Let $X$ be a noetherian scheme $U \subset X$ an open subset with complement $Z = X- U$. Assume $Z$ is cut out by the ideal sheaf $\mathcal{I} \subset \mathcal{O}\_X$. We have exact sequences:


$$0 \to \mathcal{I}^n \to \mathcal{O}\_X \to \mathcal{O}\_X/\mathcal{I}^n \to 0$$


Here's what i'd like to be able to say:



> 
> Consider it now as an exact sequence of functors:
> 
> 
> $$0 \to RHom(\mathcal{O}\_X/\mathcal{I}^n,-) \to RHom(\mathcal{O}\_X,-)
>  \to RHom(\mathcal{I}^n,-) \to 0$$
> 
> 
> Plugging in a sheaf gives a long exact sequence relating cohomologies
>  of the complexes. Assuming $X$ affine and using the fact that
>  $\lim\_nRHom(\mathcal{I}^n,-) \cong RHom(\mathcal{O}\_U,-)$ (and the long exact sequence) we see that
>  local cohomology is isomorphic (after shift) to what you get by
>  pushing-pulling (in derived sense) via the open immersion $U \to X$.
> 
> 
> 


Is this argument valid? Is the conclusion even true?



Relation between local cohomology and open immersions
This is true. The functor $RHom(\mathcal{O}\_X/\mathcal{I}^{n+1},-)$ is in fact a different name for $j^!$ when $j: Z\_n \to X$. Where $Z\_n$ is the $n$-truncated formal neighborhood of $Z$ in $X$. In the limit we get a kind of $j^!$ too only now for the inclusion of the formal scheme $\widehat X\_Z$ (formal neighborhood of $Z$ in $X$) into $X$. When $i:U \to X$ is the inclusion of the complement we have an exact triangle in the derived category


$$0 \to j\_\*j^! \to Id \to i\_\*i^\* \to 0$$


So for the case of $X$ affine we indeed have that local cohomology is identified with the shifted version of the pull push along $U \to X$.


