Editing Weyl algebra (section)

=== D-module ===
The Weyl algebra can be constructed as a [[D-module]].{{sfn | Coutinho | 1997 | pp=600–601}} Specifically, the Weyl algebra corresponding to the polynomial ring <math>R[x_1, ..., x_n]</math> with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations <math>D_{\mathbb{A}^n_R / R}</math>.{{sfn | Coutinho | 1997 | pp=600–601}}

More generally, let <math>X</math> be a smooth scheme over a ring <math>R</math>. Locally, <math>X \to R</math> factors as an étale cover over some <math>\mathbb{A}^n_R</math> equipped with the standard projection.<ref>{{Cite web |title=Section 41.13 (039P): Étale and smooth morphisms—The Stacks project |url=https://stacks.math.columbia.edu/tag/039P |access-date=2024-09-29 |website=stacks.math.columbia.edu}}</ref> Because "''étale''" means "(flat and) possessing null cotangent sheaf",<ref>{{Cite web |title=etale morphism of schemes in nLab |url=https://ncatlab.org/nlab/show/etale+morphism+of+schemes |access-date=2024-09-29 |website=ncatlab.org}}</ref> this means that every D-module over such a scheme can be thought of locally as a module over the <math>n^\text{th}</math> Weyl algebra.

Let <math>R</math> be a [[commutative algebra]] over a subring <math>S</math>. The '''ring of differential operators''' <math>D_{R/S}</math> (notated <math>D_R</math> when <math>S</math> is clear from context) is inductively defined as a graded subalgebra of <math>\operatorname{End}_{S}(R)</math>:

* <math>D^0_R=R</math>
* <math>
D^k_R=\left\{d \in \operatorname{End}_{S}(R):[d, a] \in D^{k-1}_R \text { for all } a \in R\right\} .
</math>

Let <math>D_R</math> be the union of all <math>D^k_R</math> for <math>k \geq 0</math>. This is a subalgebra of <math>\operatorname{End}_{S}(R)</math>.

In the case <math>R = S[x_1, ..., x_n]</math>, the ring of differential operators of order <math>\leq n</math> presents similarly as in the special case <math>S = \mathbb{C}</math> but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize <math>\mathbb{Z}[x_1, ..., x_n]</math>, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit <math>D_{\mathbb{A}^n_\mathbb{Z} / \mathbb{Z}}</math>. One such example is the operator <math>\partial_{x_1}^{[p]} : x_1^N \mapsto {N \choose p} x_1^{N-p}</math>.

Explicitly, a presentation is given by

:<math>D_{S[x_1, \dots, x_\ell]/S}^n = S \langle x_1, \dots, x_\ell, \{\partial_{x_i}, \partial_{x_i}^{[2]}, \dots, \partial_{x_i}^{[n]}\}_{1 \leq i \leq \ell} \rangle</math>
with the relations
:<math>[x_i, x_j] = [\partial_{x_i}^{[k]}, \partial_{x_j}^{[m]}] = 0</math>
:<math>[\partial_{x_i}^{[k]}, x_j] = \left \{ \begin{matrix}\partial_{x_i}^{[k-1]} & \text{if }i=j \\ 0 & \text{if } i \neq j\end{matrix}\right.</math>
:<math>\partial_{x_i}^{[k]} \partial_{x_i}^{[m]} = {k+m \choose k} \partial_{x_i}^{[k+m]} ~~~~~\text{when }k+m \leq n</math>
where <math>\partial_{x_i}^{[0]} = 1</math> by convention. The Weyl algebra then consists of the limit of these algebras as <math>n \to \infty</math>.<ref>{{Cite journal |last=Grothendieck |first=Alexander |date=1964 |title=Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Première partie |url=http://www.numdam.org/item/PMIHES_1964__20__5_0/ |journal=Publications Mathématiques de l'IHÉS |language=en |volume=20 |pages=5–259 |issn=1618-1913}}</ref>{{Pg|location=Ch. IV.16.II}}

When <math>S</math> is a field of characteristic 0, then <math>D^1_R</math> is generated, as an <math>R</math>-module, by 1 and the <math>S</math>-[[Derivation (differential algebra)|derivations]] of <math>R</math>. Moreover, <math>D_R</math> is generated as a ring by the <math>R</math>-subalgebra <math>D^1_R</math>. In particular, if <math>S = \mathbb{C}</math> and <math>R=\mathbb{C}[x_1, ..., x_n]</math>, then <math>D^1_R=R+ \sum_i R \partial_{x_i} </math>. As mentioned, <math>A_n = D_R</math>.{{sfn | Coutinho | 1995 | pp=20-24}}