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Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological space to $X$ and he says that one can use the 'same glueing data'. My question is two-parted:

Does this construction actually define a functor $R_{\mathbb{R}}:Sch/\mathbb{R}\to Top$ from schemes of finite type over $\mathbb{R}$ to topological spaces? (I suppose that a positive/negative answer would hold for $\mathbb{C}$ also.)

And:

Does anyone know where this is written up rigorously, I mean without just saying that one just 'uses the same glueing data' and everything is fine?

Thank you!

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  • $\begingroup$ Yes (yes) and no I don't really know a good reference. $\endgroup$
    – Donu Arapura
    Commented Apr 12, 2011 at 16:17
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    $\begingroup$ The space $R_{\mathbb{R}}(X)$ has points consisting of maps Spec $\mathbb{C} \rightarrow X$ over $\Spec \mathbb{R}$. So complex conjugation induces an involution. The natural target for $R_{\mathbb{R}}$ is $\mathbb{Z}/2$-spaces, i.e., spaces equipped with an action of $\mathbb{Z}/2$. $\endgroup$
    – Dan Isaksen
    Commented Apr 13, 2011 at 11:04

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Dear mustafa-kava, Amnon Neeman has written a rather down-to-earth book Algebraic and Analytic Geometry dedicated to a proof of Serre's celebrated GAGA theorem. Serre's article is 42 pages long and Neeman's book exactly ten times as long: 420 pages. This is because the book is extremely detailed and starts from the ground up (It comes from a fourth-year undergraduate course).

Section 4, called "The complex topology", is 20 pages long and seems to be what you want (at least in the complex case), with all details spelled out at an elementary level.
Let me add that Neeman really knows what he is talking about: he has proved a remarkable criterion to decide whether an algebraic variety is affine, given that its analytification is Stein.

As for the real algebraic geometry, the standard reference is Bochnak-Coste-Roy's Real Algebraic Geometry. What you want to study is Chapter 11 Topology of real algebraic varieties. Be warned that this is an advanced monograph and that general knowledge of schemes is insufficient : techniques like positivstellensatz, real spectra, Nash functions,...are de rigueur here. On the positive side you will be delighted by astonishing results like : every $\mathcal C^\infty $ compact manifold is diffeomorphic to an algebraic subvariety of some $\mathbb R ^p$ !

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