Let $X$ be a smooth variety, and let $\operatorname{Vec} (X)$ denote the $\mathcal{O}\_X$-module of vector fields on $X$. It is stated in several books on D-modules, for example [here](http://people.math.harvard.edu/%7Egaitsgde/grad_2009/Dmod_brav.pdf) in Corollary 6.6, that there should me an evident morphism
$$ \operatorname{Vec}(X) \rightarrow \mathcal{O}\_{T^\*X} .$$
Apparently, a pushforward along $\pi: T^\*X \rightarrow X$ is implicit on the right hand side. Moreover, this map even induces an isomorphism
$$ \bigoplus\_n \operatorname{Sym}^n(\operatorname{Vec}(X)) \cong \mathcal{O}\_{T^\*X} ,$$
but one step at a time. I couldn't really understand how the first map is even defined, and neither could I find any more indication on this, so I would be grateful if someone might explain this and maybe say a few words on why the second map is an isomorphism.



Natural map from vector fields to cotangent variety
Let $X:=Spec(A)$ be an affine scheme of finite type over a field $k$ of characteristic zero, and let $D(A):=Diff\_k(A)$ be the ring of $k$-linear polynomial differential operators on $A$. It has a filtration $D^l(A):=Diff^l\_k(A) \subseteq Diff\_k(A)$ respecting the multiplication, and $k$ is in the center of $D(A)$. The ring $A$ is not in the center of $D(A)$. Let $Der\_k(A)$ be the left $A$-module of derivations of $A$.


Example: If $A:=k[x\_1,..,x\_n]$ is the polynomial ring on $n$ variables and $\partial\_i:=\frac{\partial}{\partial\_{x\_i}}$ is the partial derivative wrto the $x\_i$-variable, it follows $Der\_k(A)\cong A\{\partial\_1,..,\partial\_n\}$ is a free $A$-module on the elements $\partial\_i$ of rank $n$.
Since the ring $D(A)$ is almost commutative, it follows the associative graded ring $gr(D(A))$ is commutative and there is a canonical isomorphism


PBW: $gr(D(A)) \cong A[\partial\_1,..,\partial\_n]\cong Sym\_A(Der\_k(A))$


of graded $A$-algebras. This is called the "PBW Theorem". I'm uncertain about your notation but, usually one writes $T^\*X$ (or $T^\*(A)$) for the cotangent bundle/module $\Omega^1\_{A/k}$, and this is the dual of the tangent bundle/module $Der\_k(A)$. It may be you should replace $T^\*X$ by $T(X)$ - the tangent bundle of $X$.


The PBW Theorem holds for a large class of such rings. In general if $\alpha: L \rightarrow Der\_k(A)$ is a Lie-Rinehart algebra/Lie algebroid ($L$ is a $k$-Lie algebra and $A$-module satisfying a set of conditions) which is locally trivial as a left $A$-module, it follows there is a "universal enveloping algebra" $U(A,L)$ and a canonical isomorphism


F1. $gr(U(A,L))\cong Sym\_A(L)$


of graded $A$-algebras. The associative ring $U(A,L)$ has a canoncial filtration similar to the enveloping algebra of a Lie algebra. There is a canonical map


F2 $U(A,Der\_k(A)) \rightarrow D(A)$


from the universal enveloping algebra to the ring of differential operators, and the ring $U(A,Der\_k(A))$ is sometimes called the ring of "crystalline differential operators" of $A$. This construction exsist in general for any extension of commutative rings $A \subseteq B$.


Example. If $k=A=B$ is a field and $L$ is a $k$-Lie algebra it, follows $U(k,L)\cong U(L)$ is the classical universal enveloping algebra of the Lie algebra $L$. Hence the construction of $U(A,L)$ gives a simultaneous generalization of the ring of differential operators on an algebraic variety and the universal enveloping algebra of a Lie algebra.


The PBW theorem for Lie-Rinehart algebras/Lie algebroids is proved in a classical paper of Rinehart. In the paper Rinehart develops the theory of cohomology/homology of a flat connection using this theorem. For this reason many people refer to this theory as "Lie-Rinehart cohomology/homology".


Rinehart, G. S.
Differential forms on general commutative algebras. (English) Zbl 0113.26204
Trans. Am. Math. Soc. 108, 195-222 (1963).


