The Tarski-Seidenberg Theorem states that the polynomial image of a semi-algebraic set is semi-algebraic. A semi-algebraic subset of a Euclidean space $\Bbb{R}^n$ is by definition a finite union of subsets of the form $$ \{P_1=\dots=P_k=0, Q_1>0,\dots,Q_l>0\} $$ where $P_i$'s and $Q_j$'s belong to $\Bbb{R}[x_1,\dots,x_n]$. I wonder is there a general coordinate-free version of this theorem for morphisms of real varieties? (By a real variety I mean the set of $\Bbb{R}$-points of a variety over $\Bbb{R}$.)

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    $\begingroup$ What does 'coordinate-free' mean? The original theorem seems coordinate-free to me …. $\endgroup$ – LSpice Apr 23 at 2:38
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    $\begingroup$ @LSpice I mean a version for abstract varieties that are not defined as subsets of a Euclidean space $\Bbb{R}^n$. $\endgroup$ – KhashF Apr 23 at 2:42

The most abstract version of the Tarski-Seidenberg theorem I know of is the following

Let $f:A\to B$ be a morphism of finite presentation of commutative rings. Then the induced map

$$f^*:\operatorname{Sper}B\to \operatorname{Sper}A$$

sends constructible sets to constructible sets.

Here $\operatorname{Sper}$ is the real spectrum of the ring, i.e. the set of all pairs $(p,<)$ where $p$ is a prime ideal and $<$ is an order on the residue field at $p$.

It is well known (e.g. theorem 7.2.3 in Bochnack-Coste-Roy Real Algebraic Geometry) that if $A$ is an algebra of finite presentation over $\mathbb{R}$, the boolean algebra of constructible subsets of $\operatorname{Sper}A$ is in natural bijection with the semialgebraic subsets of the real points of the variety $\operatorname{Spec}A$.

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  • $\begingroup$ Thanks! Out of curiosity, is there a notion of a "real scheme", a ringed space which is locally isomorphic with $Sper A$? $\endgroup$ – KhashF Apr 23 at 21:33
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    $\begingroup$ @KhashF Yes, they are called real closed spaces and they have been developed by Niels Schwartz (they are not really locally modeled on $\operatorname{Sper}A$, rather on proconstructible subsets of it, but the idea is the same) $\endgroup$ – Denis Nardin Apr 23 at 22:17

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