Nonfinite unramified extensions

Abstract

Suppose A is a normal noetherian domain containing Q and C =) A is a domain which is affine and unramified over A. Let B be the integral closure of A in C, the field of quotients of C. Now C is normal and thus A c B c C. This gives rise to four radical ideals, the Finiteness and Discriminant ideals of A, and the Zariski Complement and Branch Locus ideals of B. The purpose of the paper is to analyze these ideals and their behavior under Galois closure and tensor product. Also we give a topological proof of a theorem of Zariski: the purity of the Discriminant ideal. Since B c C is unramified, it follows from Zariski’s Main Theorem that Spec C + Spec B is an open immersion. The complement of its image is V(I) where Zc B is a radical ideal. Define the Zariski Complement ideal by ZC(C, B) = I. Define the Finiteness ideal by Fint(C, A) = {UE A; C, is finite over A,}. The statement that an ideal J of a commutative ring has pure height 1 means that every minimal prime over J has height 1. Then the Zariski Complement has pure height 1 in B, the Finiteness ideal has pure height 1 in A, and ZC(C, B) n A = Fint(C, A). Furthermore, each of these ideals commutes with localization in A. Let n be the dimension of the field extension 2 c B, and V= (U E B; the minimal polynomialf, E A[ r] of u has degree n}. Define the Branch Locus ideal BL(B, A) as the radical in B of the ideal generated by all&(u) where UE V. Let disc(u) be the discriminant of U. Define the Discriminant ideal Disc(B, A) as the radical in A of the ideal generated by all disc(u) where u E V. If p c B is a prime ideal, B is unramilied over A at p iff BL(B, A) d p. If q c A is a prime ideal, B is unramilied over A at every extension of q iff Disc(B, A) ti q. Also BL(B, A) n A = Disc( B, A), Fint(C, A) c Disc(B, A), and ZC(C, B) c BL(B, A). If A is regular, it is a theorem due to Zariski, Nagata, and Auslander that BL(B, A) and Disc(B, A) have pure height 1. Let G(C) be the normalization (or Galois closure) of C over 2, and let t7, )...) 0” . C --* G(C) be the distinct embeddings of C which are the identity on A. If R is a domain with A c R c C, define G(R) to be the ring 254 oool-8708/87 $7.50