Akizuki's Counterexample

Following [Akizuki], I construct a Noetherian local integral domain CM whose normalisation (integral closure) is not finite over CM . My proof follows closely Akizuki’s ingenious calculations. Let A be a DVR with local parameter t and residue field k = A/(t), and  the completion of A. There are no restrictions on the characteristic of A or k, but I assume that  contains a transcendental element over A. (For DVRs of interest,  usually has infinite transcendence degree over A.) The rings B, C constructed below, and their localisations, are intermediate subrings between A and Â. The construction depends on a power series z = z0 = a0 + a1t n1 + a2t n2 + · · · ∈ Â, (1) (not just on the element z). Assume: (2) Each ai ∈ A is a unit. (3) nr ≥ 2nr−1 + 2 for every r ≥ 1, where I set n0 = 0; for example, the smallest possible choice is nr = 2(2 r − 1) = 0, 2, 6, 14, 30, . . . . (4) z is transcendental over A, so that A ⊂ A[z] ⊂  is a polynomial extension. Akizuki’s construction is as follows: for r ≥ 0, let zr = z0 − first r terms tnr = ar + ar+1t mr+1 + · · · , where mr = nr − nr−1 so that (3) gives 2mr ≥ nr + 2. (5) This paper was written during a stay at the semestre “Surfaces de Riemann et fibrés vectoriels” of the Centre Emile Borel, paid for by the EEC HCM project AGE (Algebraic Geometry in Europe), contract number ERBCHRXCT 940557.