Let $X$ be a scheme and $\iota: Z\hookrightarrow X$ the embedding of a closed subscheme $Z$; let $j: U\hookrightarrow X$ be the open complement. Suppose $\mathcal{F}$ is a coherent sheaf on $X$.
Can one relate $\iota_* \mathbb{L}\iota^* \mathcal{F}$ to data about $\mathcal{F}|_U$?
In particular, I'd like to know if there's some long exact sequence relating the homology of $\iota_*\mathbb{L}\iota^*\mathcal{F}$ (that is, $\mathcal{Tor}_{\mathcal{O}_X}(\mathcal{O}_Z, \mathcal{F})$) to global data about $\mathcal{F}$ on $X$ and data about $\mathcal{F}|_U$, similar to the long exact sequence for local cohomology. This seems to be related to local homology, but unfortunately I'm having trouble working out such a long exact sequence for local homology (or finding good references about local homology at all).
If it helps, I'm happy to assume $X$ is affine and $Z$ is a closed point.