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.

  • $\begingroup$ I think what you are looking for is the distinguished triangle: $$ j_!j^!F \to F \to i_*i^*F \to +1 $$ dual to the local cohomology one, where $j_!$ is the "extension by zero" functor (while $j^!F = j^*F = F|_U$). So in terms of long exact sequences it reads $$ H^i(X,j_!F|_U) \to H^i(X,F) \to H^i(X,F_Z) \to H^{i+1}(X,j_!F_U) $$ But I don't know much about it in the framework of quasi-coherent sheaves. $\endgroup$
    – AFK
    Aug 17, 2012 at 16:47


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