Notes 07/08/2013

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 7 August 2013


The Weyl algebra is the algebra 𝒟 with generators x1,,xn, y1,,yn and relations

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

i.e. [xi,xj]=0, [yi,yj]=0 and yjxi-xiyj=δij.

Often we write j instead of yj since

xjxif- xixjf= ijf+xi fxj-xi xjf= δijf.

If 𝒪X=[x1,,xn] then elements of 𝒟X are

a0n ca(x1)a1 (xn)an ,withca𝒪X.


The Bernstein filtration is 01𝒟 with

j=span { xabwith |a|+|b| j } .

The standard filtration is Σ0Σ1𝒟 with

Σj=𝒪X -span { bwith |b|j } .


gr𝒟= j0 jj-1= [ x1,,xn, y1,,yn ] .

The characteristic variety

Let M be a 𝒟-module.

A filtration of M is M0M1M with Mj=M and

(a) iMjMi+j
(b) Mj is a finitely generated 0-module.


grMis agr 𝒟-module.

A good filtration is a filtration of M such that grM is a finitely generated gr𝒟-module.

This happens if and only if there exists a j0 with iMj=Mi+j for jj0,i0.

Anngr𝒟 (grM)= { pgr𝒟 |pm=0 for allmgrM } , Anngr𝒟(grM)= { pgr𝒟| pnAnngr𝒟 (grM)for some n>0 } .

The characteristic variety of M is

chM = { (v1,,v2n) 2n| p(v1,,v2n) =0for allp Anngr𝒟(grM) } = zero set of Anngr𝒟(grM) .

The sheaf 𝒟X

Let X be a space

𝒪X the sheaf of functions on X
𝒟X the sheaf of 𝒪X-coefficient differential operators on X.

In local coordinates, sections of 𝒟X are

a0n ca(x1)a1 (xn)an ,withca𝒪X.

A 𝒟X-module M is equivalent to a connection :MM𝒪XΩ1(X) on M via the formula

(m)= i=1n xim dxi.

The center of 𝒟X is

Z(𝒟X)=X, the constant sheaf onX.

The functors

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

have derived functors

Hom𝒟X (-,𝒪X): 𝒟+(𝒟X) 𝒟+(X) Hom𝒟X (𝒪X,-): 𝒟+(𝒟X) 𝒟+(X).

If M is a 𝒟X-module then

DR(M)= Hom𝒟X (𝒪X,M)= ( 0MM 𝒪XΩ1 (X)M 𝒪XΩ2 (X) ) with (mw)= mw- (-1)degwm dw,

is the de Rham complex of M.

The complex of holomorphic solutions of M is

Sol(M)= Hom𝒟X (M,𝒪X).

Notes and References

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

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