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?


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 $U \to X$.

  • 2
    $\begingroup$ Read Expose I of SGA2 for a version with arbitrary topological spaces (not specifically tied up with quasi-coherent sheaves on schemes). $\endgroup$
    – nfdc23
    Dec 29 '16 at 1:14
  • $\begingroup$ @nfdc23 Sounds nice, I'll try. You might have something to say about the following question: mathoverflow.net/questions/257539/… $\endgroup$ Dec 29 '16 at 1:17
  • $\begingroup$ I think it is a bit naive. Hard work is needed to make such things in various interesting cases, such as with noetherian schemes, complex-analytic spaces, suitable non-archimedean analytic spaces, etc. It is hard to imagine that some abstract nonsense would unify all of these. Is there an actual motivating reason or just idle curiosity behind that question? I recommend to learn the substance behind one of the frameworks where the 6-functor formalism exists, and then you will better appreciate the key issues involved. A general locally ringed space is too featureless for what needs to be done. $\endgroup$
    – nfdc23
    Dec 29 '16 at 1:21
  • $\begingroup$ @nfdc23 Are you claiming that noetherian schemes have six functors? $\endgroup$ Dec 29 '16 at 1:25
  • $\begingroup$ No, but for reasonably interesting classes of maps (such as proper ones) between reasonable noetherian schemes (say finite type over a regular ring) one has robust coherent duality and so on. The point is just that for various interesting classes of geometric objects there are hard theorems in the direction of "six-functors" formalism, and you should learn how some of those proofs work to appreciate what the real difficulties are in setting up the hard parts of such formalism and in particular why asking for things in the total generality of locally ringed spaces is asking for too much. $\endgroup$
    – nfdc23
    Dec 29 '16 at 19:33

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