I am sorry if this question is not appropriate for MO.
Suppose $X$ is the affine line, $i:Z\to X$ is the origin, and $j: U \to X$ is the complement to $Z$ in $X$.
I then have a distinguished triangle: $i_! i^! O \to O \to j_* j^* O \to$, where $O$ is any $D$-module on $X$, but I take it to be for this example just the structure sheaf.
I want to see "explicitly" what do I get. If I am not wrong, $j_* j^* O$ is Laurent polynomials, while $i_! i^! O$ is $\Delta = \oplus k \partial^i $, in degree $1$. Thus the distinguished triangle is equivalent to the data of an exact sequence: $0 \to O \to j_* O_U \to \Delta \to 0$.
My question is:
How in principle should I compute the arrow $j_* O_U \to \Delta$ in the s.e.s. above? It is some connecting arrow in the distinguished triangle, which seems abstract to me.
The second question I have is how to "compute" $j_! O_U$. I have two strategies, about both of which I am not sure exactly. The worse one is to compute $D (j_* O_U)$, the dual of $j_* O_U$. I don't know how to do it, a resolution seems complicated. The other method would be applying duality $D$ to the s.e.s. above, getting $0 \to \Delta \to j_! O_U \to O \to 0$. Then:
How can I compute explicitly how $j_! O_U$ is an extension of $\Delta$ and $O$?
Thank you, Sasha