\section{Introduction}
The notion of Lie algebroids play a prominent role in geometry as generalized infinitesimal symmetries of spaces related to the corresponding global symmetries of spaces, called Lie groupoids (see \cite{KM}). Lie algebroids over a $C^{\infty}$-manifold (\cite{MM}) is joint generalization of tangent vector bundle over the manifold and Lie algebras, whose algebraic analogue is known as Lie-Rinehart algebra.
In the context of complex and algebraic geometry, Lie algebroid over analytic spaces (\cite{BP}) and over algebraic varieties (\cite{UB},\cite{CV},\cite{MK}) (locally affine) respectively has considered in the sheaf theoretic language, which are joint generalization of the tangent sheaf (\cite{SR}, \cite{BP}) and the sheaf of Lie algebras. We are studying Lie algebroids over algebraic spaces (in short  $a$-spaces) $(X, \mathcal{O}_X)$ as quasicoherent sheaf of $(\mathbb{K}_X, \mathcal{O}_X)$-Lie-Rinehart algebras (\cite{AA}) where $\mathbb{K}_X$ is the constant sheaf with stalks $\mathbb{K}$ ($\mathbb{K= R, C}$ or any algebraic closed field of charecteristic $0$) and $\mathcal{O}_X $ is a sub-sheaf of algebras of the sheaf of continuous functions $C^0$ on $X$ (\cite{AM}, \cite{SR}), combines the three types of base spaces as special cases. To study geometries in a unified manner we need the notion of Lie algebroid in algebro-geometric language, allow one to treat several geometric structures, such as Poisson manifolds, foliations and their singular settings (Poisson analytic spaces and generalized involutive distributions respectively).

Associated with a Lie algebroid over an $a$-space we have two canonical generalized bialgebras, one is universal enveloping algebroid and other one is jet algebroid. Both of them played an important role in order to study homological algebra for a Lie algebroid.
The universal enveloping algebroid $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ (\cite{UB}, \cite{BP},\cite{CV},\cite{MK},\cite{DRV}) of a Lie algebroid $\mathcal{L}$ (generalization of sheaf of differential operators on a manifold (\cite{SR}, \cite{TS})) is sheafification of the presheaf of universal enveloping algebras of Lie-Rinehart algebras associated with each space of sections of $\mathcal{L}$. It has a canonical $\mathcal{O}_X/{\mathbb{K}_X}$-bialgebra structure (\cite{AA}), similar structures are present in (\cite{CV},\cite{MK}) by different names.

Moreover the dual of $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$, $\mathscr{J}(\mathcal{O}_X,\mathcal{L}) := \mathscr{H}om_{\mathcal{O}_X}(\mathscr{U}(\mathcal{O}_X,\mathcal{L}),\mathcal{O}_X )$ is the jet algebroid of $\mathcal{L}$ (\cite{AP}, \cite{CV}, \cite{BP},\cite{DRV}).
It is the sheafification of presheaf of jet algebras associated with the sheaf of Lie-Rinehart algebras. It has a canonical commutative, associative, unital $\mathcal{O}_X$-algebra structure induced from co-commutative co-associative counital coalgebra structure of $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ over $\mathcal{O}_X$ and also have a sheaf of $\mathbb{K}$-counital coalgebra structure with the canonical $\mathcal{L}$-module structure. So, together all these it has a $\mathcal{O}_X/{\mathbb{K}_X}$-bialgebra structure (\cite{AA}).

In section $2$, we consider algebraic (analytic) de Rham cohomologies associated with some principal ideal (sheaf) or principal divisors (\cite{AA}). Then in section $3$, we recall that the Lie algebroid (locally free $\mathcal{O}_X$-module of finite rank) cohomology over any Noetherian separated schemes (\cite{UB}), complex manifolds $(X,\mathcal{O}_X)$ (\cite{BP}) (generalizations of de-Rham cohomology of $(M, C^{\infty}$) (\cite{SR}), algebraic and analytic de-Rham cohomology and Lie-Rinehart algebra cohomology (\cite{GR})) expressed by derived functor $Ext$ as $\mathbb{H}^\bullet(\mathcal{L}; \mathcal{E}) \cong Ext^\bullet_{\mathscr{U}(\mathcal{O}_X,\mathcal{L})}(\mathcal{O}_X,\mathcal{E})$ for some $\mathcal{L}$-module $\mathcal{E}$. We show that we can define Lie algebroid (locally free $\mathcal{O}_X$-module but not necessarily of finite rank) cohomology combining all of the three geometric set up (including singular cases) and we produce similar result 
 
 
 
  by using ideas from the local model of Lie-Rinehart cohomology as derived functor for a projective Lie-Rinehart algebra (\cite{GR}). 
 

 

  In the first part of section $4$, we present a version of Hochschild-Kostant-Rosenbergh (HKR) theorem
  for locally free Lie algebroids over any of the special $a$-spaces (real $C^\infty$ manifolds, analytic spaces or algebraic varieties). It provides an isomorphism of sheaf of graded vector spaces between hyper Hochschild cohomology of $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ and hypercohomology of the sheaf of $\mathcal{L}$-poly vector fields, that is $\mathbb{H}H^\bullet(\mathscr{U}(\mathcal{O}_X,\mathcal{L})) \cong \mathbb{H}^\bullet(X, \wedge^\bullet_{\mathcal{O}_X} \mathcal{L})$ (\cite{CV}) by using the algebraic version locally as a projective $(\mathbb{K},R)$-Lie Rinehart algebra $L$ we have vector space isomorphism  $HH^{\bullet}(U(R,L)) \cong \wedge^\bullet_R L$ (\cite{KP}). Next we discuss about the dual version of HKR theorem in the generalize setup. Thus we show that for a locally free Lie algebroids $\mathcal{L}$ over any  

 of the special $a$-space $(X, \mathcal{O}_X)$, we get a canonical isomorphism of vector spaces between hyper Hochschild cohomology of $\mathscr{J}(\mathcal{O}_X, \mathcal{L})$ and the Lie algebroid (hyper)cohomology of $\mathcal{L}$, that is $\mathbb{HH}^\bullet(\mathscr{J}(\mathcal{O}_X,\mathcal{L})) \cong \mathbb{H}^\bullet(\mathcal{L}; \mathcal{O}_X)$ by using algebraic version locally as a finitely generated projective $(\mathbb{K},R)$-Lie Rinehart algebra $L$ we have vector space isomorphism  $HH^\bullet(J(R,L)) \cong H^\bullet(L;R)$ (\cite{KP}). 
 

 
 

  
 \section{Algebraic de Rham cohomology }
 
 

  
 
  
  We compute algebraic (analytic) de Rham cohomology groups of some family of non-singular variety and its associated singular variety (which appears as deformation of objects in algebraic geometry).
 
 $(1)$ \textbf{Rectangular Hyperbola :} We consider the affine space $X:=\mathbb{A}^2$ ($\mathbb{C}^2$ with Zariski topology) and its co-ordinate ring $\tilde{R}=\mathcal{O}_X(X)=\mathbb{C}[x,y]$. The principal ideal $I=\langle xy-t \rangle \subset \tilde{R}$ provides rectangular hyperbolas $Y_t:=V(I)$ (vanishing set of $I$) for every $t$ in $\mathbb{C}\setminus \{0\}$. For $t \in \mathbb{C} \setminus \{0\}$,  $Y_t$ is a non singular affine variety with it's co-ordinate ring $R_t:=\mathcal{O}_{Y_t}(Y_t)=\mathbb{C}[x,y]/ \langle xy-t \rangle$. 
 
  Here, $Y:=Y_1$ is homeomorphic to $\mathbb{C} \setminus \{0\}$ and $\mathbb{C} \setminus \{0\}$ is  same homotopy type of $S^1$. Thus, on considering the singular cohomology we have that 
  $$H^i_{Sing}(Y)=H^i_{Sing}(S^1)=  \mathbb{C} $$ for $i=0,1$ and $H^i_{Sing}(Y)=H^i_{Sing}(S^1)= \{0\}$ for $i \geq 2$.
  
 
  We can use algebraic de Rham theorem to get algebraic de Rham cohomology of $Y$ by the singular cohomology of $S^1$. Now we recall the computation which helps us to follow the associated singular case when $t=0$ (which is a normal crossing divisor). 
  
On $Y$  the differential $df=xdy+ydx=0$  and $\frac{1}{x} \in R$, so $dy=- \frac{dx}{x^2}\in \langle dx \rangle_{R-mod}$. Thus the space of differential $1$-forms (Kahler differentials for $Spec R$) is 
$$\Omega^1_R:=\frac{\langle \{dx,dy\} \rangle}{\langle df \rangle}=\langle dx \rangle_{R-mod}.$$ Note that $\frac{dx}{x}|_Y$ is algebraic differential $1$-form on $Y$. The space of algebraic differential $2$-forms on $Y$ is $\Omega^2_R:=\wedge^2_R \Omega^1_R$.  Now, $dx \wedge dy=dx \wedge - \frac{dx}{x^2}=0$ on $Y$ (since $y=\frac{1}{x}$ in $Y$) and thus we get $\Omega^2_R=\{0\}$ (here $\Omega^1_R$ is a free $R$-module of rank $1$).

 Thus, the algebraic de Rham complex for $Y$ (or $Spec R$) is 
 $$R  \overset{d_0}{\longrightarrow} \Omega^1_R \overset{d_1}{\longrightarrow} 0 \cdots ,$$
  where the first differential $d_0$ is given by $x\mapsto dx$ and $\frac{1}{x}\mapsto - \frac{dx}{x^2}$. Here, $\frac{dx}{x}$ is in the kernel of $d_1$ but $\frac{dx}{x}$ not in the image of  $d_0$ 

 (the only possible choice is $log x$, but it is not a polynomial or rational function). It follows that $H^1_{dR}(Y)$ is a $\mathbb{C}$-vector space of dimension $1$. Now, $H^0(Y) \cong \mathbb{C}$ and $H^n(Y)=\{0\}$ for $n \geq 2$. 


 
 The de Rham cohomology reproduces the topological invariant. This is present in \cite{AH}, but  without the computation of algebraic de Rham cohomology for the associated singular case.
 
 \begin{Rem} 
 In this case the sheaf of logarithmic vector fields associated to $Y$ is $\mathcal{T}_X(-log Y)=\langle \{x\partial_x-y\partial_y\} \rangle$ and sheaf of algebraic differential $1$-forms of $X$ with poles along the divisor $Y$ (in this case set of poles is empty) is 
 $$\Omega^1_X(log Y)=\mathscr{H}om_{\mathcal{O}_X}(\mathcal{T}_X(-log Y),\mathcal{O}_X)=\langle \{\frac{dx}{x}-\frac{dy}{y}\} \rangle.$$ 
 The sheaf of logarithmic differential $2$-forms in $X$ associated with $Y$ is $\wedge^2_{\mathcal{O}_X} \Omega^1_X(log Y)$. Now, $dx \wedge dy=dx \wedge - \frac{dx}{x^2}=0$ on any open subset $V$ in $Y$ $(y=\frac{1}{x}$ in $Y)$. Thus $ \Omega^2_X(log Y)$ is the zero sheaf (here $\Omega^1_X(log Y)$ is a free $\mathcal{O}_X$-module of rank $1)$. The hypercohomology of the (algebraic) logarithmic de Rham complex
 \begin{center}
 	$\mathcal{O}_X \overset{d_0}{\longrightarrow}\Omega^1_X(log Y)\overset{d_1}{\longrightarrow}\wedge^2_{\mathcal{O}_X}\Omega^2_X(log Y) \overset{d_2}{\longrightarrow}\cdots$
\end{center}
 is $\mathbb{H}^n(X,\Omega^{\cdot}_X(log Y))=\{\mathbb{C}\}$ for $n=0,1$ and all other higher dimensional hypercohomologies are zero (using similar kind of arguments from above).
 
 
 For $D=f_1\partial_x+ f_2 \partial_y \in \mathcal{T}_X$ such that $\langle D, \nabla f \rangle=0$ (with respect to standard inner product) we get $f_1 \cdot y+ f_2 \cdot x=0$ (since $\nabla f=y \partial_x + x \partial_y)$.  Thus $f_1=g\cdot x, f_2=-g \cdot y$ for any $g\in \mathcal{O}_X$. So, the derivation is $D=g(x\partial_x-y\partial_y)$ for some $g \in \mathcal{O}_X$ and it preserves the ideal $I$.
 
  On $Y$ we have $df=y dx+x dy=0$  implies $\frac{dx}{x}+\frac{dy}{y}=0$ ($x\neq 0, y\neq 0$) and by integrating it we get $xy=t$ are the solutions which parametrized by $t\in \mathbb{C}-\{0\}$.
 
 So, we can view the normal crossing divisor $Y_0 :=\{(x,y)\in \mathbb{C}^2|xy=0\}$ (a singular space) as deformation of family of rectangular hyperbolas $Y_t$ (appears in \cite{MP} as deformation of a scheme) and if we restrict the derivation $x\partial_x-y\partial_y$ on $Y_0$ then on $x$-axis it become $x\partial_x$ and on $y$-axis this become $y\partial_y$. 
 In this way we can take $\mathcal{T}_X(- log Y_0)=\langle \{x\partial_x,y\partial_y\} \rangle$. 
 \end{Rem}
 $(2)$ \textbf{Normal Crossing Divisor :} Here we compute the algebraic de Rham cohomology of the normal crossing divisor $Y_0$ which is a singular affine algebraic set in $X$ with the space of global section of its structure sheaf 
 $$R_0 :=\mathcal{O}_{Y_0}(Y_0)=\frac{\mathbb{C}[x,y]}{\langle xy \rangle}$$
 
  For $Y_0$ we have $d(xy)=0$. So $x dy=-y dx $, which implies $x dx \wedge_R dy=0$ or $y dx \wedge_R dy=0$. But we cannot conclude that $dx \wedge_R dy=0$ on $Y_0$. 
  
  The space of algebraic differential (Kahler differential) $1$-forms for $R$ is 
  $$\Omega^1_R=\frac{\langle \{dx,dy\} \rangle}{\langle y dx+x dy \rangle}$$
   as a $R$-module ($Y_0$ can be viewed as $Spec R$). The relation $x dy=-y dx $ will not produce $\frac{dx}{x}=-\frac{dy}{y}$ because here $xy=0$. So, $\Omega^1_X(log Y_0)=\langle \{\frac{dx}{x},\frac{dy}{y}\} \rangle$ is the $\mathcal{O}_X$-module of rational (meromorphic) differential $1$-forms on $X$ with poles along the divisor $Y_0$ and $\Omega^1_R=\langle \{\frac{dx}{x}|_{Y_0},\frac{dy}{y}|_{Y_0}\} \rangle$ is the associated $R$-module of Kahler differential $1$-forms. So, $H^1_{dR}(Y_0)$ is a two-dimensional $\mathbb{C}$-vector space (since $log x$, $log y$ both are not in $R$ and so $\frac{dx}{x}|_{Y_0}$, $\frac{dy}{y}|_{Y_0}$ are not in $Im$ $d_0$ but both are in $Ker$ $d_1$) and obviously $H^0_{dR}(Y_0)=\mathbb{C}$. Note that $d_1(x^2 dy)=0=d_1(y^2 dx)$ on $Y_0$ and $x^2 dy, y^2 dx$ both are not in $Im d_0$ but both this elements vanish on $Y_0$. 
   
   Next, we compute its second algebraic de Rham cohomology class, for that first consider the $R$-module of Kahler (algebraic) differential $2$-forms 
   $$\Omega^2_R=\frac{\langle dx \wedge dy \rangle}{\langle \{x dx\wedge dy, y dx\wedge dy\}\rangle}.$$
    Sheaf of rational (meromorphic) $2$-forms on $X$ with poles along the divisor $Y_0$ is $\Omega^2_X(log Y_0)=\langle \frac{dx}{x}\wedge \frac{dy}{y} \rangle$, a free $\mathcal{O}_X$-module of rank $1$. Note that $\Omega^k_X(log Y_0)|_{Y_0}\cong \Omega^k_R=\wedge^k_R \Omega^1_R$. So, $\Omega^2_R=\langle \frac{dx}{x}|_{Y_0}\wedge \frac{dy}{y}|_{Y_0} \rangle$ is a free $R$-module of rank $1$. Here $H^2_{dR}(Y_0)=\frac{\Omega^2_R}{Im d^1}$. Since $d_1(x dy)= dx\wedge dy \neq 0$, so $dx\wedge dy$ is a trivial choice in $H^2_{dR}(Y_0)$. Thus, $\frac{dx}{x}|_{Y_0}\wedge \frac{dy}{y}|_{Y_0}\in \Omega^2_R$ but not in $Im d_1$. So, it gives a non trivial cohomology class which generates $H^2_{dR}(Y_0)$, so $H^2_{dR}(Y_0)$ is a $\mathbb{C}$-vector space of dimension $1$. Note that $Y_0$ is of same homotopy type to a point, so singular cohomologies of $Y_0$ vanishes in all higher dimensions ($\geq 1$). 
 
 Now, we consider the associated cohomology in complex geometry. In this case $\frac{dx}{x},\frac{dy}{y}$ are holomorphic $1$-forms on $X-Y_0$ and they are meromorphic forms on $X$ with poles along the divisor $Y_0$. The hypercohomology of the logarithmic de Rham complex
 \begin{center}
 	$\mathcal{O}_X \overset{d_0}{\longrightarrow}\Omega^1_X(log Y_0)\overset{d_1}{\longrightarrow}\wedge^2_{\mathcal{O}_X}\Omega^2_X(log Y_0) \overset{d_2}{\longrightarrow}\cdots$
 \end{center}
 is the holomorphic Lie algebroid cohomology  (\cite{BRT})
 $$H^n_{holo}(\mathcal{T}_X(-log Y_0)) :=\mathbb{H}^n(X,\Omega^{\cdot}_X(log Y_0)) 
 \cong H^n_{dR}(X-Y_0;\mathbb{C})$$ 
 (by using the standard Lemma of Atiyah- Hodge \cite{MS}). For this reason in the above computation of algebraic de Rham cohomology for $A_0$ (or of $Y_0=Spec A_0$), we have computed the standard Lie-Rinehart cohomology of the $(\mathbb{C}, A_0)$- Lie-Rinehart algebra $Der_{\mathbb{K}}(A_0)$ and it provide us cohomology of the torus $\mathbb{T}^2$ in $\mathbb{R}^4 \cong \mathbb{C}^2$ (note that $X-Y_0= \{(x,y)\in \mathbb{C}^2| x\neq \bar{0}$ and $y\neq \bar{0}\}=(\mathbb{C}-{\bar{0}})\times (\mathbb{C}-{\bar{0}})$ homotopic to $S^1\times S^1= \mathbb{T}^2$).
 \begin{Note} In general for smooth algebraic variety and complex manifold, to compute its cohomology we need to consider sheaf of  de Rham complex because the space of global sections 
 cannot captures the whole informations. In these cases we can use $\check{C}$ech cohomology by considering an affine or good open covers (\cite{BRT}, \cite{MS}) accordingly. 
 \end{Note}
 \section{Lie algebroid Cohomology as derived functor}


 
 

 Lie algebroids over algebraic spaces (in short we call it as $a$-spaces) discussed in \cite{AA} by considering the sheaf of Lie-Rinehart algebras.
 Here we consider 

  special $a$- spaces $(X, \mathcal{O}_X)$ with an appropriate choice for the structure sheaf $\mathcal{O}_X$ and Lie algebroids over an $a$-space (\cite{AA}) in a slightly modified form. This defination is similar to the defination given in \cite{MK}.
  A Lie algebroid $\mathcal{L}$ over an $a$- space $(X,\mathcal{O}_X)$ is a quasicoherent sheaf of $(\mathbb{K}_X, \mathcal{O}_X)$- Lie-Rinehart algebras.
 
  
  The Chevalley-Eilenberg- de Rham cohomology  for a Lie algebroid  is presented
   as derived functor $Ext$ (\cite{UB, BP}).  In \cite{UB}, the result is proved for locally free Lie algebroid of finite rank on a Noetherian separated scheme over a field of characteristic $0$ and in \cite{BP}, the result stated for locally free Lie algebroid of finite rank over a complex manifold.  Here, we do not require  the condition of coherent $\mathcal{O}_X$-module for a Lie algebroid, and our approach for the proof is slightly different from that in\cite{UB}. Note that for analytic and algebraic geometric set up, we don't need the base space $X$ to be non-singular. But,  in that case we miss the result for the model object tangent sheaf (which become a coherent Lie algebroid but not a locally free Lie algebroid), see \cite{AA} for details.
 


 
 	
 
 	
 For a Lie algebroid $\mathcal{L}$ over $(X,\mathcal{O}_X)$, we form the sheaf of tensor algebra $\mathcal{T}_{\mathcal{O}_X}\mathcal{L}$ as the sheafification of the cannonical presheaf of tensor algebras associated with each open sets (locally gives its space of sections). Then to get universal enveloping algebroid we need to consider one of its special quotient sheaf. Thus  we find that  $\mathcal{U}(\mathcal{O}_X,\mathcal{L})$ is a  presheaf of associative unital $\mathbb{K}_X$-algebra and also a presheaf of left $\mathcal{O}_X$-modules with compatibility conditions, where $\mathcal{U}(\mathcal{O}_X,\mathcal{L})(U)= \mathcal{U}(\mathcal{O}_X(U),\mathcal{L}(U))$ and restriction morphisms are the maps $res_{UV}^{\mathcal{U}}$ for any two open sets $V, U$ with $V \subset U$.
 We consider its sheafification, and denote the resulting sheaf by $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$. The necessary algebraic structures on $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ given via structures induced on stalks of the presheaf $\mathcal{U}(\mathcal{O}_X,\mathcal{L})$.
The sheaf  $\mathscr{U}(\mathcal{O}_X, \mathcal{L})$ is called  universal enveloping algebroid of the Lie algebroid $\mathcal{L}$.

The Atiyah algebroid $ At(\mathcal{E})$ is a sheaf of derivations of the $\mathcal{O}_X$-module $\mathcal{E}$ with scaler symbols 
 (see \cite{UB,BP}). If $(\mathcal{E}, \nabla)$ is a $\mathcal{L}$-module, then the Lie algebroid morphism
 \begin{center}
 	$\nabla : \mathcal{L} \rightarrow At(\mathcal{E}) $
 \end{center}  
 extends to a $\mathcal{O}_X$-linear  homomorphism of $\mathbb{K}_X$-algebras
 	$$\tilde{\nabla} : \mathscr{U}(\mathcal{O}_X,\mathcal{L}) \rightarrow \mathcal{E}nd_{\mathbb{K}_X}(\mathcal{E}),$$
 making $\mathcal{E}$ into a left $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$-module. If $\mathcal{L}$ is locally free $\mathcal{O}_X$-module and $\mathcal{E}$ be a $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$-module, then the restriction of the action of $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ to $\mathcal{L}$ (by using the canonical embedding of $\mathcal{L}$ in $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$
  \cite{AA}), gives a $\mathcal{L}$-module structure on $\mathcal{E}$. In this case, the category of $\mathcal{L}$-modules and the category of left $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$-modules are equivalent. It helps in studying  homological algebra with $\mathcal{L}$-modules ($\mathcal{L}$ is a sheaf of non-associative $\mathbb{K}$-algebras) by treating them as modules over the associative $\mathbb{K}_X$-algebra $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$.
 
 \subsection{Chevalley-Eilenberg-de Rham complex and Koszul-Rinehart resolutions:}
 For a Lie algebroid $\mathcal{L}$ over $(X,\mathcal{O}_X)$ with a representation $(\mathcal{E},\nabla)$, we associate a cochain complex of (sheaves of) $\mathcal{O}_X$- modules $(~\mathscr{H}om_{\mathcal{O}_X}(\wedge^{\bullet}_{\mathcal{O}_X}\mathcal{L}, \mathcal{E}), d~)$ denoted simply as $\Omega^{\bullet}_{\mathcal{L}}(\mathcal{E})$, is the sheafification of the presheaf $$\{U\mapsto Hom_{\mathcal{O}_X(U)}(\wedge^{\bullet}_{\mathcal{O}_X(U)}\mathcal{L}(U), \mathcal{E}(U)), d_U)\}.$$
 The differential $d_U$ associated with an open subset $U$ of $X$ is given by
 \begin{center}
 	$d_U(\omega)(D_1\wedge \cdots \wedge D_{k+1})= \sum^k_{i=1}(-1)^{i+1}\nabla^U_{\mathfrak{a}(U)(D_i)}(\omega(D_1\wedge \cdots \wedge \hat{D_i} \wedge \cdots D_{k+1}))$\\
 	+ $ \sum_{i< j}(-1)^{i+j}\omega([D_i, D_j]\wedge D_1\wedge \cdots \wedge \hat{D_i}\cdots \wedge \hat{D_j}\wedge \cdots \wedge D_{k+1})$,
 	
 \end{center}
 where $D_i \in \mathcal{L}(U)$ for $i=1,\cdots,k+1;$ 
 $  \omega \in \Omega^k_{\mathcal{L}(U)}$, $\mathfrak{a}(U)$ is the anchor map (restricted) on $U$ and $\nabla^U$ is local operator for the flat $\mathcal{L}$ connection on $\mathcal{E}$.
 
 If $\mathcal{L}$ is a locally free $\mathcal{O}_X$-module of finite rank then
 \begin{center}
 	$\mathscr{H}om_{\mathcal{O}_X}(\wedge^{\bullet}_{\mathcal{O}_X}\mathcal{L}, \mathcal{E}) \cong \wedge^{\bullet}_{\mathcal{O}_X}\mathcal{L}^* \otimes_{\mathcal{O}_X} \mathcal{E}= \Omega^{\bullet}_{\mathcal{L}}\otimes_{\mathcal{O}_X} \mathcal{E}$
 \end{center}
 where $\Omega^{\bullet}_{\mathcal{L}}:=\Omega^{\bullet}_{\mathcal{L}}(\mathcal{O}_X)$. As a special case we consider  $\mathcal{L}=\mathcal{T}_X$, $\mathcal{E}=\mathcal{O}_X$ when $X$ is non-singular (smooth) and we get the de Rham complex $\Omega^\bullet_X$.
 
 
 Also, we have the Koszul-Rinehart resolution (locally free) for $\mathcal{O}_X$ as left $\mathscr{U}(\mathcal{O}_X, \mathcal{L})$-module given by locally exact chain complex of sheaves
 $$(\mathscr{U}(\mathcal{O}_X, \mathcal{L})\otimes_{\mathcal{O}_X} \wedge^\bullet \mathcal{L}, \partial)$$ denoted as $\mathscr{K}_{\mathcal{O}_X}^\bullet \mathcal{L}$. It is the sheafification of the presheaf 
 $$\{U \longmapsto (\mathcal{U}(\mathcal{O}_X(U), \mathcal{L}(U))\otimes_{\mathcal{O}_X(U)} \wedge^\bullet \mathcal{L}(U), \partial_U)\}$$ where the differential $\partial_U$ associated with an open subset $U$ of $X$ is given by
 \begin{center} \label{del}
 	$\partial_U(\bar{D}\otimes D_1\wedge \cdots \wedge D_{k})= \sum^k_{i=1}(-1)^{i-1}\bar{D}D_i\otimes D_1\wedge \cdots \wedge \hat{D_i} \wedge \cdots D_k$\\
 	+ $ \sum_{i< j}(-1)^{i+j}\bar{D}\otimes [D_i, D_j]\wedge D_1\wedge \cdots \wedge \hat{D_i}\cdots \wedge \hat{D_j}\wedge \cdots \wedge D_{k}$,
 \end{center}
 for $\bar{D}\in \mathcal{U}(\mathcal{O}_X(U),\mathcal{L}(U)), D_i \in \mathcal{L}(U) (i=1,\cdots,k)$. In the next section we consider similar resolutions for $\mathcal{O}_X$ in the category of left $\mathcal{S}_{{O}_X} \mathcal{L}$- (co)modules, which we denote as $\widetilde{\mathscr{K}}_{\mathcal{O}_X}^\bullet\mathcal{L}$.
 
\textbf{Notation :} From now on we use this special type of open cover $\{U_x | x\in X \}$ of $X$ for a locally free Lie algebroid $\mathcal{L}$. Here the restrictions $\mathcal{L}|_{U_x}$ are free $\mathcal{O}_X|_{U_x}$-modules for every $U_x$. These open sets $U_x$'s are called special open sets.

We now describe Lie algebroid cohomology as derived functor.
 \begin{Thm} 
 	Let $\mathcal{L}$ be a locally free Lie algebroid over $(X, \mathcal{O}_X)$ and $\mathcal{E}$ a representation of $\mathcal{L}$. Then we get an isomorphism of graded vector spaces
 	\begin{center}
 		$\mathbb{H}^\bullet (\mathcal{L}; \mathcal{E})\cong Ext^\bullet_{\mathscr{U}(\mathcal{O}_X, \mathcal{L})}(\mathcal{O}_X, \mathcal{E})$. 
 	\end{center}
 \end{Thm}
 \begin{proof}
 	For every $x\in X$, the space of section $\mathcal{L}(U_x)$ is a $(\mathbb{K},\mathcal{O}_{X}(U_x))$-Lie-Rinehart algebra and a free module over $\mathcal{O}_{X}(U_x)$. So, locally the Koszul-Rinehart resolution $\widetilde{\mathscr{K}}_{\mathcal{O}_X}^\bullet\mathcal{L}$ is a free resolution of the sheaf of local ring $\mathcal{O}_{X}$.
 	
 	
 
 
 
 
 
 	
 
 	
 	Here, we consider the two canonically associated presheaf of vector spaces
 	\begin{center}
 		$U\mapsto H^\bullet(\Omega^\bullet_{\mathcal{L}(U)}, \mathcal{E}(U))$,\\
 		$U\mapsto Ext^\bullet_{\mathcal{U}(\mathcal{O}_X(U), \mathcal{L}(U))}(\mathcal{O}_X(U), \mathcal{E}(U)))$
 	\end{center}
 	We have a cannonical $\mathcal{O}_X$-module homomorphism (sheafifying the morphism locally given in \cite{GR}) between the sheaves $\Omega^\bullet_{\mathcal{L}} (\mathcal{E})$ and $\mathscr{H}om_{\mathcal{O}_X}(\mathscr{K}_{\mathcal{O}_X}^\bullet \mathcal{L}, \mathcal{E})$, which induces a homomorphism between the corresponding cohomologies. We show that it is an isomorphism. We have canonical graded vector space isomorphism 
 	\begin{center}
 		$H^\bullet(\Omega^\bullet_{\mathcal{L}(U_x)}, \mathcal{E}(U_x)) \cong Ext^\bullet_{\mathcal{U}(\mathcal{O}_X(U_x), \mathcal{L}(U_x))}(\mathcal{O}_X(U_x), \mathcal{E}(U_x)))$
 	\end{center}
 	for every $U_x$ associated with each $x \in X$ (by using results from \cite{GR}).
 	 Thus the above two presheaves are stalk-wise isomorphic, so their sheafifications become an isomorphism (by Lemma 3.2. and Lemma 3.3. in \cite{AA}) between the associated cohomology sheaves (\cite{MS}, \cite{UB}) i.e.
 	\begin{center}
 		$\mathscr{H}^k(\Omega^\bullet_{\mathcal{L}}; \mathcal{E})\cong \mathscr{E}xt^k_{\mathscr{U}(\mathcal{O}_X, \mathcal{L})}(\mathcal{O}_X, \mathcal{E}))$, for every $k$.
 	\end{center}
 	So, the Chevalley-Eilenberg-de Rham complex of $\mathcal{L}$ with coefficient in $\mathcal{E}$ is quasi isomorphic to dual of the Koszul-Rinehart resolution of $\mathcal{O}_X$ (by $\mathcal{E}$) in the category of $\mathscr{U}(\mathcal{O}_X, \mathcal{L})$-Modules.
 	
 	Now, applying the hypercohomology functor $\mathbb{H}^\bullet(X,-)$ (\cite{VD,MS}) we get the required result (here $\mathbb{H}^\bullet (\mathcal{L}; \mathcal{E}):=\mathbb{H}^\bullet(X,\Omega^\bullet_{\mathcal{L}} (\mathcal{E}))$).
 \end{proof}
 \begin{Note} One can find the proof for tangent sheaf (tangent bundle) over a real smooth manifolds is given in\cite{GR} as a special case, and for Noetherian separated schemes discussed in \cite{UB} by locally using the ideas of \cite{GR}. 
 \end{Note}
  \begin{Rem}
  In complex geometry (algebraic geometry), for a cochain complex of coherent sheaves $\mathcal{F}^\bullet$ over an analytic space (algebraic variety or Noetherian separated scheme), we compute the hypercohomology $\mathbb{H}^\bullet(X,\mathcal{F}^\bullet)$ via $\check{C}$ech cohomology $\check{H}(\mathcal{U},\mathcal{F}^\bullet)$ associated with some good open cover $\mathcal{U}=\{U_i\}_i$ of $X$. Specifically we provide a good open cover by connected Stein spaces (affine varieties or affine schemes) and we use Leray's theorem and Cartan-Serre's vanishing theorem (\cite{FF}) for sheaf cohomology.
Thus when we consider a locally free Lie algebroid $\mathcal{L}$ of finite rank over an analytic space (algebraic variety or Noetherian separated scheme) $(X,\mathcal{O}_X)$, we can compute Lie algebroid cohomology as derived functor using $\check{C}$ech cohomology.
 
  In both cases we have
 $\mathbb{H}^\bullet (\mathcal{L}; \mathcal{E})\cong \check{H}(\mathcal{U},\Omega^\bullet_{\mathcal{L}}(\mathcal{E}))$.

 Since over Stein spaces (affine schemes) $U_i$, $\mathcal{L}(U_i)$ is a projective $\mathcal{O}_X(U_i)$-module (\cite{Mo, CW}), so we get
 \begin{center}
 	$H^\bullet(\Omega^\bullet_{\mathcal{L}(U_i)},\mathcal{E}(U_i)) \cong Ext^\bullet_{\mathcal{U}(\mathcal{O}_X(U_i),\mathcal{L}(U_i))}(\mathcal{O}_X(U_i),\mathcal{E}(U_i))$ (by \cite{GR}).
 \end{center}
 Thus, there is an graded vector space isomorphism between $\check{C}$ech cohomologies
 $$\check{H}^\bullet(\mathcal{U},\Omega^\bullet_{\mathcal{L}}(\mathcal{E})) \cong \check{H}^\bullet(\mathcal{U}, \mathscr{H}om_{\mathscr{U}(\mathcal{O}_X,\mathcal{L})}(\mathscr{K}_{\mathcal{O}_X}^\bullet \mathcal{L},\mathcal{E})).$$ 
 
 In \cite{BRT}, the Lie algebroid (hyper)cohomology of holomorphic (algebraic) Lie algebroids over a complex manifold (smooth scheme) has expressed by $\check{C}$ech cohomology using a good open cover consisting with connected Stein manifolds (affine schemes). But they have not considered the case associated with analytic spaces. The whole ideas works for the classical case of Lie algebroids over smooth manifolds (smooth real Lie algebroids) using \cite{WG,BRT}.
 \end{Rem}
 

 \section{Hochschild Cohomology of a Lie algebroid over $a$-spaces}

 


 
 In this section, we describe the Hochschild cohomology of universal enveloping algebroid and jet algebroid associated with a locally free Lie algebroid over an $a$-space. The cohomology groups can be computed using suitable standard complexes. 

 We prove a version of Hochschild-Kostant-Rosenberg (HKR) theorem and dual HKR theorem. It is done by following the local counterpart as can be found in \cite{KP}, where the authors  proved (dual) HKR theorem for (finitely generated) projective Lie-Rinehart algebras. One can find in \cite{CV} the formality for Lie algebroids in a more general approach which includes HKR isomorphism in the category of Gerstenhaber algebras.  
 
  Let $\mathcal{L}$ be a Lie algebroid over a special $a$-space $(X,\mathcal{O}_X)$ (as previously described).
 \begin{itemize}
 \item The sheaf of $\mathcal{L}$-poly-vector fields on $X$ is defined as $\mathcal{T}^{poly}_{\mathcal{L}}(\mathcal{O}_X)= \bigoplus_i \wedge^i \mathcal{L}$. This agrees with the standard definitions in case $\mathcal{L}$ is one of the variants of the tangent bundle descried for different $a$-spaces occurs in geometries. We can check that $\mathcal{T}^{poly}_{\mathcal{L}}(\mathcal{O}_X)$ is a sheaf of Gerstenhaber algebra on $X$.
 In the case for $X$ is a $C^\infty$-manifold Kontsevich introduced the sheaf of so-called poly-differential operators on $X$. This is basically a localized version of the Hochschild complex. 

 It is straightforward to construct a Lie algebroid generalization $\mathcal{D}^{poly}_{\mathcal{L}}(\mathcal{O}_X)$ (sheaf of $\mathcal{L}$-poly differential operators) of this concept as well. 
 The so-called Hochschild-Kostant-Rosenberg map is a quasi-isomorphism between $\mathcal{T}^{poly}_{\mathcal{L}}(\mathcal{O}_X)$ and $\mathcal{D}^{poly}_{\mathcal{L}}(\mathcal{O}_X)$. See \cite{CV} for details.
 
 
 \item \label{note} Using ideas from proof of HKR theorem (and canonical PBW coalgebra isomorphism) for Lie-Rinehart algebras (\cite{KP}) we get the following results. To state these results  first we need to recall some notaions associated with a $(\mathbb{K},R)$-Lie-Rinehart algebra $L$ and a $R/{\mathbb{K}}$-bialgebra $A$. The symmetric algebra of $L$ over $R$ (using underlying R-module structure) is denoted by $S_R L$ and  $S_R L^*$ is the symmetric algbra for the $R$-module $L^*:=Hom_R(L,R)$, the universal enveloping algebra of the $(\mathbb{K},R)$-Lie-Rinehart algebra $L$ is denoted by $\mathcal{U}(R,L)$ and its dual is the jet algebra $\mathcal{J}(R,L)$, the cobar resolution of $A$ (using underlying $R$-coalgebra structure) is denoted by $Cob^\cdot(A)$ and the associated cohomology (applying $Hom(-,R)$ functor) is Hochschild cohomology of the $R/{\mathbb{K}}$-bialgebra $A$ denoted by $HH^\cdot (A)$ (in our cases $A$ is either $S_RL$, $\mathcal{U}(R,L)$ or $S_R L^*$, $\mathcal{J}(R,L)$ with canonical $R/\mathbb{K}$-bialgebra structure \cite{MM,CV,DRV,KP}). 
 \end{itemize}
  
  We know that the sheaf of symmetric algebras $\mathcal{S}_{\mathcal{O}_X}\mathcal{L}$ can be seen as universal enveloping algebroid of the Lie algebroid $\mathcal{L}$ if the $\mathcal{O}_X$-module $\mathcal{L}$ is equipped with zero bracket and zero anchor (in each space of sections) (\cite{AA}). Recall that we have P.B.W. presheaf homomorphism of presheaf of $\mathcal{O}_X$-algebas given as
 \begin{center}
 	$\theta:S_{\mathcal{O}_X}\mathcal{L} \rightarrow \mathcal{U}(\mathcal{O}_{X}, \mathcal{L})$\\
 	$f \otimes D_1 \otimes \cdots \otimes D_k \mapsto  \frac{1}{k!} f \underset
 	{\sigma \in S_k}{\sum} (-1)^{\sigma} \bar{D}_{\sigma(1)} ... \bar{D}_{\sigma(k)}$
 \end{center}
 where $f$ is a section of $\mathcal{O}_X$, $D_i$ is a section of $\mathcal{L}$ and $\bar{D}_{\sigma(i)}$ is it's associated image in $\mathcal{U}(\mathcal{O}_{X}, \mathcal{L})$ $(i=1, 2, \cdots,k)$. Sheafification of this map gives the generalized PBW map between the associated sheaves (see \cite{AA}). 
 Additionally, if  $\mathcal{L}$ is locally free $\mathcal{O}_X$-module then we get following results.
 \begin{Lem}
 	If a Lie algebroid $\mathcal{L}$ is locally free $\mathcal{O}_X$-module, then 
 	the generalized PBW map (or symmetrization map)$$\mathcal{S}_{\mathcal{O}_X}\mathcal{L} \rightarrow \mathscr{U}(\mathcal{O}_X,\mathcal{L})$$ is an isomorphism of $\mathcal{O}_X$-coalgebras.
 \end{Lem}
 \begin{proof}
 First we take
 
 	the PBW map associated with space of sections of $\mathcal{L}$ over each special open subset $U_x$ in $X$. It provides an $\mathcal{O}_{X}(U_x)$- coalgebra isomorphism (between the cocommutative $\mathcal{O}_{X}(U_x)$- coalgebras)
 	$$ S_{\mathcal{O}_{X}(U_x)}\mathcal{L}(U_x) \cong \mathcal{U}(\mathcal{O}_{X}(U_x), \mathcal{L}(U_x)). $$ It induces stalkwise isomorphisms and by using the sheafification functor over the underlying presheaf homomorphism we get 
 	the generalized PBW map $$\mathcal{S}_{\mathcal{O}_X}\mathcal{L} \rightarrow \mathscr{U}(\mathcal{O}_X,\mathcal{L})$$ is an isomorphism of $\mathcal{O}_X$-coalgebras. 
 \end{proof}
  \begin{Lem}
 	If a Lie algebroid $\mathcal{L}$ is locally free $\mathcal{O}_X$-module, then there is an isomorphism of $\mathbb{K}$-vector spaces associated with each special open set $U_x$,
 	$$HH^\bullet(\mathcal{U}(\mathcal{O}_{X}(U_x), \mathcal{L}(U_x))) \cong \wedge_{\mathcal{O}_{X}(U_x)}^\bullet\mathcal{L}(U_x).$$
 \end{Lem}
 
 
 \begin{proof}
 The $(\mathbb{K},\mathcal{O}_X(U_x))$-Lie-Rinehart algebra $\mathcal{L}(U_x)$ is a free $\mathcal{O}_X(U_x)$-module for each special open set $U_x$. As in\cite{KP}, we consider the quasi isomorphism (denoted by the notation $\overset{\sim}{\rightarrow}$) of cochain complexes associated with each special open set $U_x$ given as
 $Cob^\bullet(\mathcal{U}(\mathcal{O}_{X}(U_x), \mathcal{L}(U_x)) \overset{\sim}{\rightarrow} Cob^\bullet(S_{\mathcal{O}_{X}(U_x)}\mathcal{L}(U_x)).$

  The  quasi isomorphism induces an isomorphism of graded vector spaces $$HH^\bullet(S_{\mathcal{O}_{X}(U_x)}\mathcal{L}(U_x))\cong HH^\bullet(\mathcal{U}(\mathcal{O}_{X}(U_x), \mathcal{L}(U_x)))$$ associated with every special open set $U_x$ for $x\in X$. Now for an open set $U \subset X$ we have the anti-symmetrization (or skew-symmetrization) map \label{Alt}
 \begin{center}
 	$Alt_{U} : \wedge_{\mathcal{O}_{X}(U)}^\bullet\mathcal{L}(U) \rightarrow (S_{\mathcal{O}_{X}(U)}\mathcal{L}(U))^{\otimes^\bullet}$ given by
 	
 	$D_1 \wedge \cdots \wedge D_n \mapsto \frac{1}{n!}\sum_{\sigma \in S_n} (-1)^\sigma D_{\sigma_1}\otimes \cdots \otimes D_{\sigma_n}$,
 \end{center}
 as well as the map 
 \begin{center}
 	$P_{U} : (S_{\mathcal{O}_{X}(U)}\mathcal{L}(U))^{\otimes^\bullet} \rightarrow \wedge_{\mathcal{O}_{X}(U)}^\bullet\mathcal{L}(U)$ defined as
 	
 	$\tilde{D_1} \otimes \cdots \otimes \tilde{D_n} \mapsto Pr_{U}(\tilde{D_1})\wedge \cdots \wedge Pr_{U}(\tilde{D_n})$,
 \end{center}
 Here $Pr_{U} : S_{\mathcal{O}_{X}(U)}\mathcal{L}(U) \rightarrow S^1_{\mathcal{O}_{X}(U)}\mathcal{L}(U)= \mathcal{L}(U)$ is the projection map on the direct summand $S^1_{\mathcal{O}_X(U)}\mathcal{L}(U)=\mathcal{L}(U)=\wedge^1_{\mathcal{O}_X(U)} \mathcal{L}(U)$ and $D_i \in \mathcal{L}(U)$, $\bar{D_i} \in S_{\mathcal{O}_{X}(U)}\mathcal{L}(U)$ $(i= 1,\cdots ,n)$.
 
 These morphisms defines cochain equivalence on each special open set $U_x$ in $X$ and using that we get isomorphism of vector spaces
 \begin{center}
 	$HH^\bullet(\mathcal{U}(\mathcal{O}_{X}(U_x), \mathcal{L}(U_x))) \cong \wedge_{\mathcal{O}_{X}(U_x)}^\bullet\mathcal{L}(U_x)$.
 \end{center}
  \end{proof}
 \begin{Note}
 In the sheaf theoretic settings we consider the presheaf of cobar resolutions 
 $\{U \mapsto Cob^\bullet (\mathcal{U}(\mathcal{O}_X(U), \mathcal{L}(U)))$ with canonical restrictions $\}$ provide cobar resolution for $\mathscr{U}(\mathcal{O}_X, \mathcal{L})$, denote it by $\mathscr{C}ob^\bullet(\mathscr{U}(\mathcal{O}_X, \mathcal{L}))$ and the associated hypercohomology is called Hochschild hypercohomology of $\mathscr{U}(\mathcal{O}_X, \mathcal{L})$, denoted as $ \mathbb{H}H^\bullet(\mathscr{U}(\mathcal{O}_X, \mathcal{L}))$. Similar notions applicable for the jet algebroid $\mathscr{J}(\mathcal{O}_X, \mathcal{L}).$
 
 On each open set $U$ of $X$, we can define the anti-symmetrization map 
 $$ \widetilde{Alt}_U: \wedge^\bullet_{\mathcal{O}_{X}(U)}\mathcal{L}(U) \rightarrow (\mathcal{U}(\mathcal{O}_X(U),\mathcal{L}(U)))^{\otimes^\bullet}$$
 as the earlier case and we can check it is compatible with restrictions and differentials. Applying sheafification we get homomorphism between associated cochain complex of $\mathcal{O}_X$-modules, we call it anti-symmetrization map (it depends on context). 
 \end{Note} 
 
 \begin{Lem} \label{hyper & sheaf}
 If $\mathcal{F}$ is a sheaf of $\mathcal{O}_X$-modules, then by considering the exterior algebra as a chain complex $\wedge^\bullet _{\mathcal{O}_X} \mathcal{F}:=(\oplus_i \wedge^i _{\mathcal{O}_X} \mathcal{F},0)$ we get a canonical isomorphism of $\mathbb{K}$-vector spaces
 $$\mathbb{H}^k(X, \wedge^\bullet _{\mathcal{O}_X} \mathcal{F}) \cong \underset{i+j=k}{\oplus} H^j(X, \wedge^i _{\mathcal{O}_X} \mathcal{F})$$ for every $k \geq 0$ (or $k$ is any non negative integer).
 \end{Lem}
\begin{proof} Here we apply the notion of hypercohomology from \cite{VD} (replacing usual notations by curly notations) in the special complex of sheaves.
	For each sheaf of ${\mathcal{O}_X}$-modules (or $\mathcal{O}_X$-module) $\wedge^i _{\mathcal{O}_X} \mathcal{F}$, we have a flabby Godement resolution (a quasi-isomorphism)
	$$\wedge^i _{\mathcal{O}_X} \mathcal{F} \overset{\sim}{\rightarrow} \mathscr{C}^\cdot(\wedge^i _{\mathcal{O}_X} \mathcal{F}).$$
	So, the sheaf cohomology $H^\bullet(X,\wedge^i _{\mathcal{O}_X} \mathcal{F})$ of $\wedge^i _{\mathcal{O}_X} \mathcal{F}$ ($i \in \mathbb{N})$ is isomorphic to the cohomology (in usual sense) of the complex of $\mathcal{O}_X(X)$-modules $\mathscr{C}^\bullet(\wedge^i _{\mathcal{O}_X} \mathcal{F})(X)$.
	

For the complex of sheaves $\wedge^\bullet _{\mathcal{O}_X} \mathcal{F}$, we consider the corresponding bicomplex of sheaves $\mathscr{C}^\bullet(\mathcal{F}):=(\mathscr{C}^j(\wedge^i _{\mathcal{O}_X} \mathcal{F}))$ $(i,j \geq 0)$. The original complex  can be embedded in the total complex $\mathscr{K}^\bullet:=tot(\mathscr{C}^\bullet(\mathcal{F}))$. Moreover, this embedding is a quasi-isomorphism. The cohomology of the associated complex of global sections $\mathscr{K}^\bullet(X)=tot(\mathscr{C}^\bullet(\mathcal{F}))(X)$ is the hypercohomology of $\wedge^\bullet _{\mathcal{O}_X} \mathcal{F}$. Since here differential of the complex is zero, so here computation of cohomology of the total complex is become simpler one. It is directly expressed through the sheaf cohomologies $H^j(X, \wedge^i _{\mathcal{O}_X} \mathcal{F})$ as stated.
	
\end{proof}

 \textbf{Notations :} We denote cotensor product with $\mathcal{O}_X(U)$ in the category of left $S_{\mathcal{O}_X(U)}\mathcal{L}(U)$-comodules as  $\mathcal{O}_X(U)\Box_{S_{\mathcal{O}_X(U)}\mathcal{L}(U)} -$ and the associated cohomology groups is given by the Cotor groups (functor) as $Cotor^\bullet_{S_{\mathcal{O}_X(U)}\mathcal{L}(U)} (\mathcal{O}_X(U),-)$, for an open set $U$ of $X$. Instead of $S_{\mathcal{O}_X(U)}\mathcal{L}(U)$ we use $\mathcal{J}(\mathcal{O}_X(U), \mathcal{L}(U))$ when it requires. We use all the above results in the next theorems.
 

 \begin{Thm} (Generalized HKR theorem) Let $\mathcal{L}$ be a locally free  Lie algebroid over $(X, \mathcal{O}_X)$. Then the anti-symmetrization map 
 	\begin{center}
 		$\wedge^\bullet_{\mathcal{O}_X} \mathcal{L} \rightarrow \mathscr{U}(\mathcal{O}_X, \mathcal{L})^{\otimes \bullet}$
 	\end{center}
 	is a quasi-isomorphism of sheaves, known as of HKR morphism. It induces an isomorphism of graded vector spaces 
 	\begin{center}
 		$ \mathbb{H}H^\bullet(\mathscr{U}(\mathcal{O}_X, \mathcal{L})):= \mathbb{H}^\bullet(X, \mathcal{D}^{poly}_{\mathcal{L}}(\mathcal{O}_X)) \cong \mathbb{H}^\bullet(X, \wedge^\bullet_{\mathcal{O}_X}{\mathcal{L}}))$
 	\end{center}
  \end{Thm}

 \begin{proof} Assume first $\mathcal{L}$ to be a locally free Lie algebroid of finite rank. Since for each $x \in X$ we have an open set $U_x$ containing $x$ with $\mathcal{L}|_{U_x}$ is a free $\mathcal{O}_X|_{U_x}$-module, so on each $U_x$ we use results from the proof of the HKR Theorem (\cite{KP}) for $(\mathbb{K}, \mathcal{O}_X(U_x))$- Lie-Rinehart algebra $\mathcal{L}(U_x)$.
 	Then, for each special open sets $U_x$ we have isomorphisms of cochain complexes of left $ S_{\mathcal{O}_X(U_x)}\mathcal{L}(U_x)$- comodules
 	\begin{center}
 		$ (\mathcal{O}_X (U_x)\Box_{S_{\mathcal{O}_X(U_x)}\mathcal{L}(U_x)} \tilde{K}^\bullet_{\mathcal{O}_X(U_x)}\mathcal{L}(U_x), id_{\mathcal{O}_X(U_x)} \Box \partial_{U_x}) \overset{\phi_{U_x}}{\rightarrow} (\wedge^\bullet_{\mathcal{O}_X(U_x)} \mathcal{L}(U_x), 0)$
 	\end{center}
 	compatible with restrictions, where $\tilde{K}^\bullet_{\mathcal{O}_X(U_x)}\mathcal{L}(U_x) = S_{\mathcal{O}_X(U_x)}\mathcal{L}(U_x)\otimes \wedge^\bullet_{\mathcal{O}_X(U_x)}\mathcal{L}(U_x)$ and $\partial_{U}$ is given in \ref{del}.
 	
 	Now, an open set $U$ is expressed by partition as $\cup_{x \in X}(U \cap U_x)$ and we have cochain homomorphisms of left $S_{\mathcal{O}_X(U)}\mathcal{L}(U)$- comodules
 	\begin{center}
 		$ (\mathcal{O}_X (U)\Box_{S_{\mathcal{O}_X(U)}\mathcal{L}(U)} \tilde{K}^\bullet_{\mathcal{O}_X(U)}\mathcal{L}(U), id_{\mathcal{O}_X(U)} \Box \partial_{U}) \overset{\phi_{U}}{\rightarrow} (\wedge^\bullet_{\mathcal{O}_X(U)} \mathcal{L}(U), 0)$
 	\end{center}
 	compatible with restrictions. Then it induces homomorphism on the associated presheaves which become stalkwise isomorphism. So, by applying sheafification functor we get isomorphism of cochain complexes left $\mathcal{S}_{\mathcal{O}_X}\mathcal{L}$- comodules $$ (\mathcal{O}_X \Box_{\mathcal{S}_{\mathcal{O}_X}\mathcal{L}} \tilde{K}^\bullet_{\mathcal{O}_X}\mathcal{L}, id_{\mathcal{O}_X} \Box \partial) \overset{\phi}{\rightarrow} (\wedge^\bullet_{\mathcal{O}_X} \mathcal{L}, 0).$$ Applying hypercohomology functor $\mathbb{H}^\bullet(X,-)$ we get
 	\begin{center}
 		$Cotor^\bullet_{\mathcal{S}_{\mathcal{O}_X}\mathcal{L}}(\mathcal{O}_X, \mathcal{O}_X) \cong \mathbb{H}^\bullet(X, \wedge^\bullet_{\mathcal{O}_X} \mathcal{L})$.
 	\end{center}
 	To compute $\mathbb{H}H^\bullet(\mathcal{S}_{\mathcal{O}_X} \mathcal{L})$, we consider the standard cobar resolution associated with each open set $U$, denoted as $(Cob^\bullet(S_{\mathcal{O}_X(U)} \mathcal{L}(U)), \beta_U)$ and defined by
 	\begin{center}
 		$0 \rightarrow S_{\mathcal{O}_X(U)} \mathcal{L}(U))\overset{\Delta_U}{\longrightarrow} (S_{\mathcal{O}_X(U)} \mathcal{L}(U)))^{\otimes_{\mathcal{O}_X(U)}^2}\overset{\Delta_U \otimes id}{\longrightarrow}(S_{\mathcal{O}_X(U)} \mathcal{L}(U)))^{\otimes_{\mathcal{O}_X(U)}^3}\overset{\Delta_U \otimes id}{\longrightarrow} ...$,
 	\end{center}
 where $\Delta_U$ is the comultiplication map for the $\mathcal{O}_X(U)/\mathbb{K}$-bialgebra $S_{\mathcal{O}_X(U)} \mathcal{L}(U))$ (\cite{AA},\cite{KP,DRV}) and $id$ is the identity map on $S_{\mathcal{O}_X(U)} \mathcal{L}(U))$.
 
 
 	We have two presheaf of vector spaces, one is presheaf of cobar resolutions of the sheaf $\mathcal{S}_{\mathcal{O}_X} \mathcal{L}$ of $\mathcal{O}_X/{\mathbb{K}_X}$-bialgebra, defined as
 		$$U \mapsto Cob^\bullet(S_{\mathcal{O}_X(U)} \mathcal{L}(U))$$
 	with its restriction morphism induced from the restrictions of the sheaf $\mathcal{S}_{\mathcal{O}_X}\mathcal{L}$, and other one is considering cotensor product with $\mathcal{O}_X$ to the presheaf of Koszul-Rinehart resolution of $\mathcal{O}_X$ in the category of left $S_{\mathcal{O}_X}\mathcal{L}$-comodules, defined as
 	$$U\mapsto \widetilde{K}_{\mathcal{O}_X(U)}^\bullet\mathcal{L}(U).$$ 
 	These two presheaves are quasi isomorphic on affine pieces $U_x$ for every $x \in X$ (for $C^\infty$ case these holds for every open set $U$). 
	In stalkwise, it induces quasi-isomorphism and by applying sheafification functor we get quasi-isomorphism (chain homotopy) 
 	\begin{center}
 		$\mathscr{C}ob^\bullet(\mathcal{S}_{\mathcal{O}_X} \mathcal{L}) \overset{\sim}{\rightarrow} \widetilde{\mathscr{K}}_{\mathcal{O}_X}^\bullet\mathcal{L}$
 	\end{center}
 (using the alternating map and projection map \ref{Alt}, we have quasi isomorphism $$S_{\mathcal{O}_X(U)}\mathcal{L}(U) \otimes_{\mathcal{O}_X(U)} \wedge^n_{\mathcal{O}_X(U)} (\mathcal{L}(U)) \overset{(id \otimes Alt)_{U}}{\longrightarrow} (S(\mathcal{O}_X(U),\mathcal{L}(U)))^{\otimes (n+1)})$$
 	Then on taking cotensor product by $\mathcal{O}_X$ from left (as standard left $\mathcal{S}_{\mathcal{O}_X}\mathcal{L}$-comodule) to the above cochain complexes of sheaves of left $\mathcal{S}_{\mathcal{O}_X}\mathcal{L}$-comodules we get
 	\begin{center}
 		$( \mathcal{O}_X \Box_{\mathcal{S}_{\mathcal{O}_X} \mathcal{L}} \mathscr{C}ob^\bullet(\mathcal{S}_{\mathcal{O}_X} \mathcal{L}), d) \overset{\sim}{\rightarrow} (\wedge^\bullet_{\mathcal{O}_X}\mathcal{L}, 0)$.
 	\end{center}
 	Applying hypercohomology functor $\mathbb{H}^\bullet(X,-)$ we get Hochschild hypercohomology of $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ through derived functor Cotor as
 	\begin{center}
 		$\mathbb{H}H^\bullet(\mathscr{U}(\mathcal{O}_X,\mathcal{L}))\overset{\cong}{\rightarrow} \mathbb{H}^\bullet(X, \wedge^\bullet_{\mathcal{O}_X}\mathcal{L})$.
 	\end{center}
 On the other hand, we have two presheaves of vector spaces. First one is presheaf of cobar resolutions of the structure sheaf $\mathcal{O}_X$ cotensoring with $\mathcal{O}_X$ as left  $\mathcal{S}_{\mathcal{O}_X} \mathcal{L}$-comodule ($ \mathcal{S}_{\mathcal{O}_X} \mathcal{L}$ is a $\mathcal{O}_X/{\mathbb{K}_X}$-bialgebra) i.e., for every open set $U$ in $X$ we have
 	\begin{center}
 		$U \mapsto \mathcal{O}_X(U)\Box_{S_{\mathcal{O}_X(U)} \mathcal{L}(U)}Cob^\bullet(S_{\mathcal{O}_X(U)} \mathcal{L}(U))$
 	\end{center}
 	with its restriction morphism induced from the restrictions of the sheaf $
 	\mathcal{S}_{\mathcal{O}_X}\mathcal{L}$ (using restiction of tensor products of sheaves). The second presheaf is the presheaf of Koszul-Rinehart resolution of $\mathcal{O}_X$ in the category of left $S_{\mathcal{O}_X}\mathcal{L}$-comodules  i.e., for every open set $U$ in $X$ we have
 	\begin{center}
 		$U\mapsto \mathcal{O}_X(U)\Box_{S_{\mathcal{O}_X(U)} \mathcal{L}(U)}\widetilde{K}_{\mathcal{O}_X(U)}^\bullet\mathcal{L}(U)$
 	\end{center}
 which are compatible with restriction morphisms.
 	These two presheaves are quasi isomorphic on each special open sets $U_x$ for every $x \in X$. Since we have
 	$$ Alt_{U_x}: \wedge_{\mathcal{O}_{X}(U_x)}^\bullet\mathcal{L}(U_x) \rightarrow Cob^\bullet(\mathcal{U}(\mathcal{O}_{X}(U_x), \mathcal{L}(U_x))) \cong Cob^\bullet(S_{\mathcal{O}_{X}(U_x)}\mathcal{L}(U_x),$$
  by considering their cohomologies we get, $$HH^\bullet(S_{\mathcal{O}_{X}(U_x)}\mathcal{L}(U_x))\cong HH^\bullet(\mathcal{U}(\mathcal{O}_{X}(U_x), \mathcal{L}(U_x)))\cong \wedge_{\mathcal{O}_{X}(U_x)}^\bullet\mathcal{L}(U_x).$$ 
 	
 	Let $\phi_{U_x}$ be the morphisms in the cochain level on each special open sets. Then these morphisms $\phi_{U_x}$ are quasi isomorphism, for each $x \in X$. We can canonically construct (sheafifying) a morphism $\phi$ between the associated cochain complexes such that $\phi|_{U_x}= \phi_{U_x}$ and induces stalkwise quasi isomorphism in the cochain complex of associated stalks. So the corresponding presheaves of vector spaces
 	\begin{center}
 		$U \mapsto HH^\bullet(S_{\mathcal{O}_{X}(U)}\mathcal{L}(U))$, 
 		
 		
 		$U \mapsto \wedge_{\mathcal{O}_{X}(U)}^\bullet\mathcal{L}(U)$
 	\end{center}
 	are isomorphic on special open sets by the morphism $HH^\bullet(\phi)$.
 	From this we get isomorphisms $HH_x^\bullet(\phi)$ in the cohomology of the cochain complex of associated stalks for each $x \in X$.
 	So, by considering sheafification (of these two presheaf of cohomology spaces which are isomorphic stalkwise) we get isomorphism in the associated cohomology sheaves as
 		$$\mathscr{H}H^\bullet(\mathscr{U}(\mathcal{O}_{X}, \mathcal{L}))\cong \mathscr{H}H^\bullet(\mathcal{S}_{\mathcal{O}_X}\mathcal{L}) \cong \wedge_{\mathcal{O}_X}^\bullet\mathcal{L}.$$
 Here the second isomorphism above is given by the map $\mathscr{H}H^\bullet(\phi)$.
 	
 Next on the induced hypercohomology groups we have
 	\begin{center}
 		$\mathbb{H}H^\bullet(\mathscr{U}(\mathcal{O}_{X}, \mathcal{L}))\cong \mathbb{H}H^\bullet(\mathcal{S}_{\mathcal{O}_X}\mathcal{L}) \cong \mathbb{H}^\bullet (X, \wedge_{\mathcal{O}_X}^\bullet\mathcal{L})$.
 	\end{center}
 	In general case, where $\mathcal{L}$ is locally free over $\mathcal{O}_X$ but not coherent, there exist a filtered ordered set $J$ as well as an inductive system of finitely generated projective (or, even free) $\mathcal{O}_X(U_x)$ - modules $\{{\mathcal{L}(U_x)}_j | j \in J \}$ such that $$\mathcal{L}(U_x) \cong \varinjlim_{ j \in J} {\mathcal{L}(U_x)}_j,$$ for each special open set $U_x$ (using results from \cite{KP}). Since both $\mathbb{H}H$ (which is the derived functor cotor here) as well as $\mathcal{S}$ commute with inductive limits over a filtered ordered set, we get required result using the result from locally finitely generated projective case (\cite{KP}).
 \end{proof}
  \begin{Rem}
This result is proved for a locally free Lie algebroid of finite rank over a smooth algebraic variety (or smooth scheme) in \cite{CV} with the associated isomorphism is in the category of Gerstenhaber algebras.
\end{Rem}
 \begin{Cor}
In the special case of $\mathcal{L} = \mathcal{T}_X $ when $X$ is non-singular(or smooth),
 	we get isomorphism of graded vector spaces 
 	\begin{center}
 		$ \mathbb{H}H^\bullet(\mathscr{U}(\mathcal{O}_X,\mathcal{T}_X))= \mathbb{H}^\bullet(X, \mathcal{D}^{poly,\bullet}_X) \cong \mathbb{H}^\bullet(X, \mathcal{T}^{poly,\bullet}_X)$.
 	\end{center}

\end{Cor}	
\begin{Cor} By applying the Lemma \ref{hyper & sheaf}, the above HKR theorem reduces to
		$$ \mathbb{H}H^\bullet(\mathscr{U}(\mathcal{O}_X, \mathcal{L}))\cong \underset{i,j}{\oplus} H^j(X, \wedge^i _{\mathcal{O}_X} \mathcal{L}).$$
\end{Cor}
\begin{Note}
The jet algebroid $\mathscr{J}(\mathcal{O}_X,\mathcal{L})$ of a Lie algebroid $\mathcal{L}$ over a $a$- space $(X,\mathcal{O}_X)$ is the sheafification of the presheaf $\{U \mapsto \mathcal{J}(\mathcal{O}_X(U), \mathcal{L}(U))\}$. It has a canonical  $\mathcal{O}_X/{\mathbb{K}_X}$-bialgebra structure induces from $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ (\cite{AA}).
\end{Note}
 \begin{Thm} (Generalized dual HKR theorem) Let $\mathcal{L}$ locally free Lie algebroid over $(X, \mathcal{O}_X)$ which is of finite rank. Then the anti-symmetrization map 
 	\begin{center}
 		$\wedge^\bullet_{\mathcal{O}_X} \mathcal{L^*} \rightarrow \mathscr{J}(\mathcal{O}_X, \mathcal{L})^{\otimes \bullet}$
 	\end{center}
 	is a quasi-isomorphism of sheaves, known as dual of HKR morphism. It induces in particular isomorphism of graded vector spaces 
 	\begin{center}
 		$ \mathbb{H}H^\bullet(\mathscr{J}(\mathcal{O}_X,\mathcal{L})) \cong \mathbb{H}^\bullet(X, \Omega^\bullet_{\mathcal{L}})) .$
 	\end{center}
 
 \end{Thm}
 
 \begin{proof} 
 	For a locally free Lie algebroid $\mathcal{L}$ over $(X, \mathcal{O}_X)$ of rank $r$ (for some $r \in \mathbb{N}$), the dual of $\mathcal{L}$ (denoted by $\mathcal{L^*})$ is a locally free $\mathcal{O}_X$- module of rank $r$ i.e., $\mathcal{L^*}(U_x)$ has a basis $\{w_1, \cdots, w_r\}$ (say) for some special open set $U_x$ arond $x \in X$. Since $\mathscr{J}(\mathcal{O}_X,\mathcal{L})$ is a commutative $\mathcal{O}_X$- algebra, so $\mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x))
 	\cong \mathcal{O}_X (U_x)[[w_1, \cdots, w_r]]$ as $\mathcal{O}_X(U_x)$-algebra. Thus by applying sheafification functor we get $\mathcal{O}_X$-algebras as $\mathscr{J}(\mathcal{O}_X,\mathcal{L}) \cong \widehat{\mathcal{S}_{\mathcal{O}_X}\mathcal{L^*}}$ (this sheaf of symmetric algebras is formally completed with respect to the degree) (\cite{CV,AP}). Now, consider the Koszul-Rinehart resolution of $\mathcal{O}_X$ in the category of left $\mathscr{J}(\mathcal{O}_X,\mathcal{L})$- comodules is
 	\begin{center}
 		$\mathscr{K}^\bullet_{\mathcal{O}_X}\mathcal{L}^* :  \mathcal{O}_X   \hookrightarrow (\mathscr{J}(\mathcal{O}_X,\mathcal{L}) \otimes_{\mathcal{O}_X} \wedge^\bullet_{\mathcal{O}_X}\mathcal{L^*}, \nabla) = \bar{\mathscr{K}}^\bullet_{\mathcal{O}_X}\mathcal{L}^* $
 	\end{center}
  where $\nabla$ is given by the continuation of the  Grothendieck connections (canonical left $\mathcal{L}(U)$-connection on $\mathcal{J}(\mathcal{O}_X(U),\mathcal{L}(U))$, see \cite{KP}) associated with each open set $U$ in $X$. So, it is basically the sheafification of the presheaf
 $$U \mapsto \{\mathcal{O}_X(U) \hookrightarrow \bar{K}^\cdot_{\mathcal{O}_X(U)} (\mathcal{L}(U))^* \}$$
 	with natural restrictions. The unit of $\mathscr{J}(\mathcal{O}_X,\mathcal{L})$ as $\mathcal{O}_X$-algebra gives the morphism from $\mathcal{O}_X$ to $\bar{\mathscr{K}}^\bullet_{\mathcal{O}_X}\mathcal{L}^*$.
 	
 	Applying cotensor product with $\mathcal{O}_X$ (with canonical $\mathscr{J}(\mathcal{O}_X,\mathcal{L})$- comodule structure, which induces from the $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$- module structure on $\mathcal{O}_X$) to the Koszul-Rinehart complex of $\mathcal{L}^*$, we get isomorphism of sheaf of cochain complexes of  $\mathscr{J}(\mathcal{O}_X,\mathcal{L})$- comodules as
 	\begin{center}
 		$ (\mathcal{O}_X \Box_{\mathscr{J}(\mathcal{O}_X,\mathcal{L})} \tilde{K}^\bullet_{\mathcal{O}_X}\mathcal{L}^*, id_{\mathcal{O}_X} \Box \nabla) \cong (\wedge^\bullet_{\mathcal{O}_X} \mathcal{L^*}, d_{\mathcal{L}})$.
 	\end{center}
 	
 	Since $\mathcal{L}$ is locally free $\mathcal{O}_X$-module of finite rank, so there exists open sets $U_x$ around each point $x\in X$ where $\mathcal{L}|_{U_x}$ is free $\mathcal{O}_X|_{U_x}$-module of finite rank and  so for each $U_x$ we can use results from the proof of the dual HKR Theorem (\cite{KP}) for the $(\mathbb{K}, \mathcal{O}_X(U_x))$- Lie-Rinehart algebra $\mathcal{L}(U_x)$. Thus we get
    $$\wedge^\bullet_{\mathcal{O}_X(U_x)} (\mathcal{L}(U_x))^* \cong (\wedge^\bullet_{\mathcal{O}_X(U_x)} \mathcal{L}(U_x))^*.$$
 So, here $\mathscr{H}om_{\mathcal{O}_X}(\wedge^\bullet_{\mathcal{O}_X} \mathcal{L}, \mathcal{O}_X)\cong \wedge^\bullet_{\mathcal{O}_X} \mathcal{L^*}$ and the Chevally-Eilenberg-de Rham complex of $\mathcal{L}$ is $ \Omega^\bullet_{\mathcal{L}} := (\wedge^\bullet_{\mathcal{O}_X} \mathcal{L^*}, d_{\mathcal{L}})$.
 	Also, for each open set $U$ of $X$ we have
 		$$ \mathcal{O}_X (U)\Box_{\mathcal{J}(\mathcal{O}_X(U),\mathcal{L}(U))} \tilde{K}^\bullet_{\mathcal{O}_X(U)}(\mathcal{L}(U))^* \cong\wedge^\bullet_{\mathcal{O}_X(U)} (\mathcal{L}(U))^*$$
 	and since on the open set $U$ the unit $1_U \in \mathcal{J}(\mathcal{O}_X(U),\mathcal{L}(U))$ is given by the left counit $\epsilon_U : \mathcal{U}(\mathcal{O}_X(U),\mathcal{L}(U))
 	\rightarrow \mathcal{O}_X(U)$, so on $U$'s the induced differential is exactly the Lie-Rinehart coboundary $d_{\mathcal{L}(U)}$.
 	So, for each special open set $U_x$ we have isomorphisms of cochain complexes of left $\mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x))$- comodules
 	\begin{center}
 	$ (\mathcal{O}_X (U_x)\Box_{\mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x))} \tilde{K}^\bullet_{\mathcal{O}_X(U_x)}(\mathcal{L}(U_x))^*, id_{\mathcal{O}_X(U_x)} \Box \nabla_{U_x}) \overset{\phi_{U_x}}{\longrightarrow} (\wedge^\bullet_{\mathcal{O}_X(U_x)} (\mathcal{L}(U_x))^*, d_{\mathcal{L}(U_x)})$
 	\end{center}
 	which are compatible with restrictions.
 	
 	Now consider the canonical presheaves from the above complexes associated with each open sets, which are isomorphic on each special open sets and compatible with the natural restrictions. Then considering there sheafifications and using the above isomorphisms, we get isomorphism of corresponding complexes of sheaves.
 	
 	Applying hypercohomology functor $\mathbb{H}^\bullet(X,-)$ we get
 	\begin{center}
 		$Cotor^\bullet_{\mathscr{J}_{\mathcal{O}_X}\mathcal{L}}(\mathcal{O}_X, \mathcal{O}_X) = \mathbb{H}^\bullet(X, \Omega^\bullet_{\mathcal{L}})$.
 	\end{center}
 	
 	To get Hochschild hypercohomology of $\mathscr{J}(\mathcal{O}_X, \mathcal{L})$ we need to consider standard cobar resolution of $\mathcal{O}_X$ as $\mathscr{J}(\mathcal{O}_X, \mathcal{L})$ comodules and cotensor it with $\mathcal{O}_X$. 
 	
 	We know that for an special open cover $\{U_x| x \in X \}$ of $X$, we have finitely generated projective (in fact free) $(\mathbb{K}, \mathcal{O}_X(U_i))$- Lie-Rinehart algebras $\mathcal{L}(U_x)$ for which we get quasi-isomorphisms  
 	$$ (\mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x))))^{\otimes (n+1)} \overset{(id \otimes P)_{U_i}}{\longrightarrow} \mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x)) \otimes_{\mathcal{O}_X(U_x)} \wedge^n_{\mathcal{O}_X(U_x)} (\mathcal{L}(U_x)^*),$$
 	$$\mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x)) \otimes_{\mathcal{O}_X(U_x)} \wedge^n_{\mathcal{O}_X(U_x)} (\mathcal{L}(U_x)^*) \overset{(id \otimes Alt)_{U_x}}{\longrightarrow} (\mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x)))^{\otimes (n+1)}$$
 	for each $U_x$ associated with $x \in X$, where $P_{U_x} $ is given by the cannonical projections
 	$pr_1 : \mathcal{J}(\mathcal{O}_X(U_x),\mathcal{L}(U_x)) \cong S_{\mathcal{O}_X (U_x)}(\mathcal{L}(U_x))^* \rightarrow (\mathcal{L}(U_x))^*$  and the anti-symmetrization map
 	$Alt_{U_x} : \wedge^n_{\mathcal{O}_X(U_x)} (\mathcal{L}(U_x)^*) \rightarrow (S_{\mathcal{O}_X (U_x)}(\mathcal{L}(U_x))^*)^{\otimes n} $ for every special open set $U_x$ of $ X$ (for smooth manifold we get these quasi isomorphisms for each open sets).
 	Sheafification of the corresponding presheaves gives the quasi-isomorphism between the cobar resolution $Cob^\bullet(\mathscr{J}_{\mathcal{O}_X} \mathcal{L})$ and the Koszul-Rinehart resolution $\mathscr{K}^\bullet_{\mathcal{O}_X}\mathcal{L}^*$.
 		Applying hypercohomology functor $\mathbb{H}^\bullet(X,-)$ we get 
 	\begin{center}
 		$\mathbb{H}H^\bullet(\mathscr{J}(\mathcal{O}_X,\mathcal{L})) = Cotor^\bullet_{\mathscr{J}_{\mathcal{O}_X}\mathcal{L}}(\mathcal{O}_X, \mathcal{O}_X)$.
 	\end{center}
 	So, the Hochschild hypercohomology groups of jet algebroid is expressed by the Chevally-Eilenberg-de Rham hypercohomology (\cite{UB,BRT}) of $\mathcal{L}$.
 \end{proof}
 \begin{Cor}
 	In the special case when $\mathcal{L} = \mathcal{T}_X $ and $X$ is non-singular (smooth manifold, complex manifold, smooth algebraic variety or smooth scheme over the field $\mathbb{C})$,
 	we get isomorphism of graded vector spaces 
 	\begin{center}
 		$ \mathbb{H}H^\bullet(\mathscr{J}(\mathcal{O}_X,\mathcal{T}_X)) \cong \mathbb{H}^\bullet(X, \Omega^\bullet_X) \cong H^\bullet (X; \mathbb{K})$ or $H^\bullet(X^{an}; \mathbb{K})$ 
 	\end{center}
 by applying de Rham theorems in different settings (smooth, analytic, algebraic) \cite{MS, VD},
 where $ \mathbb{K = R}$ or $ \mathbb{C}$ accordingly and $X^{an}$ is the analytification of the algebraic variety (scheme) $X$.
\end{Cor}






 \textbf{Acknowledgment.} I am thankful to my thesis supervisor Dr. Ashis Mandal for his immense support and valuable suggestions.
 
 