_X^{loc}/ \cA_X)[-1] \simeq \gr_\hbar \cA_X$. Hence, $ \gr_\hbar(\cM^{cc})\simeq \gr_\hbar \cM$.
\end{proof}



%\begin{cor} \label{grcomp}
%For every $\mathcal{M} \in \Der(\cA_X)$,
%\begin{equation*}
%\gr_\hbar \mathcal{M}^{\cc} \simeq \gr_\hbar \mathcal{M}.
%\end{equation*}
%\end{cor}

\begin{defi}
 Let $(\mathcal{M}_i)_{i \in I}$ be a family of objects of $\Der_{\cc}(\cA_X)$. We set 
\begin{equation*}
\obplus \mathcal{M}_i= \left(\bigoplus_{i \in I} \mathcal{M}_i \right)^{\cc}
\end{equation*}\index{1obplus@$\obplus$}
where $\bigoplus$ denotes the direct sum in the category $\Der(\cA_X)$. 
\end{defi}

\begin{prop}\label{sumcc}
The category $\Der_{\cc}(\cA_X)$ admits direct sums. The direct sum of the family $(\mathcal{M}_i)_{i \in I}$ is given by $\obplus \mathcal{M}_i$.
\end{prop}

\begin{proof}
 Let $(\mathcal{M}_i)_{i \in I}$ be a family of elements of $\Der_{\cc}(\cA_X)$.
By Proposition \ref{ccfact} $(ii)$, $\obplus \mathcal{M}_i$ is cohomologically complete.

Using the natural transformation (\ref{morcc}) we obtain a morphism

\begin{equation*}
cc:  \bigoplus_{i \in I} \mathcal{M}_i  \to \obplus \mathcal{M}_i.
\end{equation*}

It remains to shows that for all $\mathcal{F} \in \Der_{\cc}(\cA_X)$, the morphism (\ref{morcc}) induces an isomorphism

\begin{equation} \label{iso_int}
 \Hom_{\cA_X}( \obplus \mathcal{M}_i, \mathcal{F})\stackrel{\sim}{\to}  \Hom_{\cA_X}( \bigoplus_{i \in I} \mathcal{M}_i, \mathcal{F}).
\end{equation}

It is enough to prove the isomorphism

\begin{equation}\label{isoplus}
\fRHom_{\cA_X}( \obplus \mathcal{M}_i, \mathcal{F})\to \fRHom_{\cA_X}( \bigoplus_{i \in I} \mathcal{M}_i, \mathcal{F}).
\end{equation}

Since both terms of (\ref{isoplus}) are cohomologically complete by Lemma \ref{Homcc}, it remains to check the isomorphism on the associated graded map. Applying $\gr_\hbar$ to (\ref{isoplus}) and using Lemma \ref{grhtens} $(ii)$ and Proposition \ref{cciso} , we obtain an isomorphism

\begin{equation*}
\fRHom_{\gr_\hbar\cA_X}( \gr_\hbar(\obplus  \mathcal{M}_i), \gr_{\hbar} \mathcal{F})  \stackrel{\sim}{\to}  \fRHom_{\gr_\hbar \cA_X}( \gr_\hbar(\bigoplus_{i \in I}  \mathcal{M}_i),\gr_\hbar \mathcal{F}).
\end{equation*}

which proves the isomorphism (\ref{isoplus}).
%
%Moreover by definition of the direct sum, we have
%
%\begin{equation} \label{final}
% \Hom_{\cA_X}(\bigoplus_{i \in I} \mathcal{M}_i, \mathcal{F}) \stackrel{\sim}{\to} \prod_{i \in I}  \Hom_{\cA_X}( \mathcal{M}_i, \mathcal{F}).
%\end{equation}
%
%Composing the isomorphisms (\ref{iso_int}) and (\ref{final}), we obtain the following functorial isomorphism
%
%\begin{equation*}
% \Hom_{\cA_X}( \obplus \mathcal{M}_i, \mathcal{F}) \stackrel{\sim}{\longrightarrow}\prod_{i \in I} \Hom_{\cA_X}( \mathcal{M}_i, \mathcal{F})
%\end{equation*}
%which proves the proposition.
\end{proof}

\begin{prop}\label{cocoqcc}
The category $\Der_{\qcc}(\cA_X)$ admits direct sums. The direct sum of the family $(\mathcal{M}_i)_{i \in I}$ is given by $\obplus \mathcal{M}_i$.
\end{prop}

\begin{proof}
We know by Proposition \ref{sumcc}, that $\Der_{\cc}(\cA_X)$ admits direct sums and it is given by $\obplus[]$. Let $(\mathcal{M}_i)_{i  \in I} \in \Der_{\qcc}(\cA_X)$. Then, by Proposition \ref{cciso}, $\gr_\hbar \obplus \mathcal{M}_i =\bigoplus_{i \in I} \gr_\hbar \mathcal{M}_i$. It follows that $\obplus \mathcal{M}_i \in \Der_{\qcc}(\cA_X)$.
\end{proof}


\subsection{Compact objects and generators in $\Der_{\qcc}(\cA_X)$}
In this subsection, we show that $\Der_{\qcc}(\cA_X)$ is generated by a compact generator and we describe its compact objects. We start by proving some additional properties on the functors $\gr_\hbar$ and  $\iota_g$ which are defined in subsection \ref{sectiongr}. 

Concerning $\iota_g$, recall that there is a functor of stacks $\cA_X \to \gr_{\hbar}(\cA_X) \simeq \cO_X$ inducing

\begin{equation*}
\iota_g : \Der(\cO_X) \to \Der(\cA_X)
\end{equation*}

Notice that $\cO_X$ can be endowed with a structure of left $\cO_X$-module and right $\cA_X$-module. When endowed with such structures we denote it by $\cO_{XA}$. The module $\cO_{XA }$ belongs to $\Der^b(\cO_X \otimes_\C \op{\cA_{X}})$. Similarly we have ${_A}\cO_X \in \Der^{b}(\cA_X \otimes_{\C} \op{\cO_X})$. When $\cO_X$ is endowed with its structure of $\cA_X \otimes \op{\cA_X}$-module we denote it by ${_A} \cO_{XA} \in \Der^b(\cA_X \otimes_C \op{\cA_X})$.\\
With these notation, we have for $\cM \in \Der(\cA_X)$ and $\cN \in \Der(\cO_X)$
\begin{align*}
\gr_\hbar(\mathcal{M})= \cO_{XA} \Lte_{\cA_X} \mathcal{M}&& \textnormal{and}&&\iota_g(\mathcal{N})=& {_A}\cO_X \otimes_{\cO_X} \mathcal{N}\\
 && && =&{_A}\cO_X \Lte_{\cO_X} \mathcal{N}.
\end{align*}
Hence 
\begin{align*}
\iota_g \circ \gr_\hbar(\mathcal{M})&= {_A}\cO_X \Lte_{\cO_X} \cO_{XA} \Lte_{\cA_X} \mathcal{M}\\
                                                         &\simeq{_A}\cO_{XA} \Lte_{\cA_X} \mathcal{M}.
\end{align*}

\begin{prop}\label{iotafaith}
For every $\mathcal{N} \in \Der(\cO_X)$,
\begin{equation*}
 \iota_g \circ \gr_\hbar \circ \iota_g (\mathcal{N}) \simeq \iota_g(\mathcal{N}) \oplus \iota_g(\mathcal{N})[1]
\end{equation*}
\end{prop}

\begin{proof}
 We have the exact sequence of $\cA_X \otimes \op{\cA_X}$-modules
\begin{equation*}
\xymatrix{0 \ar[r]& \cA_X \ar[r]^-{\hbar}& \cA_X \ar[r]& {_A}\cO_{XA} \ar[r] & 0.}
\end{equation*}
Thus, for every $\mathcal{M} \in \Der(\cA_X)$, we have $\iota_g \circ \gr_\hbar(\mathcal{M}) \simeq  (\cA_X[1] \stackrel{\hbar}{\to} \cA_X) \Lte_{\cA_X} \mathcal{M}$. Hence, for $\mathcal{N} \in \Der(\cO_X)$, $ \iota_g \circ \gr_\hbar \circ \iota_g (\mathcal{N}) \simeq \iota_g(\mathcal{N}) \oplus \iota_g(\mathcal{N})[1]$. 
\end{proof}

\begin{cor}\label{conq}
If $\mathcal{N} \in \Der_{\qcoh}(\cO_X)$, then $\iota_g(\mathcal{N}) \in \Der_{\qcc}(\cA_X)$.
\end{cor}

\begin{proof}
Let $\mathcal{N}$ in $\Der_{\qcoh}(\cO_X)$ and consider $\gr_\hbar \circ \iota_g(\mathcal{N})$. We compute $\Hn^{i}(\gr_\hbar \circ \iota_g(\mathcal{N}))$.
\begin{align*}
\iota_g(\Hn^{i}(\gr_\hbar \circ \iota_g(\mathcal{N})))& \simeq \Hn^{i}(\iota_g \circ \gr_\hbar \circ \iota_g(\mathcal{N}))& \text{(exactness of $\iota_g$)}\\
& \simeq \Hn^{i}(\iota_g(\mathcal{N}) \oplus \iota_g(\mathcal{N})[1])& \text{(Proposition \ref{iotafaith})}\\
&\simeq \iota_g(\Hn^{i}(\mathcal{N}) \oplus \Hn^{i+1}(\mathcal{N}))& \text{(exactness of $\iota_g$)}.
\end{align*} 
The functor $\iota_g: \Mod(\cO_X) \to \Mod(\cA_X)$ is fully faithful. Thus,
\begin{equation*}
\Hn^{i}(\gr_\hbar \circ \iota_g(\mathcal{N}))\simeq \Hn^{i}(\mathcal{N}) \oplus \Hn^{i+1}(\mathcal{N}).
\end{equation*}
It follows that $\iota_g(\mathcal{N})$ is in $\Der_{\gqcoh}(\cA_X)$ and it is cohomologicaly complete by Proposition \ref{torsion comp}.
\end{proof}

\begin{prop}\label{genqcc}
If $\mathcal{G}$ is a generator of $\Der_{\qcoh}(\cO_X)$, then $\iota_g(\mathcal{G})$ is a generator of $\Der_{\qcc}(\cA_X)$
\end{prop}


\begin{proof}
By Corollary \ref{conq}, $\iota_g(\mathcal{G})$ is in $\Der_{\qcc}(\cA_X)$.

Let $\mathcal{M} \in \Der_{\qcc}(\cA_X)$ with $\RHom_{\cA_X}(\iota_g(\mathcal{G}),\mathcal{M})=0$. 
By Proposition \ref{double_adj}, we have
\begin{equation*}
 \RHom_{\cA_X}(\iota_g(\mathcal{G}),\mathcal{M}) \simeq  \RHom_{\cO_X}(\mathcal{G}, \gr_\hbar(\mathcal{M})[-1]).
\end{equation*}
Thus, $\RHom_{\cO_X}(\mathcal{G}, \gr_\hbar(\mathcal{M})[-1]) \simeq 0$ and $ \gr_\hbar(\mathcal{M})[-1]$ is in $\Der_{\qcoh}(\cO_X)$ thus $ \gr_\hbar(\mathcal{M})[-1] \simeq 0$.
 Since $\mathcal{M}$ is cohomologically complete, $\mathcal{M} \simeq 0$.
\end{proof}

\begin{lemme}\label{impli}
If $\mathcal{F} \in \Der^{b}_{\coh}(\cA_X)$ satisfies $\cA_X^{loc} \Lte_{\cA_X} \mathcal{F}=0$ then $\mathcal{F}$ is compact in $\Der_{\qcc}(\cA_X)$.
\end{lemme}

\begin{proof}
Let $(\mathcal{M}_i)_{i \in I}$ be a family of objects of $\Der_{\qcc}(\cA_X)$. By the adjunction between $(\cA_{X}^{loc}/ \cA_X)[-1] \Lte_{\cA_X} \cdot$ and $\fRHom_{\cA_X}(\cA_{X}^{loc}/ \cA_X)[-1],\cdot)$, we have
\begin{equation*}
\Hom_{\cA_X}(\mathcal{F}, \obplus \mathcal{M}_i) \simeq \Hom_{\cA_X}((\cA_{X}^{loc}/ \cA_X)[-1] \Lte_{\cA_X} \mathcal{F}, \bigoplus_{i\in I} \mathcal{M}_i).  
\end{equation*}

In $\Mod(\cA_X \otimes \op{\cA_{X}})$, we have the exact sequence (\ref{suiteloc}). Tensoring by $\mathcal{F}$, we obtain  that $\cA_{X}^{loc} / \cA_X[-1] \Lte_{\cA_X} \mathcal{F}$ is isomorphic to $\mathcal{F}$. It follows that

\begin{equation}\label{step1}
\Hom_{\cA_X}(\mathcal{F}, \obplus \mathcal{M}_i) \simeq \Hom_{\cA_X}(\mathcal{F}, \bigoplus_{i\in I} \mathcal{M}_i).
\end{equation}

The module $\mathcal{F}$ belongs to $\Der_{\coh}^{b}(\cA_X)$. Using Proposition \ref{resloclibre}, and the fact that $X$ is a Noetherian topological space of finite dimension, we have by \cite[Corollary 3.9.3.2]{Lipman}\footnote{In \cite{Lipman} the results is stated for a concentraded scheme map. On a Noetherian topological space $X$ of finite dimension, $\Rg(X, \cdot)$ is of finite cohomological dimension on $\Mod(\Z_X)$. Thus the proof of \cite[Corollary 3.9.3.2]{Lipman} goes without any changes for abelian sheaves on $X$ .}
 
\begin{equation}\label{step2}
\Hom_{\cA_X}(\mathcal{F}, \bigoplus_{i\in I} \mathcal{M}_i) \simeq \bigoplus_{i\in I} \Hom_{\cA_X}(\mathcal{F},  \mathcal{M}_i)
\end{equation}
 which together with (\ref{step1}) proves the lemma.
\end{proof}   

\begin{cor}
If $\mathcal{F}$ is compact in $\Der_{\qcoh}(\cO_X)$ then $\iota_g(\mathcal{F})$ is compact in $\Der_{\qcc}(\cA_X)$.
\end{cor}

\begin{proof}
Obviously, $\cA_X^{loc} \Lte_{\cA_X} \iota_g(\mathcal{F})=0$. Since $\mathcal{F}$ is compact in $\Der_{\qcoh}(\cO_X)$, it has coherent cohomology. Thus, its image by $\iota_g$ belongs to $\Der_{\coh}(\cA_X)$. The result follows from Lemma \ref{impli}.  
\end{proof}

\begin{cor}\label{compge}
If $\mathcal{G}$ is a compact generator of $\Der_{\qcoh}(\cO_X)$ then $\iota_g(\mathcal{G})$ is a compact generator of $\Der_{\qcc}(\cA_X)$. In particular, the category $\Der_{\qcc}(\cA_X)$ is compactly generated.
\end{cor}

\begin{proof}
By Theorem \ref{BV} due to Bondal and Van den Bergh, $\Der_{\qcoh}(\cO_X)$ has a compact generator. Then, the second claim of the corollary is a direct consequence of the first one. 
\end{proof}

\begin{lemme}\label{compcohdq}
Let $X$ be a smooth complex algebraic variety endowed with a DQ-algebroid.
\begin{enumerate}[(i)]
\item If $\mathcal{G}$ is a compact generator of $\Der_{\qcoh}(\cO_X)$ then $\gr_\hbar \iota_g \mathcal{G}$ is still a compact generator of $\Der_{\qcoh}(\cO_X)$.
\item One has $\Der^b_{\coh}(\cO_X)= \langle \gr_\hbar \iota_g(\mathcal{G}) \rangle $.
\end{enumerate}
\end{lemme}

\begin{proof}
\begin{enumerate}[(i)]
\item Let $\cM \in \Der_{\qcoh}(\cO_X)$ and assume that $\RHom_{\cO_X}(\gr_\hbar \iota_g \mathcal{G},\cM)\simeq 0$. We have
\begin{align*}
\RHom_{\cO_X}(\gr_\hbar \iota_g \mathcal{G},\cM) &\simeq \RHom_{\cA_X}( \iota_g \mathcal{G}, \iota_g \cM)\\
&\simeq 0.
\end{align*}
By Corollary \ref{compge} it follows that $\iota_g(\cM) \simeq 0$. Hence, for every $i \in \Z, \, \Hn^i(\iota_g \cM) \simeq 0$. Since $\iota_g:\Mod(\cO_X) \to \Mod(\cA_X)$ is fully faithful and exact, we have $\Hn^i(\cM)\simeq 0$. It follows that $\cM \simeq 0$. Moreover, $\gr_\hbar \iota_g \mathcal{G}$ is coherent which proves the claim. 
\item On a complex smooth algebraic variety the category of compact objects is $\Der^b_{\coh}(\cO_X)$. Hence the results follows from Theorem \ref{Rav_Nee}.
\end{enumerate}
\end{proof}


\begin{thm}\label{qccompact}
An object $\mathcal{M}$ of $\Der_{\qcc}(\cA_X)$ is compact if and only if $ \mathcal{M} \in \Der^{b}_{\coh}(\cA_X)$ and $\cA_X^{loc} \otimes_{\cA_X} \mathcal{M}=0$.
\end{thm}



\begin{proof}
The condition is sufficient by Lemma \ref{impli}.
Let $\mathcal{G}$ be a compact generator of $\Der_{\qcoh}(\cO_X)$. By Theorem \ref{Rav_Nee}, we know that the set  of compact objects of $\Der_{\qcc}(\cA_X)$ is equivalent to the thick envelope $\langle \iota_g (\mathcal{G}) \rangle$ of $\iota_g (\mathcal{G})$. Let us show that if $\mathcal{F} \in \langle \iota_g (\mathcal{G}) \rangle$ then $\mathcal{F} \in \Der^{b}_{\coh}(\cA_X)$ and $\cA_{X}^{loc} \otimes_{\cA_X} \mathcal{F} =0$. 

Let $\cK er$ be the full subcategory of $\Der_{qcc}(\cA_X)$ such that
\begin{equation*}
\Ob(\cK er)= \lbrace \mathcal{F} \in \Der_{\qcc}(\cA_X) \; | \; \cA_X^{loc} \Lte_{\cA_X} \mathcal{F} \simeq 0 \rbrace. 
\end{equation*} 
By Proposition \ref{thickiso}, $\cK er$ is a thick subcategory of $\Der_{qcc}(\cA_X)$. Moreover, the category $\Der^{b}_{\coh}(\cA_X)$ is also a thick subcategory of $\Der_{\qcc}(\cA_X)$. Then, the category $\cK o=\Der^{b}_{\coh}(\cA_X) \cap \cK er$ is a thick subcategory of $\Der_{\qcc}(\cA_X)$ containing $\iota_g(\mathcal{G})$. Since $\langle \iota_g (\mathcal{G}) \rangle$ is the smallest thick subcategory of $\Der_{\qcc}(\cA_X)$ containing $\iota_g(\mathcal{G})$, it follows that $\langle \iota_g (\mathcal{G}) \rangle \subset \cK o$ which proves the claim.
%
%
%We will proceed by induction.\\
%
%Recall that $\langle \iota_g(\mathcal{G}) \rangle_1=\smd(\add( \iota_g(\mathcal{G}))$ (cf. subsection \ref{envsec}) where $\smd(\add( \iota_g(\mathcal{G}))$ denote the smallest full subcategory of $\Der_{\qcc}(\cA_X)$ containing $\add(\iota_g (\mathcal{G}))$ and closed under taking direct summand. 
%
%The category $\add(\iota_g (\mathcal{G}))$ is the smallest full subcategory of $\Der_{\qcc}(\cA_X)$ which contains $\iota_g (\mathcal{G})$ and is closed under taking finite direct sums and shifts.
%
%It is clear that if $\mathcal{F} \in \add(\iota_g (\mathcal{G}))$, then $\mathcal{F} \in \Der_{\coh}^b(\cA_X)$ and $\cA_{X}^{loc} \otimes_{\cA_X} \mathcal{F}=0$. If $\mathcal{F} \in \smd(\add(\iota_g(\mathcal{G})))$, then there exist $\mathcal{M} \in \smd(\add(\iota_g(\mathcal{G})))$ such that $\mathcal{F} \oplus \mathcal{M} \in \add(\iota_g(\mathcal{G}))$. Hence, $\cA_X^{loc} \otimes_{\cA_X} \mathcal{F}=0$. We still need to check that for every $i \in \Z$, $\Hn^i(\mathcal{F}) \in \Mod_{\coh}(\cA_X)$. The question is local, so we can choose an open set $U$ of $X$ such that $\cA_X |_U$ is trivial. The sheaf $\Hn^i(\mathcal{F}|_U)$ is a direct summand of the coherent sheaf $\Hn^i((\mathcal{F} \oplus \mathcal{M})|_U)$, thus $\Hn^i(\mathcal{F}|_U)$ is coherent.\\
%
%
%Assume  that for every $k \leq n, \langle \mathcal{G} \rangle_k$ is a subcategory of $\Der_{\coh}^{b}(\cA_X)$ and that for every $\mathcal{F} \in  \langle \mathcal{G} \rangle_k, \; \cA_{X}^{loc} \otimes_{\cA_X} \mathcal{F}=0$. Let $\mathcal{F}$ in $ \langle \mathcal{G} \rangle_{n+1}$. By  Lemma \ref{rec}, $\langle \mathcal{G} \rangle_{n+1}=\smd(\langle \mathcal{G} \rangle_{n} \star \langle \mathcal{G} \rangle_{1})$. Since $\Der^b_{\coh}(\cA_X)$ is a triangulated category, the induction hypothesis implies that  $\langle \mathcal{G} \rangle_{n} \star \langle \mathcal{G} \rangle_{1} \subset  \Der^b_{\coh}(\cA_X)$. It follows that $\mathcal{F}$ is a direct summand of an object of the category $\langle \mathcal{G} \rangle_{n} \star \langle \mathcal{G} \rangle_{1}$. Then, $\mathcal{F} \in \Der^{b}_{\coh}(\cA_X)$ and $\cA_{X}^{loc} \otimes_{\cA_X} \mathcal{F} =0$.
\end{proof}

\begin{Rem}
In the case of the ring $\C^\hbar$, Theorem \ref{qccompact} implies that $\C^\hbar$ is not a compact object of $\Der_{\cc}(\C^\hbar)$. This can be checked directly as follow. Notice that
\begin{equation*}
\Hom_{\C^\hbar}(\C^\hbar, \obplus[n \in \Z] \C^\hbar) \simeq \Ext^1_{\C^\hbar}(\C^{\hbar,loc}/ \C^\hbar, \underset{n \in \Z}{\bigoplus} \C^\hbar).
\end{equation*}
Then by \cite[�1.5]{KS3} it follows that $\Ext^1_{\C^\hbar}(\C^{\hbar,loc}/ \C^\hbar, \underset{n \in \Z}{\bigoplus} \C^\hbar)$ is isomorphic to the $\hbar$-adic completion of $\underset{n \in \Z}{\bigoplus} \C^\hbar$.
Now, assume that $\C^\hbar$ is compact in $\Der_{\cc}(\C^\hbar)$. Then
\begin{equation*}
\Hom_{\C^\hbar}(\C^\hbar, \obplus[n \in \Z] \C^\hbar) \simeq \underset{n \in \Z}{\bigoplus} \C^\hbar.
\end{equation*}
But $\underset{n \in \Z}{\bigoplus} \C^\hbar$ is not $\hbar$-adically complete. Thus, $\C^\hbar$ is not compact in $\Der_{\cc}(\C^\hbar)$.
\end{Rem}

\subsection{DG Affinity of DQ-modules}

In this subsection, we prove that the category of $\qcc$ DQ-modules is DG affine. 
\begin{thm}\label{affinity}
Assume $X$ is a smooth complex algebraic variety endowed with a deformation algebroid $\cA_X$. Then, $\Der_{\qcc}(\cA_X)$ is equivalent to $\Der(\Lambda)$ for a suitable dg algebra $\Lambda$ with bounded cohomology.
\end{thm}

\begin{proof}
By Proposition \ref{qcc_tri}, $\Der_{\qcc}(\cA_X)$ is a $\C^{\hbar}$-linear triangulated subcategory of $\Der(\cA_X)$ which is algebraic by Proposition \ref{deralg}. It follows, by Proposition \ref{trialg}, that $\Der_{\qcc}(\cA_X)$ is algebraic. By  Proposition \ref{cocoqcc}, $\Der_{\qcc}(\cA_X)$ is a cocomplete category. Moreover, by Corollary \ref{compge}, $\Der_{\qcc}(\cA_X)$ has a compact generator. It follows from Theorem \ref{equidg} that $\Der_{\qcc}(\cA_X)$ is equivalent to the derived category of a dg algebra $\Lambda$ such that 
\begin{equation*}
\Hn^n(\Lambda)\simeq \Hom_{\cA_X}(\iota_g(\mathcal{G}),\iota_g(\mathcal{G})[n]), \; n \in \Z.
\end{equation*}
Using the adjunction between $\iota_g$ and $\gr_\hbar[-1]$ and \cite[Lemma 3.3.8]{BVdB}, we get that the cohomology of $\Lambda$ is bounded.
\end{proof}

\begin{example}
Let us compute such a dg algebra $\Lambda$ in the case of $X=\pt$. In this setting, $\cA_X=\C^\hbar$ and we get an equivalence $\Der_{\cc}(\C^\hbar) \simeq \Der(\Lambda)$.
The $\C^\hbar$-module $\C$ is a compact generator of the category $\Der_{\cc}(\C^\hbar)$. The complex $\mathscr{F}^\bullet:=0 \to \C^\hbar \stackrel{\hbar}{\to} \C^\hbar \to 0$ is a free resolution of $\C$ as a $\C^\hbar$-module. Then, we can chose 
\begin{equation*}
\Lambda \simeq \RHom_{\C^\hbar}(\mathscr{F}^\bullet,\mathscr{F}^\bullet).
\end{equation*}
It follows that 
\begin{align*}
\Lambda^{-1} \simeq \C^\hbar& &\Lambda^{0} \simeq \C^\hbar \oplus \C^\hbar & &\Lambda^{1} \simeq \C^\hbar
\end{align*}%
and that
\begin{align*}
d^{-1}= \begin{pmatrix} \hbar \\ \hbar \end{pmatrix} && d^0= \begin{pmatrix}
\hbar & -\hbar
\end{pmatrix}.
\end{align*}
\end{example}


\section{QCC Sheaves on Affine Varieties}
We assume that $X$ is a smooth algebraic affine variety and that $\cA_X$ is a sheaf of formal deformations (see Proposition \ref{deforsheaf}).  We set $A=\Gamma(X,\cA_X)$, $B=\Gamma(X, \cO_X)$ and $a_X: X \to \lbrace pt \rbrace$. As usual we denote by $A_X$ (resp. $B_X$) the constant sheaf with stalk $A$ (resp. $B$).

\subsection{Preliminary results}

\begin{lemme}
The $A_X$-module $\cA_X$ is flat.
\end{lemme}

\begin{proof}
It is a direct consequence of \cite[Theorem 1.6.6]{KS3}.
\end{proof}

\begin{lemme}\label{Commutation_Rf}
Let $f:X \to Y$ be a morphism of varieties and  let $\mathcal{M} \in \Der(f^{-1}\cA_Y)$. Then we have 
\begin{equation*}
\dR f_{\ast} \mathcal{M}^{cc} \simeq (\dR f_{\ast} \mathcal{M})^{\cc} \; in \; \Der(\cA_Y). 
\end{equation*} 
\end{lemme}

\begin{proof}
It is a slight modification of \cite[1.5.12]{KS3}
\end{proof}

We recall the following classical result.

\begin{lemme}\label{comcom}
Let $M \in \Der(B)$. The canonical morphism 
\begin{equation} \label{morpecom}
 M \to \Rg(X,\cO_X \otimes_{B_X} a_X^{-1} M)
\end{equation}
is an isomorphism.
\end{lemme}

\begin{proof}
If $M$ is concentrated in degree zero, the result follows directly from the equivalence of categories between $Qcoh(\cO_X)$ and $\Mod(B)$. The result extends immediately to the derived category because $\cO_X \otimes_{B_X}\cdot$ is an exact functor and  because $\Gamma(X, \cdot)$ is exact on $Qcoh(\cO_X)$ since $X$ is affine.
\end{proof}


%If $\cR \to \cR^{'}$ is a morphism of sheaves of rings, we denote by $for_{\cR^{'}}^{\cR}$ the forgetful functor from 
%$\Der(\cR^{'})$ to $ \Der(\cR)$, $\gr_\hbar ^{\C_X}$ the functor $\Der(\C^{\hbar}_X) \to \Der(\C_X)$,  $\mathcal{M} \mapsto \C_X \Lte_{\C^{\hbar}_X} \mathcal{M}$, $\gr_{\hbar}^{B_X} \mathcal{M}$ for $B_X \Lte_{A_X} \mathcal{M}$.\\

The functors $for_{(\cdot)}^{(\cdot)}$, below, are forgetful functors. We set 

\begin{align*}
\begin{aligned}
\gr_{\hbar}^{B_X}:\Der(A_X) & \to \Der(B_X)\\ 
\mathcal{M}  & \mapsto B_X \Lte_{A_X} \mathcal{M}
\end{aligned}
& \hspace{0.3cm} 
\begin{aligned}
\gr_\hbar^{\C_X}:\Der(\C^{\hbar}_X) & \to  \Der(\C_X)\\
\mathcal{M} & \mapsto \C_X \Lte_{\C^{\hbar}_X} \mathcal{M} 
\end{aligned}
&
\begin{aligned}
\gr_\hbar:\Der(\cA_X) & \to \Der(\cO_X)\\ 
\cM & \mapsto \cO_X \Lte_{\cA_X} \cM.
\end{aligned}
\end{align*}

\begin{prop}\label{oubli}
The diagrams below are quasi-commutative.
\begin{align*}
(i)\xymatrix{\Der^{}(A_X) \ar[r]^-{for_{A_X}}& \Der^{}(\C^{\hbar}_X)\\
\Der^{}(\cA_X) \ar[r]^-{for_{\cA_X}} \ar[u]^-{for_{\cA_X}^{A_X}} \ar[d]^-{\gr_\hbar}& \Der^{}(\C^{\hbar}_X) \ar @{=}[u] \ar[d]^-{\gr_\hbar^{\C_X}}\\
\Der^{}(\cO_X) \ar[r]^-{for_{\cO_X}} \ar[d]_-{for_{\cO_X}^{B_X}}& \Der^{}(\C_X) \ar @{=}[d]\\
\Der^{}(B_X) \ar[r]^-{for_{B_X}}& \Der^{}(\C_X) 
}
&(ii) \xymatrix{\Der(A_X) \ar[r]^-{for_{A_X}} \ar[d]_-{\gr_\hbar^{B_X}} & \Der(\C^\hbar_X) \ar[d]^-{\gr_\hbar^{\C_X}} \\
\Der(B_X)\ar[r]^-{for_{B_X}}  & \Der(\C_X)
.}\\
\end{align*}
 

\end{prop}

\begin{proof}
\begin{enumerate}[(i)]
\item We start by proving that $for_{\cO_X} \circ \gr_\hbar= \gr_{\hbar}^{\C_X} \circ for_{\cA_X}$. Let $\mathcal{M} \in \Der(\cA_X)$. We have  
\begin{align*}
for_{\cO_X} \circ \gr_\hbar(\mathcal{M})&=\cO_X \Lte_{\cA_X} \mathcal{M}\\
                                                                  &\simeq \C \Lte_{\C^{\hbar}} \cA_X \Lte_{\cA_X} \mathcal{M}\\
                                                                 &\simeq \C \Lte_{\C^{\hbar}} for_{\cA_X}(\mathcal{M})\\
                                                                &\simeq \gr_\hbar^{\C} \circ for_{\cA_X}(\mathcal{M}).
\end{align*}
The other commutation relations are similarly proved and are left to the reader.\\ 

\item Similar to (i).
\end{enumerate} 
\end{proof}


\begin{prop}
Let $\mathcal{M} \in \Der^{+}(\cA_X)$. There is an isomorphism in $\Der^{+}(B)$
\begin{equation*}
B \Lte_{A} \Rg(X,\mathcal{M}) \simeq \Rg(X, \cO_X \Lte_{\cA_X} \mathcal{M}).
\end{equation*}
\end{prop}

\begin{proof}

Let $\cM$ be an object of $\Der^{+}(\cA_X)$. The morphism of rings $B_X \to \cO_X$ gives us a morphism in $\Der^{+}(B_X)$
\begin{equation}\label{piece1}
\alpha: B_X \Lte_{A_X} \mathcal{M} \to \cO_X \Lte_{\cA_X} \mathcal{M}
\end{equation}
and by Proposition \ref{oubli}, $for_{B_X}(\alpha)= \id_{\C \Lte_{\C^\hbar} \cM}$. Since $for_{B_X}$ is conservative, it follows that $\alpha$ is an isomorphism in $\Der^{+}(B_X)$.

For the sake of brevity, we write $a_{X \ast}$ instead of $\dR a_{X \ast}$
The adjunction between $a_X^{-1}$ and $a_{X \ast}$ gives a morphism in $\Der^{+}(B)$

\begin{equation} \label{projinter}
pr: B \Lte_{A} a_{X \ast}(\mathcal{M}) \to a_{X \ast}( a_X^{-1}B \Lte_{A_X} \mathcal{M}).
\end{equation}

Applying $for_{B}$ to the morphism $pr$, we obtain

\begin{equation} \label{projinteroubli}
\C \Lte_{\C^{\hbar}}\dR a_{X\ast} \mathcal{M} \to \dR a_{X\ast}( \C_X \Lte_{\C^{\hbar}_X} \mathcal{M}).
\end{equation}

In the category $\Der^{+}(\C^{\hbar}_X)$, $\C_X$ admits a free resolution given by $\C^{\hbar}_X \stackrel{\hbar}{\to}\C^{\hbar}_X$. Hence, by the projection formula, we get that the morphism (\ref{projinteroubli}) is an isomorphism in $\Der(\C_X)$. Since $for_{B}$ is conservative, $pr$ is an isomorphism. The composition $a_{X \ast}(\alpha) \circ pr$ gives us the desired isomorphism.

%\begin{enumerate}[(i)]
%\item Let $\mathcal{M} \in \Der^{+}(\C^{\hbar}_X)$. The functor
%$\gr_\hbar: \Der^{+}(\C^{\hbar})\to\Der^{+}(\C)$ is given by $\C_X \Lte_{\C^{\hbar}_X}\cdot$ where $\C_X$ is in $\Der^{+}(\C^{\hbar}_X)$. In the category $\Der^{+}(\C^{\hbar}_X)$, $\C_X$ admits a free resolution given by $\C^{\hbar}_X \stackrel{\hbar}{\to}\C^{\hbar}_X$. Hence, by the projection formula, we get the isomorphism
%\begin{equation*}
%p:\C \Lte_{\C^{\hbar}}\dR a_{X\ast} \mathcal{M} \stackrel{\sim}{\to}\dR a_{X\ast}( \C_X \Lte_{\C^{\hbar}_X} \mathcal{M}).
%\end{equation*}
%
%
%\item Let $\cM \in \Der^{+}(\cA_X)$. For the sake of legibility we write $a_{X \ast}$ instead of $\dR a_{X \ast}$ and $_{B}\gr_\hbar \cM$ for $for_{\cO_X}^{B_X}(\gr_\hbar \cM)$. The morphism $B_X \to \cO_X$ gives us a morphism in $\Der(B_X)$
%\begin{equation}\label{piece1}
%\alpha: \gr_\hbar^{B_X} \mathcal{M} \to _{B}\gr_\hbar \mathcal{M}
%\end{equation}
%such that $for_{B_X}(\alpha)= \id_{\gr_\hbar^{\C_X}(\cM)}$.
%
%
%The coevaluation $a^{-1}_{X} a_{X\ast} \mathcal{M} \to \mathcal{M}$ of the adjunction between $a^{-1}_{X}$ and $ a_{X\ast}$ induces a morphism
%\begin{equation} \label{piece2}
% \Hom_{B_X}(\gr_\hbar^{B_X} \mathcal{M}, _{B}\gr_\hbar \mathcal{M}) \to
% \Hom_{B_X}( \gr_\hbar^{B_X} a^{-1}_X  a_{X \ast} \mathcal{M},_{B}\gr_\hbar \mathcal{M}).
%\end{equation}
%The adjunction between $a_{X}^{-1}$ and $a_{X \ast}$ gives the following isomorphism
%
%\begin{equation} \label{piece3}
%\scalebox{0.95}{$
%\Hom_{B_X}( \gr_\hbar^{B_X} a^{-1}_X a_{X \ast} \mathcal{M},_{B}\gr_\hbar \mathcal{M}) \simeq \Hom_{B}(\gr_\hbar^{B_X} a_{X \ast}\mathcal{M}, a_{X \ast}(_{B}\gr_\hbar \mathcal{M})).
%$}%
%\end{equation}%
%%
%%and by Proposition \ref{oubli} (ii), there is an isomorphism in $\Der^+(B)$, 
%%\begin{equation*}
%%\Rg(X, for_{\cO_X}^{B_X}(\gr_\hbar \mathcal{M})) \simeq \Rg(X, %\gr_\hbar \mathcal{M}).
%%\end{equation*}
%Hence, the image of the morphism (\ref{piece1}) by the composition of the maps (\ref{piece2}) and (\ref{piece3}) give a morphism
%\begin{equation*}
%pr:\gr_\hbar^B \Rg(X,\mathcal{M}) \to \Rg(X, \gr_\hbar \mathcal{M})
%\end{equation*}
%It follows from Proposition \ref{oubli} that 
%\begin{align*}
%for_{B_X}(\gr_\hbar^{B_X} \cM) &\simeq \gr_\hbar^{\C_X} \cM\\  for_{B_X} \circ for_{\cO_X}^{B_X} \gr_\hbar \cM &\simeq \gr_\hbar^{\C_X} \cM\\
%for_B(\Rg(X, for_{\cO_X}^{B_X}\gr_\hbar \cM)& \simeq \Rg(X,\gr_\hbar^{\C_X} \cM)
%\end{align*}
%The diagram below commutes.
%\begin{equation*}
%\xymatrix{
%\Hom_{B_X}(\gr_\hbar^{B_X} \mathcal{M}, _{B}\gr_\hbar \mathcal{M}) \ar[r]^-{for_{B_X}} \ar[d] & \Hom_{\C_X}(\gr_\hbar^{\C} \cM, \gr_\hbar^{\C} \cM) \ar[d] \\
% \Hom_{B_X}( \gr_\hbar^{B_X} a^{-1}_X  a_{X \ast} \mathcal{M},_{B}\gr_\hbar \mathcal{M}) \ar[r]^-{for_{B_X}} \ar[d]^{\wr} & \Hom_{\C_X}(\gr_\hbar^{\C} a^{-1}_X  a_{X \ast} \cM, \gr_\hbar^{\C} \cM)\ar[d]^{\wr}\\
%\Hom_{B}( \gr_\hbar^{B}  a_{X \ast} \mathcal{M},a_{X \ast}(_{B}\gr_\hbar \cM)) \ar[r]^-{for_{B}}  & \Hom_{\C}(\gr_\hbar^{\C} a_{X \ast} \cM, a_{X \ast}(\gr_\hbar^{\C} \cM))
%}
%\end{equation*}
%It follows that $for_{B}(pr)=p$. Since $for_{B}$ is conservative, $pr$ is an isomorphism.
%\end{enumerate}
\end{proof}
\subsection{QCC sheaves on affine varieties}
We define the two functors:
\begin{equation*} 
\Phi: \Der_{\qcc}^+(\cA_X) \to \Der_{\cc}^{+}(A), \; \Phi(\mathcal{M})=\Rg(X,\mathcal{M}) 
\end{equation*}
and
\begin{equation*}
\Psi:\Der_{\cc}^{+}(A) \to \Der_{\qcc}^+(\cA_X), \Psi(M)=(\cA_X \otimes_{A_X} a_X ^{-1}M)^{\cc}.
\end{equation*}

\begin{thm}\label{deforq}
Let X be a smooth affine variety. The functors $\Phi$ and $\Psi$ are equivalences of triangulated categories, are inverses one to each other and the diagram below is quasi-commutative
\begin{equation*}
\xymatrix{ \Der_{\qcc}^+(\cA_X) \ar@<.4ex>[r]^-{\Phi} \ar[d]^-{\gr_\hbar}&  \Der^+_{\cc}(A) \ar@<.4ex>[l]^-{\Psi} \ar[d]^-{\gr_\hbar}\\
\Der_{\qcoh}^+(\cO_X) \ar@<.4ex>[r]^-{\Rg(X,\cdot)} &  \Der^+(B) \ar@<.4ex>[l]^-{\cO_X \otimes_{B_X} \cdot}.}
\end{equation*}
\end{thm}

\begin{proof}
Let $\mathcal{M} \in  \Der_{\qcc}^+(\cA_X)$. By definition, 
\begin{equation*}
\Psi \circ \Phi(\mathcal{M})=\fRHom_{\cA_X}((\cA^{loc}_X / \cA_X) \lbrack -1 \rbrack, \cA_X \Lte_{\cA_X} a_X^{-1}\Rg(X, \mathcal{M})).
\end{equation*}
By adjunction, we have the morphism of functors $a_X^{-1} \circ \dR a_{X\ast} \to \id$. It follows that we have a morphism $ a_X^{-1} \Rg(X,\mathcal{M}) \to \mathcal{M}$. Tensoring by $\cA_X \Lte_{A_X} \cdot$ we get  

\begin{equation}\label{piece}
 \cA_X \otimes_{A_X} a_X^{-1} \Rg(X,\mathcal{M}) \to \cA_X \otimes_{A_X} \mathcal{M}.
\end{equation}
Moreover, 
\begin{equation*}
\begin{split}
\Hom_{\cA_X}( \cA_X \otimes_{A_X} \mathcal{M},\mathcal{M}) &\simeq \Hom_{A_X}( \mathcal{M}, \fHom_{\cA_X}(\cA_X,\mathcal{M}))\\
&\simeq \Hom_{A_X}( \mathcal{M},\mathcal{M}).
\end{split}
\end{equation*}

Consequently the image of the identity gives a morphism $\cA_X \otimes_{A_X} \mathcal{M} \to \mathcal{M}$. By composing with (\ref{piece}), one obtains a morphism
\begin{equation*}
 \cA_X \otimes_{A_X} a_X^{-1} \Rg(X,\mathcal{M}) \to \mathcal{M}.
\end{equation*}
Applying the functor $( \cdot)^{\cc}$ to the preceding morphism we obtain 
\begin{equation*}
(\cA_X \otimes_{A_X} a_X^{-1} \Rg(X,\mathcal{M}))^{\cc} \to \mathcal{M}^{\cc}.
\end{equation*}
Since $\mathcal{M}$ is cohomologically complete, $\mathcal{M}^{\cc}\simeq \mathcal{M}$. Thus

\begin{equation}\label{premier_iso}
( \cA_X \otimes_{A_X} a_X^{-1} \Rg(X,\mathcal{M}))^{\cc} \to \mathcal{M}.
\end{equation}
Applying $\gr_\hbar$ to the preceding formula, and using the well known equivalence 

\begin{equation*}
\xymatrix{\Der_{\qcoh}^+(\cO_X) \ar@<.4ex>[r]^-{\Rg(X,\cdot)} &  \Der^+(B) \ar@<.4ex>[l]^-{\cO_X \otimes_{B_X} -}},
\end{equation*}

 we obtain the isomorphism

\begin{equation*}
\cO_X \otimes_{B_X} \Rg(X, \gr_\hbar \mathcal{M}) \stackrel{\sim}{\longrightarrow} \gr_\hbar \mathcal{M}.
\end{equation*}

Since $( \cA_X \otimes_{A_X} a_X^{-1} \Rg(X,\mathcal{M}))^{\cc}$ and $\mathcal{M}$ are cohomologically complete modules, it follows that (\ref{premier_iso}) is an isomorphism.\\



Let $M \in \Der^+_{\cc}(A)$. By definition,
\begin{equation*}
\Phi \circ \Psi(M)=\Rg(X,(\cA_X \otimes_{A_X} a_X^{-1}M)^{\cc}).
\end{equation*}
and using Lemma \ref{Commutation_Rf} we get that
\begin{equation*}
 \Phi \circ \Psi(M) \simeq (\Rg(X,\cA_X \otimes_{A_X} M))^{\cc}.
\end{equation*}

We have a morphism

\begin{equation*}
\dR\! a_{X\ast} (\cA_X) \otimes_{A} M \to \dR a_{X\ast}( \cA_X \otimes_{A_X} a_X^{-1} M).
\end{equation*}

Since $X$ is affine we obtain $\dR a_{X\ast} \cA_X \simeq A$ thus

\begin{equation*}
M \to \Rg(X, \cA_X \otimes_{A_X} M).
\end{equation*}

We have a map

\begin{equation}\label{equi2}
M^{\cc} \to  (\Rg(X, \cA_X \otimes_{A_X} M))^{\cc}.   
\end{equation}

Applying the functor $\gr_{\hbar}$, we obtain

\begin{equation}\label{gr_proj}
\gr_{\hbar}M \to \Rg(X, \cO_X \otimes_{B_X}  \gr_{\hbar} M).
\end{equation}

Using Lemma \ref{comcom}, we deduce that the map (\ref{gr_proj}) is an isomorphism. It follows by Corollary \ref{isogr} that the morphism (\ref{equi2}) is an isomorphism. This proves the announced equivalence.
\end{proof}

%\acks  I would like to express my gratitude to my advisor
%Pierre Schapira for his patience and enthusiasm in sharing his mathematical knowledge. I would also like to thank Masaki Kashiwara for enlightening discussions and Gr�gory Ginot for his constant support.


