Exercise 13, § 2

\* We keep the general hypotheses and the notations of exerc. 12.

a) Let L be a free sub-A-module of E of dimension n = dim E. For all x ∈ E \ {0}, the set of the a ∈ A such that ax ∈ L is a fractional ideal of K of the form m^h (h a positive or negative integer); putting α(x) = q^h and α(0) = 0, show that α is an ultranorm on E. It is said to be associated with the free A-module L.

b) Conversely, if α is an ultranorm on E, the set L_α of the x ∈ E such that α(x) ≤ 1 is a free A-module of dimension n. If [α] is the norm associated with L_α, we have α ≤ [α] ≤ qα, and [α] is the lower bound of the norms associated with free A-modules and which are ≥ α. We have [qα] = q.[α], and α(x) = inf(q^{-t}[q^tα] (x)) for all x ∈ E, where t varies in the interval [0, 1]. Further, the function t ↦ [q^tα] (x) is left continuous in this interval.

c) With the same notations, show that for 0 ≤ t ≤ 1, there are at most n distinct ultranorms among the [q^tα]. Conversely, let L be the set of ultranorms associated with the free A-modules of dimension n, and let (α_t)_{0 ≤ t ≤ 1} be an increasing family of ultranorms of L such that α_1 = qα_0. Show that there exists a basis of E relative to which all the α_t have the standard form (if u ∈ A is an element of valuation 1, and L_t the free A-module of the x ∈ E such that α_t(x) ≤ 1, consider the vector spaces L_t/uL_0 on k). Deduce further, that if, for all x ∈ E, t ↦ α_t(x) is left-continuous in [0, 1] then there exists a unique ultranorm α such that α_t = [q^tα] for all t ∈ [0, 1].

d) The linear group GL(E) operates continuously in Ω_0(E); show that it operates properly. For all α ∈ Ω_0(E), the stabiliser S_α of α in GL(E) is the intersection of the stabilisers of the [q^tα] for 0 ≤ t ≤ 1; deduce that S_α is an open compact subgroup of GL(E), and hence that the orbit of each α ∈ Ω_0(E) is a closed, discrete subspace of Ω_0(E).

e) For every ultranorm α ∈ Ω_0(E) consider the decreasing sequence of the dimensions of the vector k-spaces L_t/uL_0, where L_t is the A-module of the x ∈ E such that [q^tα] (x) ≤ 1, and t varies from 0 to 1; we call this sequence, the sequence of invariants of α. In order that α and β belong to the same orbit in Ω_0(E), it is necessary and sufficient that X_α = X_β (exerc. 12, e)) and that the sequence of the invariants of α and of β should be the same (use exerc. 12, b)).

f) Deduce from e) that the space of the orbits Ω_0(E)/GL(E) is isomorphic with the space of the orbits T^n/Σ_n, where the symmetric group operates on the right on T^n by (z_1, ..., z_n) ↦ (z_{σ(1)}, ..., z_{σ(n)}). \*