Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ be a finitely generated $R$-module and $\widetilde M$ be its associated sheaf on $\text{Spec} (R)$. We have the closed immersion $\pi: \text{Spec } (R/I)\to \text{Spec} (R)$. My question is: Can we explicitly compute what the derived pullback $\mathbf L\pi^* \widetilde M$ is ?
I am a beginner in derived category methods in algebraic geometry, so your help and guidance is kindly appreciated. Many thanks in advance.