18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 Even more on affinoid algebras

Ruochuan caught me last time using the following fact without justification, in the proof that spectral norms are Banach norms. In fact, this is [FvdP, Theorem 3.5.1], so I might as well explain this. (Actually, I only used part (b) in that proof, so the annoying part (a) below wasn’t really needed. But let’s explain it anyway.)

Theorem 1. Let A be a reduced integral affinoid algebra. Then the integral closure of A in its fraction field is finitely generated as an A-module. (Corollary: the integral closure of A in any finite extension of its fraction field is also finite over A.)

Proof. We can write A as a finite integral extension of some T_{d} by Noether normalization, so it is enough to show that the integral closure of T_{d} in any finite extension of its fraction field is finite over T_{d}. As in the previous proof, this breaks down into two steps.

(a) If L is a finite purely inseparable extension of T_{d}, then the integral closure of T_{d} in L is finite over T_{d}.

(b) If A is an affinoid algebra which is a normal (integrally closed in its fraction field) domain, and L is a finite Galois extension of Frac A, then the integral closure of A in L is finite over A.

Part (b) is easy: let B be the integral closure of A in L, choose e_{1}, …, e_{n} ∈ B which form a basis of L over Frac A, and let e_{1}^{∗}, …, e_{n}^{∗} be the dual basis for the trace pairing. Now simply pick a ∈ A such that ae_{1}^{∗}, …, ae_{n}^{∗} ∈ B; then note that af ∈ Ae_{1} + ⋯ + Ae_{n} for any f ∈ B. So B is contained in a finitely generated A-module; since A is noetherian, A is finitely generated.

Part (a) is a bit more annoying. It’s clear if K is perfect, as in that case we can pass from L to the fraction field of K⟨x_{1}^{1/p^{n}}, …, x_{d}^{1/p^{n}}⟩ for some n, and the integral closure in that field is clearly the bigger Tate algebra, which is visibly finite over T_{d}. You can still argue like this if [K : K^{p}] < ∞, but otherwise it gets messy. Let K′ be the completed algebraic closure of K; the point is that whatever generators you get of the integral closure of K′⟨x_{1}, …, x_{d}⟩ in K′⟨x_{1}^{1/p^{n}}, …, x_{d}^{1/p^{n}}⟩ can be approximated by generators which are actually integral over T_{d}. (Compare the argument in the proof of the “closed submodule principle”, i.e., [FdvP, Lemma 1.2.3], or just look this up in [FvdP, Theorem 3.5.1].)