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I understand Serre's GAGA theorem as saying that equations over algebraically closed fields can be studied equally from the algebraic and analytic points of view, at least with respect to cohomology.

Real algebra and real analysis are means to study inequalities over real closed fields. Firstly, I'm wondering if there is a cohomology for real algebraic varieties and real semialgebraic varieties?

Given that there is such a cohomology, has there be a comparison of the algebraic cohomology and the analytic cohomology, in analogy to GAGA's theorem? I would expect that if there is such a theorem, it would deny the equivalences of these two approaches. Instead, I would believe that such a theorem would say that algebra can detect strictly less than analysis. For after all, cohomology is about local-global obstructions.

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I'm a little confused about what is being asked here. For an algebraic variety over any field at all you can do things like algebraic de Rham cohomology, which produces vector spaces over the ground field "of the expected dimension", and you can do ell-adic etale cohomology, which produces vector spaces over ell-adic fields, also "of the expected dimension". But if you have an algebraic variety over the real numbers and you want to do some analytic construction which depends only on the real points with their real analytic topology, then you're in real trouble... – Kevin Buzzard Feb 27 '11 at 13:26
...because there's no reason that the variety should have any real points at all. For example consider the conic in projective two-space defined by $x^2+y^2+z^2=0$. This has no real points, but its complex points are isomorphic to the projective line, so the "right answer" is that $H^0$ and $H^2$ are 1-dimensional, but if you only look at the real points then you have the empty set so can't reconstruct the cohomology. – Kevin Buzzard Feb 27 '11 at 13:28
+1 to Kevin's comment for the "[...] then you are in real trouble". :) – Qfwfq Feb 27 '11 at 16:16
The tiniest baby case of GAGA is that a complex variety is connected in the analytic topology if and only if it is connected in the Zariski topology. But it's a difficult problem, as far as I know, to determine whether the real points of a variety are connected (even if you know they're nonempty). – Tom Church Feb 27 '11 at 18:55
@Colin and Kevin: There is certainly a cohomology theory for semialgebraic varieties -- in fact, there is at least the beginning of a sheaf cohomology theory for any o-minimal structure. If you do a google search "cohomology semialgebraic varieties" you see get some interesting links. For cohomology theory in the o-minimal setting see the homepage of Mario Edmundo – Ramin Feb 27 '11 at 18:55

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