My aim is to understand how to describe the localization of modules over the Weyl algebra. I want to be able to do simple examples by hand. I wrote the following code in Macaulay2.
W = QQ[x,Dx, WeylAlgebra => {x=>Dx}]
I = ideal(Dx)
f = x
Mf = Dlocalize(W^1/I, f)
I wrote the above code in Macaulay2.
i11 : Mf
o11 = cokernel | xDx+2 |
1
o11 : W-module, quotient of W
I am trying to understand what this means. The output seems to suggest that $\mathbb{Q}[x][\frac{1}{f}]$, where $f = x$, as a module over $\mathbb{Q}[x, \mathrm{d}x]$ is just $\frac{\mathbb{Q}[x, \mathrm{d}x]}{<x\mathrm{d}x + 2>}$. Am I understanding this wrong?
I tried to work it out myself. We have that $(p_1, f_1) \sim (p_2, f_2)$, where $p_i \in \mathbb{Q}[x]$, and $f_i$ is a power of $x$, when $f_2 \cdot p_1 = f_1 \cdot p_2$. I am not sure what to do after this. I'd be very grateful if someone can also give me the explanation behind this example.
f = x^2
Mf = Dlocalize(W^1/I, f)
This gives output
i11 : Mf
o11 = cokernel | xDx+4 |
1
o11 : W-module, quotient of W