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.