Let $f: X \rightarrow Y$ be a morphism between schemes. Then, under mild hypothesis on $f, X$ and $Y$, we have Grothendieck duality. This gives an isomorphism
$\mathcal{R}\mathcal{H}om_{Y}(\mathcal{R}f_{x}\mathcal{F},\mathcal{G}) \cong \mathcal{R}f_{x}\mathcal{R}\mathcal{H}om_{X}(\mathcal{F},f^{!}\mathcal{G})$.
Here, $\mathcal{R}f_{x}: D^{b}(Qcoh(X)) \rightarrow D^{b}(Qcoh(Y))$ is the right derived functor of push forward (sorry, the latex is being a bit funny, hence the unusual notation), and $f^{!}: D^{b}(Qcoh(Y)) \rightarrow D^{b}(Qcoh(X))$ is the right adjoint of the right derived push forward, as given by Grothendieck duality. People usually call $f^{!}$ the Exceptional inverse image functor.
My question is that I would like to compute $f^{!}$ for some simple closed immersions.
More specifically, consider the following closed immersions, given by the canonical embeddings:
$i : \mathbb{A}^{n} \rightarrow \mathbb{A}^{n+1}$
$i : \mathbb{P}^{n} \rightarrow \mathbb{P}^{n+1}$
$i : Gr^{k,n} \rightarrow Gr^{k,n+1}$.
What are
$i^{!}(\mathcal{O}_{\mathbb{A}^{n+1}})$,
$i^{!}(\mathcal{O}_{\mathbb{P}^{n+1}})$,
$i^{!}(\mathcal{O}_{Gr^{k,n+1}})$ ?
The stacks project gives an explicit description on how to compute the Exceptional inverse image functor in the case of closed immersions, see https://stacks.math.columbia.edu/tag/0A74.
The problem is that I don't know how to `compute' with this explicit description, and I feel as if, for the maps I have described above, one should be able to say something explicit!
I am not an Algebraic Geometer by training, so if someone could take my hand and walk me through the technicalities, that would be much appreciated!