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.)
Best Answer
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?"