Notes 07/08/2013

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

$xixj=xj xi, yiyj=yj yiand [yj,xi]= δij.$

i.e. $\left[{x}_{i},{x}_{j}\right]=0,$ $\left[{y}_{i},{y}_{j}\right]=0$ and ${y}_{j}{x}_{i}-{x}_{i}{y}_{j}={\delta }_{ij}\text{.}$

Often we write ${\partial }_{j}$ instead of ${y}_{j}$ since

$∂∂xjxif- xi∂∂xjf= ∂ijf+xi ∂f∂xj-xi ∂∂xjf= δijf.$

If ${𝒪}_{X}=ℂ\left[{x}_{1},\dots ,{x}_{n}\right]$ then elements of ${𝒟}_{X}$ are

$∑a∈ℤ≥0n ca(∂∂x1)a1 …(∂∂xn)an ,with ca∈𝒪X.$

Filtrations

The Bernstein filtration is ${ℬ}_{0}\subseteq {ℬ}_{1}\subseteq \dots \subseteq 𝒟$ with

$ℬj=span { xa∂b with |a|+|b| ≤j } .$

The standard filtration is ${\Sigma }_{0}\subseteq {\Sigma }_{1}\subseteq \dots \subseteq 𝒟$ with

$Σj=𝒪X -span { ∂b with |b|≤j } .$

Then

$gr 𝒟= ⨁j∈ℤ≥0 ℱjℱj-1= ℂ [ x1,…,xn, y1,…,yn ] .$

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

$gr M is a gr 𝒟-module.$

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

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

$Anngr 𝒟 (gr M)= { p∈gr 𝒟 | pm=0 for all m∈gr M } , Anngr 𝒟(gr M)= { p∈gr 𝒟 | pn∈Anngr 𝒟 (gr M) for some n∈ℤ>0 } .$

The characteristic variety of $M$ is

$ch M = { (v1,…,v2n) ∈ℂ2n | p(v1,…,v2n) =0 for all p∈ Anngr 𝒟(gr M) } = zero set of Anngr 𝒟(gr M) .$

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

$∑a∈ℤ≥0n ca(∂∂x1)a1 …(∂∂xn)an ,with ca∈𝒪X.$

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

$∇(m)= ∑i=1n ∂∂xim⊗ dxi.$

The center of ${𝒟}_{X}$ is

$Z(𝒟X)=ℂX, the constant sheaf on X.$

The functors

$Hom𝒟X (-,𝒪X): Mod(𝒟X)⟶ Mod(𝒪X) Hom𝒟X (𝒪X,-): Mod(𝒟X)⟶ Mod(𝒪X)$

have derived functors

$ℛHom𝒟X (-,𝒪X): 𝒟+(𝒟X)⟶ 𝒟+(ℂX) ℛHom𝒟X (𝒪X,-): 𝒟+(𝒟X)⟶ 𝒟+(ℂX).$

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

$DR(M)=ℛ Hom𝒟X (𝒪X,M)= ( 0→M→∇M ⊗𝒪XΩ1 (X)→∇M ⊗𝒪XΩ2 (X)→… ) with∇ (m⊗w)=∇ m∧w- (-1)deg wm ∧dw,$

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

The complex of holomorphic solutions of $M$ is

$Sol(M)=ℛ Hom𝒟X (M,𝒪X).$

Notes and References

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