Notes 07/08/2013

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 7 August 2013

$D\text{-modules}$

The Weyl algebra is the algebra $𝒟$ with generators $x_1,\dots ,x_n,$ $y_1,\dots ,y_n$ and relations

$$x_i{x}_j=x_j{x}_i,y_i{y}_j=y_j{y}_i\text{and}[y_j,x_i]={\delta }_{ij}\text{.}$$

i.e. $[x_i,x_j]=0,$ $[y_i,y_j]=0$ and $y_j{x}_i-x_i{y}_j={\delta }_{ij}\text{.}$

Often we write ${\partial }_j$ instead of $y_j$ since

$$\frac{\partial }{\partial x_j}{x}_{i}f-x_i\frac{\partial }{\partial x_j}f={\partial }_{ij}f+x_i\frac{\partial f}{\partial x_j}-x_i\frac{\partial }{\partial x_j}f={\delta }_{ij}f\text{.}$$

If ${𝒪}_X=ℂ[x_1,\dots ,x_n]$ then elements of ${𝒟}_X$ are

$$\sum _{a\in {\mathbb{Z}}_{\ge 0}^n}{c}_a{\left(\frac{\partial }{\partial x_1}\right)}^{a_1}\dots {\left(\frac{\partial }{\partial x_n}\right)}^{a_n},\text{with}\hspace{0.17em}{c}_a\in {𝒪}_X\text{.}$$

Filtrations

The Bernstein filtration is ${ℬ}_0\subseteq {ℬ}_1\subseteq \dots \subseteq 𝒟$ with

$${ℬ}_j=\text{span}\{x^a{\partial }^b\hspace{0.17em}\text{with}\hspace{0.17em}\left|a\right|+\left|b\right|\le j\}\text{.}$$

The standard filtration is ${\Sigma }_0\subseteq {\Sigma }_1\subseteq \dots \subseteq 𝒟$ with

$${\Sigma }_j={𝒪}_X\text{-span}\{{\partial }^b\hspace{0.17em}\text{with}\hspace{0.17em}\left|b\right|\le j\}\text{.}$$

Then

$$\text{gr}\hspace{0.17em}𝒟=\underset{j\in {\mathbb{Z}}_{\ge 0}}{⨁}\raisebox{1ex}{${ℱ}_j$}\!\left/ \!\raisebox{-1ex}{${ℱ}_{j-1}$}\right.=ℂ[x_1,\dots ,x_n,y_1,\dots ,y_n]\text{.}$$

The characteristic variety

Let $M$ be a $𝒟\text{-module.}$

A filtration of $M$ is $M_0\subseteq M_1\subseteq \dots \subseteq M$ with $\bigcup M_j=M$ and

(a)	${ℱ}_i{M}_j\subseteq M_{i+j}$
(b)	$M_j$ is a finitely generated ${ℱ}_0\text{-module.}$

Then

$$\text{gr}\hspace{0.17em}M\hspace{0.17em}\text{is a}\hspace{0.17em}\text{gr}\hspace{0.17em}𝒟\text{-module.}$$

A good filtration is a filtration of $M$ such that $\text{gr}\hspace{0.17em}M$ is a finitely generated $\text{gr}\hspace{0.17em}𝒟\text{-module.}$

This happens if and only if there exists a $j_0$ with ${ℱ}_i{M}_j=M_{i+j}$ for $j\ge j_0,i\ge 0\text{.}$

$$\begin{array}{c}{\text{Ann}}_{\text{gr}\hspace{0.17em}𝒟}\left(\text{gr}\hspace{0.17em}M\right)=\{p\in \text{gr}\hspace{0.17em}𝒟\hspace{0.17em}|\hspace{0.17em}pm=0\hspace{0.17em}\text{for all}\hspace{0.17em}m\in \text{gr}\hspace{0.17em}M\},\\ \sqrt{{\text{Ann}}_{\text{gr}\hspace{0.17em}𝒟}\left(\text{gr}\hspace{0.17em}M\right)}=\{p\in \text{gr}\hspace{0.17em}𝒟\hspace{0.17em}|\hspace{0.17em}{p}^n\in {\text{Ann}}_{\text{gr}\hspace{0.17em}𝒟}\left(\text{gr}\hspace{0.17em}M\right)\hspace{0.17em}\text{for some}\hspace{0.17em}n\in {\mathbb{Z}}_{>0}\}\text{.}\end{array}$$

The characteristic variety of $M$ is

$$\begin{array}{ccc}\text{ch}\hspace{0.17em}M& =& \{(v_1,\dots ,v_{2n})\in {ℂ}^{2n}\hspace{0.17em}|\hspace{0.17em}p(v_1,\dots ,v_{2n})=0\hspace{0.17em}\text{for all}\hspace{0.17em}p\in \sqrt{{\text{Ann}}_{\text{gr}\hspace{0.17em}𝒟}\left(\text{gr}\hspace{0.17em}M\right)}\}\\ & =& \text{zero set of}\hspace{0.17em}\sqrt{{\text{Ann}}_{\text{gr}\hspace{0.17em}𝒟}\left(\text{gr}\hspace{0.17em}M\right)}\text{.}\end{array}$$

The sheaf ${𝒟}_X$

Let $X$ be a space

	${𝒪}_X$ the sheaf of functions on $X$
	${𝒟}_X$ the sheaf of ${𝒪}_X\text{-coefficient}$ differential operators on $X\text{.}$

In local coordinates, sections of ${𝒟}_X$ are

$$\sum _{a\in {\mathbb{Z}}_{\ge 0}^n}{c}_a{\left(\frac{\partial }{\partial x_1}\right)}^{a_1}\dots {\left(\frac{\partial }{\partial x_n}\right)}^{a_n},\text{with}\hspace{0.17em}{c}_a\in {𝒪}_X\text{.}$$

A ${𝒟}_X\text{-module}$ $M$ is equivalent to a connection $\nabla :M\to M{\otimes }_{{𝒪}_X}{\Omega }^1\left(X\right)$ on $M$ via the formula

$$\nabla \left(m\right)=\sum _{i=1}^n\frac{\partial }{\partial x_i}m\otimes d{x}_i\text{.}$$

The center of ${𝒟}_X$ is

$$Z\left({𝒟}_X\right)={ℂ}_X,\hspace{0.17em}\text{the constant sheaf on}\hspace{0.17em}X\text{.}$$

The functors

$$\begin{array}{c}{\text{Hom}}_{{𝒟}_X}(-,{𝒪}_X):\text{Mod}\left({𝒟}_X\right)⟶\text{Mod}\left({𝒪}_X\right)\\ {\text{Hom}}_{{𝒟}_X}({𝒪}_X,-):\text{Mod}\left({𝒟}_X\right)⟶\text{Mod}\left({𝒪}_X\right)\end{array}$$

have derived functors

$$\begin{array}{c}ℛ{\text{Hom}}_{{𝒟}_X}(-,{𝒪}_X):{𝒟}^{+}\left({𝒟}_X\right)⟶{𝒟}^{+}\left({ℂ}_X\right)\\ ℛ{\text{Hom}}_{{𝒟}_X}({𝒪}_X,-):{𝒟}^{+}\left({𝒟}_X\right)⟶{𝒟}^{+}\left({ℂ}_X\right)\text{.}\end{array}$$

If $M$ is a ${𝒟}_X\text{-module}$ then

$$\begin{array}{c}\text{DR}\left(M\right)=ℛ{\text{Hom}}_{{𝒟}_X}({𝒪}_X,M)=(0\to M\stackrel{\nabla }{\to }M{\otimes }_{{𝒪}_X}{\Omega }^1\left(X\right)\stackrel{\nabla }{\to }M{\otimes }_{{𝒪}_X}{\Omega }^2\left(X\right)\to \dots )\\ \text{with}\nabla (m\otimes w)=\nabla m\wedge w-{(-1)}^{\text{deg}\hspace{0.17em}w}m\wedge dw,\end{array}$$

is the de Rham complex of $M\text{.}$

The complex of holomorphic solutions of $M$ is

$$\text{Sol}\left(M\right)=ℛ{\text{Hom}}_{{𝒟}_X}(M,{𝒪}_X)\text{.}$$

Notes and References

This is a typed copy of handwritten notes by Arun Ram.

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