Natural map from vector fields to cotangent variety

Question

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 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.

Answer 1

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).