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Is there a relative version of sheaf cohomology? EDIT: I rather mean the cohomology of pairs.

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Yes (in algebraic geometry, at least): the derived functors of the pushforward functor from sheaves on the top the to sheaves on the bottom. – Kevin Buzzard Jan 22 '10 at 18:55
Let's make sure we are talking about the same thing. The word `relative' has (at least) two meanings: - (AG meaning) defined for families: this leads to derived direct images; - (topological meaning) defined for pairs: this leads to cohomology with support. Which one are we talking about here? – t3suji Jan 22 '10 at 23:43
up vote 5 down vote accepted

It turns out that my previous answer dealt with the wrong question. The answer to the new question is also yes: local cohomology $H\_Z(X,\mathcal F)$ corresponds to cohomology of the pair $(X,X\setminus Z)$.

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It seems to be exactly what I am looking for. I want to understand the local cohomology and look for a more 'geometrical' definition. Do you know of a good text on local cohomology where this correspondence is treated? – Rootof Jan 24 '10 at 16:03
One basic text is (if I remember correctly) the Springer Lecture Notes <I>Local cohomology</I> by Hartshorne (it's early in the series, maybe in the first 100), based on lectures of Grothendieck. But it is focused on algebraic geometry. There are some texts on sheaf theory that are more focussed on the usual (rather than Zariski) topology, e.g. Borel's <I> Intersection homology </I>. Verdier duality is treated in that book, I think, which generalizes Poincare and Alexander-Lefschetz duality, so at least implicity it should treat this correspondence. – Emerton Jan 24 '10 at 19:13
One could also consult the references in Borel and see where they lead. Local cohomology also plays an important role in Sato's theory of hyperfunctions (which is treated in one or more Springer Lecture Notes volumes, among other places). Maybe looking at that literature would give an interesting perspective. Some of Kashiwara's books on sheaf theory might also help (and they probably have overlap with the books already mentioned). I'm sorry not to be able to give more specific references. Maybe someone else can? – Emerton Jan 24 '10 at 19:16
This overlaps somewhat with Matt's answer, but SGA2 also has a nice and comprehensive discussion of local cohohomology and its various incarnations in topology, algebraic geometry, and commutative algebra, and relations among all three. The book of Freitag-Kiehl on etale cohomology discuss the analogous theory in that setting, including various important/useful "purity theorems" when Z sits nicely inside a nice X. – BCnrd Mar 1 '10 at 16:16

To elaborate on Kevin's comment: If $f: X \to S$, and $\mathcal F$ is a sheaf on $X$, then $f\_*\mathcal F$ is the sheaf on $S$ defined by $H^0(U,f\_*\mathcal F) := H^0(f^{-1}(U),\mathcal F).$

Taking the derived functors of $f\_*$ gives functors $R^if\_*$, and it turns out (fairly easily) that $R^if\_*(\mathcal F)(U)$ is the sheaf associated to the presheaf $U \mapsto H^i(f^{-1}(U),\mathcal F)$. If $i > 0,$ then this presheaf may not be a sheaf (unlike the $i = 0$ case), and this is related to the fact that it can be a little subtle to compute the stalks of $R^if\_*\mathcal F$ in general; for example, it need not be the case in general that the stalk $(R^if\_*\mathcal F)_s$ is equal to $H^i(f^{-1}(s),\mathcal F)$. (E.g. think about the case when $f$ is the inclusion of a punctured disk into a disk, $\mathcal F$ is the constant sheaf ${\mathbb Z}$, and $s$ is the centre of the disk (so that $f^{-1}(s)$ is empty).)

In other words, $R^if\_*\mathcal F$ does not always literally interpolate the cohomology of the fibres.

There is one case where one knows that $R^if\_*\mathcal F$ does interpolate the cohomology of the fibres: if the map $f$ is proper, than the proper base-change theorem says that the stalk of $R^if\_*\mathcal F$ at $s$ is the cohomology of $\mathcal F$ along the fibre of $s$. (One good place for these kinds of facts is the beginning of Borel's book on Intersection Cohomology.)

Also, in the context of maps of varieties, if $f$ is proper and $\mathcal F$ is a coherent sheaf, then the completed stalk of $R^if\_* \mathcal F$ at $s$ coicides with the cohomology of the pull-back of $\mathcal F$ to the formal completion of $f^{-1}(s)$ in $X$. (This is Grothendieck's proper base-change theorem, proved in some form in Hartshorne, Ch. III.)

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I would think -- but this is for the moment a thought, not meant as an authorative answer -- that every notion of cohomology whatsoever has a relative version in this sense.

I am thinking here of the general abstract definition of cohomology as exposed at nLab:cohomology. This includes in particular the special case of sheaf cohomology as described there in some detail.

What I expect the fully general notion of relative cohomology from this point of view to be I have now briefly indicated at relative cohomology. I'd believe this does in particular reproduce the definition given in Emerton's answer.

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