Real Algebra of Excellent Rings
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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.
KeywordsPrime Ideal Local Ring Valuation Ring Residue Field Finite Extension
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