Let $N$ be the nilpotent variety of $\mathcal{sl}_2$, and $\pi :\tilde{N}\to N$ its Grothendieck resolution. I.e. $$N=\left\{\begin{pmatrix}a&b\\c&-a\end{pmatrix}:a^2+bc=0\right\}$$ $$N\times \mathbb{P}^1\supset \tilde{N}=\left\{\left (\begin{pmatrix}a&b\\c&-a\end{pmatrix},[u:v]\right ):\begin{pmatrix}a&b\\c&-a\end{pmatrix}\begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\right\},$$ and $\pi :\tilde{N}\to N$ is the projection.
Let $i$ be the inclusion of $\{0\}$ into $N$. I want to compute
$i^!\pi_*\mathbb{Z}_{\tilde{N}}\to i^*\pi_*\mathbb{Z}_{\tilde{N}}$
(All functors are derived!)
I think I know what is $\pi_*\mathbb{Z}_{\tilde{N}}$, its zeroth cohomology is the constant sheaf $\mathbb{Z}_N$, its second cohomology is the constant sheaf $\mathbb{Z}_{\{0\}}$ supported on $\{0\}$ , and other cohomologies vanish.
So, $i^*\pi_*\mathbb{Z}_{\tilde{N}}$ is two copies of $\mathbb{Z}$, one in degree $0$ and one in degree $2$. And $i^!\pi_*\mathbb{Z}_{\tilde{N}}=$$i^*\mathbb{D}(\pi_*\mathbb{Z}_{\tilde{N}})$ is two copies of $\mathbb{Z}$, one in degree $2$, and one in degree $4$. ($\mathbb{D}$ is the Verdier dual functor).
So I only need to compute
$H^2(i^!\pi_*\mathbb{Z}_{\tilde{N}})\to H^2(i^*\pi_*\mathbb{Z}_{\tilde{N}})$,
which must be multiplication by an integer, and I want to know what this integer is. Thank you in advanced.
