Quotients of semi-algebraic spaces
We consider semi-algebraic spaces over a fixed real closed field R. For M such a space and \(E\subset M\times M\) a semi-algebraic equivalence relation on M we say that the (geometric) quotient M/E exists if there is an identifying semi-algebraic map \(M\to N\) whose fibres are the E- equivalence classes. In many cases, categorical quotients (i.e. effective epimorphisms) are also geometric quotients, for example if M is locally complete or if also non-affine spaces are allowed. Assume now that M is locally complete and E is closed. Then a necessary and sufficient condition for M/E to exist is that there should be a (closed) subspace \(K\subseteq M\) such that \(p_ 1| p_ 2^{-1}(K): p_ 2^{- 1}(K)\to M\) is proper and surjective (where \(p_ 1,p_ 2: E\rightrightarrows M\) are the projections). If this is satisfied, then M/E ``is'' the proper quotient \(K/E_ K\) \((E_ K:=E\cap (K\times K))\). On the other hand, M/E does always exist if \(p_ 1, p_ 2\) are open. Examples show that the local completeness assumptions cannot be dropped. The proofs make use of the real spectrum and of elimination of quantifiers in the theory of real closed fields with non-trivial compatible valuation.
scientific article; zbMATH DE number 4055757