Real Algebra of Excellent Rings

This chapter is devoted to excellent rings, and contains the results that allow to extend what we have already seen for semialgebraic sets to semianalytic sets. In Sections 1 and 2 we collect the commutative algebra needed later. Very few proofs are given, since almost everything can be found in our general references [Mt] and the more elementary [At-Mc], [Bs-Is-Vg]; an important exception is our proof that local-ind-etale limits of excellent rings are again excellent. In addition, we state without proof the fundamental Rotthaus’s theorem on M. Artin’s approximation property. In Section 3 we characterize the extension of prime cones under completion, a crucial result for all that follows. The curve selection lemma which is proved in Section 4 has many important applications: existence theorems for valuations and fans (Section 5), and constructibility of closures (Section 6) are some. It is also needed in Section 7 for the proof of another key theorem: the real going-down for regular homomorphisms. After this, we characterize local constructibility of connected components in Section 8.