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In algebraic topology, relative (co)homology is very useful. For example, we have a long exact sequence which is often helpful for lots of calculations.

In algebraic geometry, we have local cohomology, which is basically the same thing and has the same long exact sequence. However, while the commutative algebra community seems to use this a lot, it seems to be rarely used in algebraic geometry. Is indeed local cohomology more useful in commutative algebra than it is in algebraic geometry? If so, why?

(I'm primarily talking about the Zariski topology, but we also have local cohomology in any context where we have six functors; in étale cohomology, in de Rham cohomology, etc. I would also like to know something about local cohomology in these contexts.)

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    $\begingroup$ Well, I've used it at least once, when thinking about Hodge theory of open varieties. But, I agree with you with for the most part, that algebraic geometers don't use it much. Probably because we mostly like to think about proper schemes. $\endgroup$ Nov 14, 2021 at 17:03
  • $\begingroup$ @DonuArapura I don't know a lot about local cohomology, but doesn't it work for locally closed subschemes? $\endgroup$
    – Gabriel
    Nov 14, 2021 at 17:20
  • $\begingroup$ Suppose that $X$ is a closed scheme, and $U$ an open subscheme. Then there's some sheaf $\mathcal{I}$ of ideals of $\mathcal{O}_X$ such that $U$ is the complement of the vanishing locus of $\mathcal{I}$. I recall finding the local cohomology of quasicoherent $\mathcal{O}_X$-modules, with respect to $\mathcal{I}$, quite useful for calculating cohomology of quasicoherent sheaves on $U$. $\endgroup$
    – user164898
    Nov 14, 2021 at 17:23
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    $\begingroup$ @Gabriel It seems to me that the six functors formalism (which is much commonly used in algebraic geometry than in other fields) subsumes relative cohomology by far. On what grounds do you claim that relative versions of cohomology is less used in algebraic geometry? (I mean no disrespect nor any provocation, but I sincerely do not see the situation this way) $\endgroup$ Nov 14, 2021 at 19:42
  • $\begingroup$ Dear @D.-C.Cisinski, after reading the answers to this questions, it has become clear to me that my claim is based only on my small experience in algebraic geometry. $\endgroup$
    – Gabriel
    Nov 15, 2021 at 8:25

2 Answers 2

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I don't agree with the premise of this question. Local cohomology (per se and not in the wider context of the six functor formalisms) and its consequences are still very much used in algebraic geometry. If you just glance at SGA2, you will find that the following results are proved using local cohomology:

$\bullet$ Lefschetz Theorems for the Picard groups and étale fundamental groups,

$\bullet$ Samuel conjecture on factoriality for complete intersections with small singular loci,

$\bullet$ Comparison Theorems between formal and algebraic geometry, which lead to Grothendieck's proof of Zariski Main Theorem.

$\bullet$ Grothendieck and Fulton-Hansen connectedness results, which have been at the origin of the whole industry studying (higher) secant varieties of projective varieties.

And more recently:

$\bullet$ Bounds for the étale cohomological dimensions of toroidal and determinantal varieties, which imply new bounds on the arithmetical rank of such varieties,

$\bullet$ Improved bounds for Castelnuevo-Mumford regularity of specific projective verieties.

Hence, I would be rather inclined to reformulate your question as "Are there significant areas in Algebraic Geometry where people do not use local cohomology in any way?"

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    $\begingroup$ Local cohomology is also essential for Grothendieck duality as formulated in”Residues and Duality”. $\endgroup$ Nov 15, 2021 at 3:13
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    $\begingroup$ In addition, one interpretation of algebraic $D$-modules uses local cohomology with supports in the diagonal (via residues). $\endgroup$
    – Kapil
    Nov 15, 2021 at 16:03
  • $\begingroup$ @Kapil do you know some reference where I can learn more about that? $\endgroup$
    – Gabriel
    Nov 20, 2021 at 10:41
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(answer in the context of étale cohomology)

Why use purpose-built notation for a particular special case of the six functors formalism when you have access to the full power of the six functors formalism?

I'm sure at some point in (e.g.) my work, I've needed to consider the cohomology of the mapping cone of the adjunction $\mathcal F \to j_* j^* \mathcal F$ for $j$ an open immersion. But I've also needed to consider the compactly-supported cohomology, or the mapping cone of the adjunction $i_! i^* \mathcal F \to \mathcal F$, or the case when $j$ is not an open immersion. It doesn't make sense to package a specific construction as its own concept.

In particular, this doesn't make sense when, as it often is, the first step in calculating the cohomology of this mapping cone is considering the cone as its own object, a complex of sheaves, and studying the sheaf by evaluating, for example, its stalk at particular points. Thus the sheaf plays the starring role in our notation and not its cohomology groups.

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