On points at infinity of real spectra of polynomial rings

F. Lucas Département de Mathématiques Université d’Angers 2, bd Lavoisier 49045 Angers Cedex, France

D. Schaub Département de Mathématiques Université d’Angers 2, bd Lavoisier 49045 Angers Cedex, France

M. Spivakovsky CNRS-Institut de Mathématiques de Toulouse Université Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex 9, France.

Dedicated to Mel Hochster on the occasion of his sixty-fifth birthday.

Abstract Let R be a real closed field and A = R[x_{1}, …, x_{n}]. Let Sper A denote the real spectrum of A. There are two kinds of points in Sper A: finite points (those for which all of |x_{1}|,…,|x_{n}| are bounded above by some constant in R) and points at infinity. In this paper we study the structure of the set of points at infinity of Sper A and their associated valuations. Let T be a subset of \{1, …, n\}. For j ∈ \{1, …, n\}, let y_{j} = x_{j} if j∈/T and y_{j} = \frac{1}{x_{j}} if j ∈ T. Let B_{T} = R[y_{1}, …, y_{n}]. We construct a finite partition Sper A = ∐_{T} U_{T} and a homeomorphism of each of the sets U_{T} with a subspace of the space of finite points of Sper B_{T}. For each point δ at infinity in U_{T}, we describe the associated valuation ν_{δ^{∗}} of its image δ^{∗} ∈ Sper B_{T} in terms of the valuation ν_{δ} associated to δ. Among other things we show that the valuation ν_{δ^{∗}} is composed with ν_{δ} (in other words, the valuation ring R_{δ} is a localization of R_{δ^{∗}} at a suitable prime ideal).

1 Introduction

Let R be a real closed field and z_{0}, …, z_{n} independent variables. A basic fact of life in mathematics is the way the n-dimensional projective space Proj R[z_{0}, …, z_{n}] and other rational projective schemes such as (ℙ_{R}^{1})^{n} are glued together from affine charts of the form Spec R[x_{1}, …, x_{n}]. Given two such coordinate charts Spec R[x_{1}, …, x_{n}] and Spec R[y_{1}, …, y_{n}], it is often easy to write down formulae describing the coordinate transformation from the x to the y coordinates. The subject of this paper is a part of the analogous story for real spectra (see Definition 1.1 below), which is more interesting, because the real spectrum Sper R[x_{1}, …, x_{n}] already contains much information “at infinity”.

To explain this in more detail, we first recall the definition of real spectrum and other related objects, studied in this paper.

Notation and conventions. All the rings in this paper will be commutative with 1. For a prime ideal 𝔭 in a ring B, κ(𝔭) will denote the residue field of the local ring B_{𝔭}: κ(𝔭) = \frac{B_{𝔭}}{𝔭B_{𝔭}}.

Let B be a ring. A point α in the real spectrum of B is, by definition, the data of a prime ideal 𝔭 of B, and a total ordering ≤ of the quotient ring B/𝔭, or, equivalently, of the field of fractions of B/𝔭. Another way of defining the point α is as a homomorphism from B to a real closed field, where two homomorphisms are identified if they have the same kernel 𝔭 and induce the same total ordering on B/𝔭.

The ideal 𝔭 is called the support of α and denoted by 𝔭_{α}, the quotient ring B/𝔭_{α} by B[α], its field of fractions by B(α) and the real closure of B(α) by k(α). The total ordering of B(α) is denoted by ≤_{α}. Sometimes we write α = (𝔭_{α}, ≤_{α}).

Definition 1.1 The real spectrum of B, denoted by Sper B, is the collection of all pairs α = (𝔭_{α}, ≤_{α}), where 𝔭_{α} is a prime ideal of B and ≤_{α} is a total ordering of B/𝔭_{α}.

Given a point δ ∈ Sper(A) and an element f ∈ A, the notation |f|_{δ} will mean f if f ≥_{δ} 0, −f if f ≤_{δ} 0. When no confusion is possible, we will write simply |f|, with δ understood.

Two kinds of points occur in Sper B: finite points and points at infinity.

Definition 1.2 Let B be an R-algebra and α a point of Sper B. We say that α is finite if for each y ∈ B[α] there exists N ∈ R such that |y|_{α} <_{α} N. Otherwise, we say that α is a point at infinity.

Notation: The subset of Sper B consisting of all the finite points will be denoted by Sper^{∗}B.

It is known, as we explain in detail in §2, that Sper B is closely related to the space ⋃_{𝔭∈Spec B} S_{𝔭}, where S_{𝔭} denotes the Zariski–Riemann surface of the residue field κ(𝔭). Namely,

one can associate to every point δ ∈ Sper B a valuation ν_{δ} of κ(𝔭_{δ}) (where 𝔭_{δ} is the support of δ) with totally ordered residue field k_{δ}. Conversely, given a prime ideal 𝔭 ⊂ B and a valuation ν of κ(𝔭) with totally ordered residue field, one can define a point δ ∈ Sper R[x_{1}, …, x_{n}] with 𝔭_{δ} = 𝔭 and ν_{δ} = ν by specifying the signs of finitely many elements of κ(𝔭) with respect to the total ordering ≤_{δ} (see Remark 2.2 below).

The real spectrum Sper B is endowed with the spectral (or Harrison) topology. By definition, this topology has basic open sets of the form

U(f_{1}, …, f_{k}) = {α | f_{1}(α) > 0, …, f_{k}(α) > 0}

with f_{1}, ..., f_{k} ∈ B. Here and below, we commit the following standard abuse of notation: for an element f ∈ B, f(α) stands for the natural image of f in B[α] and the inequality f(α) > 0 really means f(α) >_{α} 0.

Denote by Maxr(A) the set of points α ∈ Sper(A) such that 𝔭_{α} is a maximal ideal of A. We view Maxr(A) as a topological subspace of Sper(A) with the spectral (respectively, constructible) topology. We may naturally identify R^{n} with Maxr(A): a point (a_{1}, …, a_{n}) ∈ R^{n} corresponds to the point α = (𝔭_{α}, ≤_{α}) ∈ Sper(A), where 𝔭_{α} is the maximal ideal

𝔭_{α} = (x_{1} − a_{1}, …, x_{n} − a_{n})

and ≤_{α} is the unique order on R. The spectral topology on Sper(A) induces the euclidean topology on R^{n}.

Let A = R[x_{1}, …, x_{n}]. Take a point δ ∈ Sper A. In §3 we associate to δ three disjoint subsets I_{δ}, F_{δ}, G_{δ} ⊂ {1, …, n}, as follows. By definition, the set I_{δ} ∐ F_{δ} ∐ G_{δ} is the set of all j ∈ {1, …, n} such that

ν_{δ}(x_{j}) = 0.  (1)

The set G_{δ} consists of all j such that |x_{j}|_{δ} is bounded below by all the elements of R, I_{δ}—of all j such that (1) holds and |x_{j}|_{δ} is smaller than any strictly positive constant in R and F_{δ} of all j such that |x_{j}|_{δ} is bounded both above and below by strictly positive constants from R. We show that I_{δ} = ∅ whenever G_{δ} = ∅.

Let T be a set such that G_{δ} ⊂ T ⊂ G_{δ} ∪ F_{δ}. For j ∈ {1, …, n}, let y_{j} = x_{j} if j∈/T and y_{j} = \frac{1}{x_{j}} if j ∈ T. Let B_{T} = R[y_{1}, …, y_{n}]. We associate to δ a point δ^{∗} in Sper^{∗}B_{T} such that A(δ) = B_{T} (δ^{∗}). We show that R_{δ} is a localization of R_{δ^{∗}} at a prime ideal.

Let I, F, G be three disjoint subsets of {1, …, n}, such that if G = ∅ then I = ∅. Let U_{I,F,G} denote the set of all points of Sper A such that I = I_{δ}, F = F_{δ} and G = G_{δ}. The main theorem, Theorem 3.1, describes a homeomorphism between U_{I,F,G} and a certain explicitly described subspace U_{I,F,G}^{∗} ⊂ Sper^{∗}(B_{T}), where T is a set satisfying G ⊂ T ⊂ F ∪ G. At the end of section §3, we describe a partition

Sper(A) = ∐_{I,F,G} U_{I,F,G},  (2)

where I, F, G runs over all the triples of disjoint subsets of {1, …, n} such that I = ∅ whenever G = ∅, and each U_{I,F,G} is homeomorphic to a subspace U_{I,F,G}^{∗} ⊂ Sper^{∗}(B_{T}), as above.

This paper originally grew out of the authors’ joint work with J.J. Madden [7] on the Pierce– Birkhoff conjecture. Certain definitions and constructions only worked for finite points of Sper A, so a need naturally arose to cover SperA by subspaces, each of which is homeomorphic to a subspace of Sper^{∗}B for some other polynomial ring B. Eventually, we found another way of getting around this difficulty and were able to deal in a uniform way with all the points of Sper A, whether finite or infinite. However, we hope that the decomposition (2) may some day come in useful to someone who is faced with finiteness problems similar to ours. Also, since in [7] we are interested in proving connectedness of certain subsets of Sper A, we gave a variation of the decomposition (2) into sets which are not disjoint; we derive it as an easy consequence of (2).

2 The valuation associated to a point in the real spectrum

Let B be a ring and α a point in Sper B. In this section we define the valuation ν_{α} of B(α), associated to α. We also give a geometric interpretation of points in the real spectrum as semi-curvettes.

Terminology: If B is an integral domain, the phrase “valuation of B” will mean “a valuation of the field of fractions of B, non-negative on B”. Also, we will sometimes commit the following abuse of notation. Given a ring B, a prime ideal 𝔭 ⊂ B, a valuation ν of \frac{B}{𝔭} and an element x ∈ B, we will write ν(x) instead of ν(x mod 𝔭), with the usual convention that ν(0) = ∞, which is taken to be greater than any element of the value group.

For a point α in Sper B, we define the valuation ring R_{α} by

R_{α} = {x ∈ B(α) | ∃z ∈ B[α], |x|_{α} ≤_{α} z}.

That R_{α} is, in fact, a valuation ring, follows because for any x ∈ B(α), either x ∈ R_{α} or \frac{1}{x} ∈ R_{α}. The maximal ideal of R_{α} is M_{α} = \{x ∈ B(α) | |x|_{α} < \frac{1}{|z|_{α}} , ∀z ∈ B[α] \ \{0\}\}; its residue field k_{α} comes equipped with a total ordering, induced by ≤_{α}. For a ring B let U(B) denote the

multiplicative group of units of B. Recall that Γ_{α} ≅ \frac{B(α) \ \{0\}}{U(R_{α})} and that the valuation ν_{α} can

be identified with the natural homomorphism

B(α) \ \{0\} → \frac{B(α) \ \{0\}}{U(R_{α})}.

By definition, we have a natural ring homomorphism

B → R_{α}  (3)

whose kernel is 𝔭_{α}. The valuation ν_{α} has the following properties:

(1) ν_{α}(B[α]) ≥ 0

(2) If B is an R-algebra then for any positive elements y, z ∈ B(α),

ν_{α}(y) < ν_{α}(z) ⟹ y > Nz, ∀N ∈ R  (4)

(an example at the end of the paper shows that the converse implication in (4) is not true in general).

Remark 2.1 Let B be an R-algebra and take a point α ∈ Sper^{∗}B (see Definition 1.2). Then

R_{α} = {x ∈ B(α) | ∃N ∈ R, |x| ≤_{α} N } .  (5)

Thus for points α ∈ Sper^{∗}B the valuation ν_{α} of B(α) depends on the ordering ≤_{α} but not on the ring B[α] (this means that given another R-algebra B̃, a point α̃ ∈ Sper^{∗}B̃ and an orderpreserving isomorphism ϕ : B(α) ≅ B̃(α̃), we have ϕ(R_{α}) = Rα̃).

Remark 2.2 ([1], [6], [2] 10.1.10, p. 217) Conversely, the point α can be reconstructed from the ring R_{α} by specifying a certain number of sign conditions (finitely many conditions when B is noetherian), as we now explain. Take a prime ideal 𝔭 ⊂ B and a valuation ν of κ(𝔭) := \frac{B_{𝔭}}{𝔭B_{𝔭}}, with value group Γ. Let

r = dim_{𝔽_{2}}(Γ/2Γ)

(if B is not noetherian, it may happen that r = ∞). Let x_{1}, …, x_{r} be elements of κ(𝔭) such that ν(x_{1}), …, ν(x_{r}) induce a basis of the 𝔽_{2}-vector space Γ/2Γ. Then for every x ∈ κ(𝔭), there exist f ∈ κ(𝔭) and a unit u of R_{ν} such that x = ux_{1}^{ϵ_{1}} ⋯ x_{r}^{ϵ_{r}}f^{2} with ϵ_{i} ∈ {0, 1} (to see this, note that for a suitable choice of f and ϵ_{j} the value of the quotient u of x by the product x_{1}^{ϵ_{1}} ⋯ x_{r}^{ϵ_{r}}f^{2} is 0, hence u is invertible in R_{ν}). Now, specifying a point α ∈ Sper B supported at 𝔭 amounts to specifying a valuation ν of \frac{B}{𝔭}, a total ordering of the residue field k_{ν} of R_{ν}, and the sign data sgn x_{1}, …, sgn x_{r}. For x∈/𝔭, the sign of x is given by the product sgn(x_{1})^{ϵ_{1}} ⋯ sgn(x_{r})^{ϵ_{r}}sgn(u), where sgn(u) is determined by the ordering of k_{ν}.

Points of Sper B admit the following geometric interpretation (we refer the reader to [3], [4], [8], p. 89 and [9] for the construction and properties of generalized power series rings and fields).

Definition 2.1 Let k be a field and Γ an ordered abelian group. The generalized formal power series ring k [[t^{Γ}]] is the ring formed by elements of the form ∑_{γ} a_{γ}t^{γ}, a_{γ} ∈ k such that the set

{γ | a_{γ} ≠ 0} is well ordered.

The ring k [[t^{Γ}]] is equipped with the natural t-adic valuation v with values in Γ, defined by v(f) = inf{γ | a_{γ} ≠ 0} for f = ∑_{γ} a_{γ}t^{γ} ∈ k [[t^{Γ}]]. Specifying a total ordering on k and

dim_{𝔽_{2}}(Γ/2Γ) sign conditions defines a total ordering on k [[t^{Γ}]]. In this ordering |t| is smaller than any positive element of k. For example, if t^{γ} > 0 for all γ ∈ Γ then f > 0 if and only if a_{v(f)} > 0.

For an ordered field k, let k̄ denote the real closure of k. The following result is a variation on a theorem of Kaplansky ([4], [5]) for valued fields equipped with a total ordering.

Theorem 2.1 ([9], p. 62, Satz 21) Let K be a real valued field, with residue field k and value group Γ. There exists an injection K ↪ k̄ ((t^{Γ})) of real valued fields.

Let α ∈ Sper B and let Γ_{α} be the value group of ν_{α}. In view of (3) and the Remark above, specifying a point α ∈ Sper B is equivalent to specifying a total order of k_{α}, a morphism

B[α] → k̄_{α} [[t^{Γ_{α}}]]

and dim_{𝔽_{2}}(Γ_{α}/2Γ_{α}) sign conditions as above.

We may pass to usual spectra to obtain morphisms

Spec (k̄_{α} [[t^{Γ_{α}}]]) → Spec B[α] → Spec B.

In particular, if Γ_{α} = ℤ, we obtain a formal curve in Spec B (an analytic curve if the series are convergent). This motivates the following definition:

Definition 2.2 Let k be an ordered field. A k-curvette on Sper(B) is a morphism of the form

α : B → k [[t^{Γ}]] ,

where Γ is an ordered group. A k-semi-curvette is a k-curvette α together with a choice of the sign data sgn x_{1},..., sgn x_{r}, where x_{1}, ..., x_{r} are elements of B whose t-adic values induce an 𝔽_{2}-basis of Γ/2Γ.

We have thus explained how to associate to a point α of Sper B a k̄_{α}-semi-curvette. Conversely, given an ordered field k, a k-semi-curvette α determines a prime ideal 𝔭_{α} (the ideal of all the elements of B which vanish identically on α) and a total ordering on B/𝔭_{α} induced by the ordering of the ring k [[t^{Γ}]] of formal power series. These two operations are inverse to each other. This establishes a one-to-one correspondence between semi-curvettes and points of Sper B.

Below, we will sometimes describe points in the real spectrum by specifying the corresponding semi-curvettes.

Example: Consider the curvette R[x, y] → R[[t]] defined by x ↦ t^{2}, y ↦ t^{3}, and the semicurvette given by declaring, in addition, that t is positive. This semi-curvette is nothing but the upper branch of the cusp.

Later in the paper, we will need, for a certain number p ∈ {0, 1, …, n} and two points δ, δ^{∗} living in different spaces, to compare (n − p)-tuples of elements such as (ν_{δ}(x_{p+1}), …, ν_{δ}(x_{n})) ∈ Γ_{δ}^{n−p} and (ν_{δ^{∗}}(y_{p+1}), …, ν_{δ^{∗}}(y_{n})) ∈ Γ_{δ^{∗}}^{n−p} and to be able to say that they are in some sense “equivalent”. To do this, we need to embed Γ_{δ} in some “universal” ordered group.

Notation and convention: Let us denote by Γ the ordered group ℝ_{lex}^{n}. This means that elements of Γ are compared as words in a dictionary: we say that (a_{1}, …, a_{n}) < (a′_{1}, …, a′_{n}) if and only if there exists j ∈ {1, …, n} such that a_{q} = a′_{q} for all q < j and a_{j} < a′_{j}.

The reason for introducing Γ is that by Abhyankar’s inequality we have rank ν_{δ} ≤ dim A = n for all δ ∈ Sper A, so the value group Γ_{δ} of ν_{δ} can be embedded into Γ as an ordered subgroup (of course, this embedding is far from being unique). Let Γ_{+} be the semigroup of non-negative elements of Γ.

Fix a strictly positive integer ℓ. In order to deal rigourously with ℓ-tuples of elements of Γ_{δ} despite the non-uniqueness of the embedding Γ_{δ} ⊂ Γ, we introduce the category OGM(ℓ), as follows. An object in OGM(ℓ) is an ordered abelian group G together with ℓ fixed generators a_{1}, …, a_{ℓ} (such an object will be denoted by (G, a_{1}, …, a_{ℓ})). A morphism from (G, a_{1}, …, a_{ℓ}) to (G′, a′_{1}, …, a′_{ℓ}) is a homomorphism G → G′ of ordered groups which maps a_{j} to a′_{j} for each j.

Given (G, a_{1}, …, a_{ℓ}), (G′, a′_{1}, …, a′_{ℓ}) ∈ Ob(OGM(ℓ)), the notation

(a_{1}, …, a_{ℓ}) ∼_{∘} (a′_{1}, …, a′_{ℓ})  (6)

will mean that (G, a_{1}, …, a_{ℓ}) and (G′, a′_{1}, …, a′_{ℓ}) are isomorphic in OGM(ℓ).

Take an element

a = (a_{1}, …, a_{ℓ}) ∈ Γ_{+}^{ℓ}.

Let G ⊂ Γ be the ordered group generated by a_{1}, …, a_{ℓ}. Then (G, a_{1}, …, a_{ℓ}) ∈ Ob(OGM(ℓ)). For each δ ∈ Sper(A), let Γ_{δ} denote the value group of the associated valuation ν_{δ} and Γ_{δ}^{∗} the subgroup of Γ_{δ} generated by ν_{δ}(x_{1}), …, ν_{δ}(x_{n}). In this way, we associate to δ an object (Γ_{δ}^{∗}, ν_{δ}(x_{1}), …, ν_{δ}(x_{n})) ∈ Ob(OGM(n)).

Notation. Let Γ be an ordered group. Consider an ℓ-tuple a = (a_{1}, …, a_{ℓ}) ∈ Γ^{ℓ}. We denote by Rel(a) the set

Rel(a) = {(m_{1}, …, m_{ℓ}, m_{ℓ+1}, …, m_{2ℓ}) ∈ ℤ^{2ℓ} | ∑_{j=1}^{ℓ} m_{j}a_{j} > 0 and ∑_{j=ℓ+1}^{2ℓ} m_{j}a_{j−ℓ} = 0} .

Remark 2.3 Let Γ and a be as above and let G be the subgroup of Γ generated by a_{1}, …, a_{ℓ}, so that (G, a_{1}, …, a_{ℓ}) ∈ Ob(OGM(ℓ)). The set Rel(a) completely determines the isomorphism class of (G, a_{1}, …, a_{ℓ}) in OGM(ℓ) and vice-versa; the set Rel(a) and the isomorphism class of (G, a_{1}, …, a_{ℓ}) are equivalent sets of data.

3 Points at infinity of Sper(A)

In this section, we study the structure of the set of points at infinity in Sper(A). Take a point δ ∈ Sper(A). Renumbering the coordinates if necessary, we may assume there exists p, 0 ≤ p ≤ n, such that

ν_{δ}(x_{j}) = 0 for 1 ≤ i ≤ p and ν_{δ}(x_{j}) > 0 for j > p.  (7)

For a subset T of {1, …, p}, let B_{T} = R[y_{1}, …, y_{n}] where

y_{j} = x_{j}  if j ∈ {1, …, n} \ T  (8)
= 1/x_{j} if j ∈ T .  (9)

For certain subsets T ⊂ {1, …, p} we will associate to δ a point δ^{∗} in Sper^{∗}(B_{T}) such that A(δ) = B_{T} (δ^{∗}). We will define a new valuation ν_{δ^{∗}} of A(δ), such that R_{δ} is a localization of R_{δ^{∗}} at a suitable prime ideal. At the end of this section, we will use these results to cover Sper(A) by sets, each of which is homeomorphic to a certain subspace of Sper^{∗}(B_{T}) for some T.

First, take any subset T ⊂ {1, …, n} whatsoever. Let B_{T} be defined as in (8)–(9).

Notation : The notation A_{f} stands for the localization of A by f, the ring A[1/f].

Remark : We have a natural homeomorphism

Sper(A) \ {f = 0} →^{∼} Sper(A_{f})  (10)

Consider the natural isomorphism A_{∏ x_{j},j∈T} ≅ (B_{T})_{∏ y_{j},j∈T}. It induces a homeomorphism

ψ : Sper(A) \ {∏_{j∈T} x_{j} = 0} → Sper(B_{T}) \ {∏_{j∈T} y_{j} = 0}
∥  ∥
Sper(A_{∏_{j∈T} x_{j}})  →  Sper((B_{T})_{∏_{j∈T} y_{j}})

(11)

which we describe explicitly for future reference. Take a point δ ∈ Sper(A_{∏_{j∈T} x_{j}}). We will now describe the point δ^{∗} = ψ(δ) in Sper(B_{T}), as follows. The ideal 𝔭_{δ^{∗}} is the prime ideal of B_{T} such that

𝔭_{δ}A_{∏ x_{j},j∈T} ≅ 𝔭_{δ^{∗}}(B_{T})_{∏ y_{j},j∈T}  (12)

Then (12) implies the existence of a canonical isomorphism

ϕ : κ(𝔭_{δ}) ≅ κ(𝔭_{δ^{∗}}).  (13)

The total order ≤_{δ^{∗}} is the order induced by δ on κ(𝔭_{δ^{∗}}) via the isomorphism (13). This describes ψ; the inverse map ψ^{−1} is described in a completely analogous way.

Here and below, R_{>0} will denote the set of strictly positive elements of R.

Take a δ ∈ Sper(A) and let p be as in (7). We associate to δ a partition

{1, …, p} = I_{δ} ∐ F_{δ} ∐ G_{δ},

as follows:

j ∈ I_{δ} ⟺ |x_{j}|_{δ} <_{δ} ϵ, ∀ϵ ∈ R_{>0}  (14)
j ∈ F_{δ} ⟺ ∃c_{1}, c_{2} ∈ R_{>0} such that c_{1} <_{δ} |x_{j}|_{δ} <_{δ} c_{2}  (15)
j ∈ G_{δ} ⟺ |x_{j}|_{δ} >_{δ} N, ∀N ∈ R.  (16)

Remark 3.1 We have δ ∈ Sper^{∗}(A) if and only if G_{δ} = ∅. Below, we show that in this case necessarily I_{δ} = ∅.

Take a set T such that

G_{δ} ⊂ T ⊂ G_{δ} ∪ F_{δ}.  (17)

Let B_{T} be the ring defined by (8) and (9). It follows from (15), (16) and (17) that x_{j}∈/𝔭_{δ} for j ∈ T. Let δ^{∗} = ψ(δ). It is immediate from the definition that δ^{∗} is finite in Sper(B_{T}).

Proposition 3.1 The valuation ν_{δ^{∗}} of B_{T} (δ^{∗}) associated to δ^{∗} has the following properties: (1) ν_{δ^{∗}}(y_{j}) = 0 for j ∈ F_{δ};

(2) ν_{δ^{∗}}(y_{j}) > 0 for j ∈ I_{δ} ∪ G_{δ};

(3) there exists q ∈ G_{δ} and a strictly positive integer N such that, for all j ∈ I_{δ},

Nν_{δ^{∗}}(y_{q}) > ν_{δ∗}(y_{j}).  (18)

In particular, if I_{δ} ≠ ∅ then G_{δ} ≠ ∅.

(4) The valuation ring R_{δ} is the localization of R_{δ^{∗}} at a prime ideal; this gives rise to a surjective order-preserving group homomorphism ϕ̃ : Γ_{δ^{∗}} → Γ_{δ} whose kernel is an isolated subgroup.

(5) For all j ∈ {1, …, n}, ϕ̃(ν_{δ^{∗}}(y_{j})) = ν_{δ}(x_{j}).

(6) For j ∈ {1, …, p}, ν_{δ^{∗}}(y_{j}) ∈ ker(ϕ̃). In particular, given any j ∈ {1, …, p}, t ∈ {p + 1, …, n} and N′ ∈ ℕ, we have N′ν_{δ^{∗}}(y_{j}) < ν_{δ^{∗}}(y_{t}).

(7) Assume that ν_{δ}(x_{p+1}), …, ν_{δ}(x_{n}) are ℚ-linearly independent. Then

(ν_{δ^{∗}}(y_{p+1}), …, ν_{δ^{∗}}(y_{n})) ∼_{∘} (ν_{δ}(x_{p+1}), …, ν_{δ}(x_{n}))

in OGM(n − p).

Proof : (1) Take j ∈ F_{δ}. We have 1/|y_{j}|_{δ^{∗}} <_{δ^{∗}} c for some c ∈ R by definition of F_{δ} (15). Hence \frac{1}{y_{j}} ∈ R_{δ^{∗}} and the result follows.

(2) Take j ∈ I_{δ} ∪ G_{δ}. By definition of I_{δ} (14), G_{δ} (16) and (17), |y_{j}|_{δ^{∗}} <_{δ^{∗}} ϵ for every ϵ ∈ R_{>0}, so 1/|y_{j}|_{δ^{∗}} >_{δ^{∗}} N for every N ∈ R. By the boundedness of δ^{∗}, for each f ∈ B_{T}, we have |f(y_{1}, …, y_{n})|_{δ^{∗}} <_{δ^{∗}} N′ for some N′ ∈ R. Hence 1/|y_{j}|_{δ^{∗}} >_{δ^{∗}} f(y_{1}, …, y_{n}) for each f ∈ B_{T}, so 1/y_{j}∈/R_{δ^{∗}}. This proves that ν_{δ^{∗}}(y_{j}) > 0.

(3) Take a j ∈ I_{δ}. Since ν_{δ}(x_{j}) = 0, we have 1/x_{j} ∈ R_{δ}. This means that there exists z ∈ A[δ] such that 1/|x_{j}|_{δ} <_{δ} |z|_{δ}. Now, z is a polynomial in the x_{k}, k = 1, …, n, and taking x_{q}, q ∈ G_{δ}, such that |x_{q}|_{δ} ≥_{δ} |x_{k}|_{δ} for all k ∈ G_{δ} (and hence |x_{q}|_{δ} ≥_{δ} |x_{k}|_{δ} for all k ∈ {1, …, n}), there exists N > 0 such that 1/|x_{j}|_{δ} <_{δ} |x_{q}|_{δ}^{N} for all j ∈ I_{δ}, so that |y_{j}|_{δ^{∗}} >_{δ^{∗}} |y_{q}|_{δ^{∗}}^{N}. Then Nν_{δ^{∗}}(y_{q}) ≥ ν_{δ^{∗}}(y_{j}) by equation (4). Replacing N by N + 1, we can make the inequality (18) strict.

(4) It is well known that every homomorphism between two valuation rings having the same field of fractions is a localization at a prime ideal. Thus it is sufficient to show that R_{δ^{∗}} ⊂ R_{δ}. Take f ∈ R_{δ^{∗}}. By definition, this means that |f|_{δ^{∗}} is bounded above by a polynomial in the y_{j} with respect to ≤_{δ^{∗}}, and hence also by a monomial ω in the y_{j}. Then ϕ^{−1}(ω) is bounded above with respect to ≤_{δ} by a monomial in the x_{j}, in which x_{j} with j ∈ T appear with non-positive exponents. Since each \frac{1}{|x_{j}|_{δ}}, j ∈ T, is bounded above by a constant in R, replacing factors of

the form x_{j}^{−γ_{j}}, j ∈ T, γ_{j} ∈ ℕ by a suitable constant in R, we obtain that ϕ^{−1}(f) is bounded above with respect to ≤_{δ^{∗}} by a monomial in y with non-negative exponents. This proves that ϕ^{−1}(f) ∈ R_{δ} as desired.

The last statement of (4) follows immediately by the general theory of composition of val-

uations ([11], Chapter VI, §10, p. 43). Alternatively, recall that Γ_{δ} ≅ \frac{A(δ) \ \{0\}}{U(R_{δ})} and that the

valuation ν_{δ} can be identified with the natural homomorphism

A(δ) \ \{0\} → \frac{A(δ) \ \{0\}}{U(R_{δ})}.

Similarly, ν_{δ^{∗}} can be thought of as

B_{T} (δ^{∗}) \ \{0\} → \frac{B_{T} (δ^{∗}) \ \{0\}}{U(R_{δ^{∗}})} ≅ Γ_{δ^{∗}}.

From the isomorphism ϕ and the inclusion R_{δ^{∗}} ↪ R_{δ}, we obtain a natural surjective homomorphism of ordered groups

ϕ̃: \frac{B_{T} (δ^{∗}) \ \{0\}}{U(R_{δ^{∗}})} → \frac{A(δ) \ \{0\}}{U(R_{δ})}.  (19)

(5) If j∈/T, the fact that ϕ(x_{j}) = y_{j} implies that

ϕ̃(y_{j} mod U(R_{δ^{∗}})) = x_{j} mod U(R_{δ}).

If j ∈ T, we have ϕ(x_{j}) = 1/y_{j}, hence

ϕ̃(ν_{δ^{∗}}(y_{j})) = ν_{δ}(1/x_{j}) = 0 = ν_{δ}(x_{j}).

(6) is an immediate consequence of (5) and the fact that ν_{δ}(x_{1}) = ⋯ = ν_{δ}(x_{p}) = 0.

(7) By Remark 2.3 at the end of the previous section, it suffices to prove that

Rel(ν_{δ^{∗}}(y_{ℓ+1}), …, ν_{δ^{∗}}(y_{n})) = Rel(ν_{δ}(x_{ℓ+1}), …, ν_{δ}(x_{n})).  (20)

The fact that ν_{δ}(x_{p+1}), …, ν_{δ}(x_{n}) are ℚ-linearly independent and (5) of the Proposition imply that so are ν_{δ^{∗}}(y_{p+1}), …, ν_{δ^{∗}}(y_{n}). Hence, using (5) of the Proposition again, for any (n − p)-tuple, (m_{p+1}, …, m_{n}) ∈ ℤ^{n−p}, we have ∑_{i=p+1}^{n} m_{j}ν_{δ}(x_{j}) > 0 if and only if ∑_{j=p+1}^{n} m_{j}ν_{δ^{∗}}(y_{j}) > 0. Together with the linear independence of ν_{δ}(x_{p+1}), …, ν_{δ}(x_{n}) and of ν_{δ^{∗}}(y_{p+1}), …, ν_{δ^{∗}}(y_{n}), this proves the desired equality (20). □

Let G be an ordered group of rank r and ℓ a positive integer. Take ℓ elements a_{1}, …, a_{ℓ} ∈ G. Let (0) = Δ_{r} ⫋ Δ_{r−1} ⫋ ⋯ ⫋ Δ_{0} = G be the isolated subgroups of G. Renumbering the a_{j} if necessary, we may assume that there exist integers i_{0}, i_{1}, …, i_{r} with

ℓ = i_{0} ≥ i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{r} = 0,

such that a_{i_{q+1}}, …, a_{i_{q}} ∈ Δ_{q} − Δ_{q+1} for q ∈ {0, …, r − 1}.

Definition 3.1 We say that a_{1}, …, a_{ℓ} are scalewise ℚ-linearly independent if, for each q ∈

\{0, …, r − 1\}, the images of a_{i_{q+1}}, …, a_{i_{q}} in \frac{Δ_{q}}{Δ_{q+1}} are ℚ-linearly independent.

Remark 3.2 Let the notation be as above and assume that a_{1}, …, a_{ℓ} are scalewise ℚ-linearly independent. Let λ : G → G′ be a homomorphism of ordered groups. Then λ(a_{1}), …, λ(a_{ℓ}) are scalewise ℚ-linearly independent if and only if they are ℚ-linearly independent if and only if all of them are non-zero. This is precisely the form in which we will use scalewise ℚ-linear independence in the sequel.

Fix an integer p ∈ {1, …, n} and two decompositions

{1, …, p} = H ∐ T = I ∐ F ∐ G,  (21)

where I = ∅ whenever G = ∅,

I ⊂ H and  (22)
G ⊂ T.  (23)

Fix an n-tuple (a_{1}, …, a_{n}) ∈ Γ_{+}^{n} such that a_{1} = ⋯ = a_{p} = 0. Let

U_{I,F,G} = \{δ ∈ Sper(A) | \begin{matrix}∀j ∈ I, ∀c ∈ R_{>0}, |x_{j}|_{δ} <_{δ} c \\ ∀j ∈ F ∃c_{1}, c_{2} ∈ R_{>0}, c_{1} <_{δ} |x_{j}|_{δ} <_{δ} c_{2} \\ ∀j ∈ G, ∀N ∈ R, |x_{j}|_{δ} >_{δ} N \\ ν_{δ}(x_{1}) = … = ν_{δ}(x_{p}) = 0 \\ ν_{δ}(x_{p+1}) > 0, …, ν_{δ}(x_{n}) > 0 \end{matrix} \} ,  (24)

U_{I,F,G}^{∗} = \{δ^{∗} ∈ Sper^{∗}(B_{T}) | \begin{matrix}∀j ∈ F ∃c ∈ R_{>0}, |y_{j}|_{δ^{∗}} >_{δ^{∗}} c \\ ∃q ∈ G, N ∈ ℕ s.t. ∀j ∈ I, Nν_{δ^{∗}}(y_{q}) > ν_{δ∗}(y_{j}) \\ ∀j ∈ \{1, …, p\}, ∀t ∈ \{p + 1, …, n\}, ∀N′ ∈ ℕ, \\ N′ν_{δ^{∗}}(y_{j}) < ν_{δ^{∗}}(y_{t}) \\ ν_{δ^{∗}}(y_{j}) > 0 ∀j ∈ I ∪ G \end{matrix} \} ,  (25)

U_{a,I,F,G} = {δ ∈ U_{I,F,G} | (ν_{δ}(x_{1}), …, ν_{δ}(x_{n})) ∼_{∘} (a_{1}, …, a_{n})} ,  (26)
U_{a,I,F,G}^{∗} = {δ^{∗} ∈ U_{I,F,G}^{∗} | (ν_{δ^{∗}}(y_{p+1}), …, ν_{δ^{∗}}(y_{n})) ∼_{∘} (a_{p+1}, …, a_{n})} ,  (27)

U_{H,T} = \{δ ∈ Sper(A) | \begin{matrix}∃c ∈ R, |x_{j}|_{δ} <_{δ} c, ∀j ∈ H \\ ∃ϵ ∈ R_{>0}, |x_{j}|_{δ} >_{δ} ϵ, ∀j ∈ T \\ ν_{δ}(x_{1}) = … = ν_{δ}(x_{p}) = 0 \\ ν_{δ}(x_{p+1}) > 0, …, ν_{δ}(x_{n}) > 0 \end{matrix} \}  (28)

and

U_{H,T}^{∗} = \{δ^{∗} ∈ Sper^{∗}(B) | \begin{matrix}∃q ∈ T, N ∈ ℕ s.t. ∀j ∈ H, Nν_{δ^{∗}}(y_{q}) > ν_{δ∗}(y_{j}) \\ ∀j ∈ \{1, …, p\}, ∀t ∈ \{p + 1, …, n\}, ∀N′ ∈ ℕ, \\ N′ν_{δ^{∗}}(y_{j}) < ν_{δ^{∗}}(y_{t}) \end{matrix} \} . (29)

We view U_{I,F,G}, U_{a,I,F,G} and U_{H,T} (resp. U_{I,F,G}^{∗}, U_{a,I,F,G}^{∗} and U_{H,T}^{∗}) as topological subspaces of Sper(A) (resp. Sper^{∗}(B_{T})) with the spectral topology. Clearly, for each I and G satisfying (21) we have

U_{I,F,G} = ⋃ ; \begin{matrix}a ∈ Γ_{+}^{n} \\ a_{1} = ⋯ = a_{p} = 0 \\ a_{p+1} > 0, …, a_{n} > 0 \end{matrix} U_{a,I,F,G}

and

U_{I,F,G}^{∗} = ⋃ ; \begin{matrix}a ∈ Γ_{+}^{n} \\ a_{1} = ⋯ = a_{p} = 0 \\ a_{p+1} > 0, …, a_{n} > 0 \end{matrix} U_{a,I,F,G}^{∗}

Also, we have

U_{H,T} = ∐ ; \begin{matrix}\{1, …, p\} = I ∐ F ∐ G \\ I ⊂ H, G ⊂ T \end{matrix} U_{I,F,G},  (30)

and

U_{H,T}^{∗} = ∐ ; \begin{matrix}\{1, …, p\} = I ∐ F ∐ G \\ I ⊂ H, G ⊂ T \end{matrix} U_{I,F,G}^{∗}.  (31)

Theorem 3.1 The map ψ which sends δ to δ^{∗}, defined above, induces homeomorphisms

U_{I,F,G} ˜→U_{I,F,G}^{∗}  (32)

and

U_{H,T} ˜→U_{H,T}^{∗} .  (33)

If, in addition, a_{p+1}, …, a_{n} are scalewise ℚ-linearly independent, we also have a homeomorphism

U_{a,I,F,G} ˜→U_{a,I,F,G}^{∗}.  (34)

Proof: To show (32) and (34), we have to prove that

ψ(U_{I,F,G}) ⊂ U_{I,F,G}^{∗}.  (35)
ψ(U_{a,I,F,G}) ⊂ U_{a,I,F,G}^{∗},  (36)
ψ^{−1}(U_{I,F,G}^{∗}) ⊂ U_{I,F,G}  (37)

and

ψ^{−1}(U_{a,I,F,G}^{∗}) ⊂ U_{a,I,F,G}.  (38)

First, take a point δ ∈ U_{I,F,G}. By definitions, we have I_{δ} = I, F_{δ} = F and G_{δ} = G. For all j ∈ F = F_{δ} there exists c ∈ R_{>0} such that |y_{j}|_{δ^{∗}} >_{δ^{∗}} c by definition of F_{δ} and B_{T}. The condition

∀j ∈ {1, …, p}, ∀t ∈ {p + 1, …, n}, ∀N′ ∈ ℕ, N′ν_{δ^{∗}}(y_{j}) < ν_{δ^{∗}}(y_{t})  (39)

is nothing but Proposition 3.1 (6).

By Proposition 3.1 (3), there exist q ∈ G and N ∈ ℕ, N > 0 such that for all j ∈ I we have

Nν_{δ^{∗}}(y_{q}) > ν_{δ∗}(y_{j}).  (40)

By Proposition 3.1 (2), we have ν_{δ^{∗}}(y_{j}) > 0 for all j ∈ I ∪ G. This completes the proof of the inclusion (35).

Next, assume that δ ∈ U_{a,I,F,G} and that a_{p+1}, …, a_{n} are ℚ-linearly independent. The isomorphism

(ν_{δ^{∗}}(y_{p+1}), …, ν_{δ^{∗}}(y_{n})) ∼_{∘} (a_{p+1}, …, a_{n})

is given by Proposition 3.1 (7). This proves the inclusion (36).

To prove the opposite inclusions, take any δ^{∗} ∈ U_{I,F,G}^{∗}. The existence of c, N ∈ R_{>0} such that |x_{j}|_{δ} <_{δ} c for all j ∈ I and

|x_{j}|_{δ} >_{δ} N, for all j ∈ G  (41)

follow from the facts that δ^{∗} is bounded, x_{j} = y_{j} for j ∈ I and x_{j} = 1/y_{j} for j ∈ G. For j ∈ F we have either x_{j} = y_{j} or x_{j} = \frac{1}{y_{j}}, but in both cases the fact that δ^{∗} ∈ U_{I,F,G}^{∗} implies the existence of c_{1}, c_{2} ∈ R_{>0} such that

c_{1} <_{δ} |x_{j}|_{δ} <_{δ} c_{2}.  (42)

To prove the inclusion (37), it remains to prove that

ν_{δ}(x_{1}) = … = ν_{δ}(x_{p}) = 0  (43)

and

ν_{δ}(x_{t}) > 0 for all t ∈ {p + 1, …, n}.  (44)

Equation (43) is equivalent to saying that

1/|x_{j}|_{δ} ∈ R_{δ}  for 1 ≤ j ≤ p.  (45)

First, if j ∈ G, |x_{j}|_{δ} = \frac{1}{|y_{j}|_{δ^{∗}}} is bounded below by a positive constant by (41), hence (45) holds for j ∈ G.

If j ∈ I, the assumed existence of q ∈ G and a positive N ∈ ℕ such that for all j ∈ I we have Nν_{δ^{∗}}(y_{q}) > ν_{δ∗}(y_{j}) implies that |y_{j}|_{δ^{∗}} >_{δ^{∗}} |y_{q}|_{δ^{∗}}^{N} by equation (4), in other words, |x_{j}|_{δ} >_{δ} 1/|x_{q}|_{δ}^{N} or, equivalently, 1/|x_{j}|_{δ} <_{δ} |x_{q}|_{δ}^{N}. This proves (45) for j ∈ I. For j ∈ F, (45) follows from (42). Thus (45) holds for all j ∈ {1, …, p}, which proves (43).

Take an index t ∈ {p + 1, …, n}. To prove (44), it suffices to show that

1/x_{t}∈/R_{δ},  (46)

that is, that 1/|x_{t}|_{δ} is not bounded above (with respect to ≤_{δ}) by any polynomial in x_{1}, …, x_{n}. By the triangle inequality, this is equivalent to saying that 1/|x_{t}|_{δ} is not bounded above by any monomial in x_{1}, …, x_{n}, or, equivalently, by any element of the form cx_{j}^{N} with j ∈ {1, …, n}, N ∈ ℕ and c ∈ R. We prove this last statement by contradiction. Suppose there was an inequality of the form

1/|x_{t}|_{δ} <_{δ} cx_{j}^{N}  (47)

with N ∈ ℕ, c ∈ R and j ∈ {1, …, n}. Since ν_{δ^{∗}}(y_{t}) > 0, we have |y_{t}|_{δ^{∗}} <_{δ^{∗}} ϵ for all positive ϵ ∈ R, so |x_{t}|_{δ} <_{δ} ϵ and 1/|x_{t}|_{δ} > 1/ϵ for all positive ϵ ∈ R. On the other hand, if j ∈ I ∪ {p + 1, …, n}, we have ν_{δ^{∗}}(y_{j}) > 0, hence |x_{j}|_{δ} = |y_{j}|_{δ^{∗}} <_{δ^{∗}} θ for all positive θ ∈ R and if j ∈ F then |x_{j}|_{δ} is bounded above by a constant from R by (42). This proves that j ∈ G in (47).

Now, the hypotheses (39) implies that for any constant d ∈ R and any N′ ∈ ℕ we have d|y_{j}|_{δ^{∗}}^{N′} >_{δ^{∗}} |y_{t}|_{δ^{∗}}, so d/|x_{j}|_{δ}^{N′} >_{δ} |x_{t}|_{δ}, which contradicts (47). This completes the proof of (46) and (44). The inclusion (37) is proved.

Assume that δ^{∗} ∈ U_{a,I,G}^{∗}. To prove the inclusion (38), it remains to prove the isomorphism

(ν_{δ}(x_{1}), …, ν_{δ}(x_{n})) ∼_{∘} (a_{1}, …, a_{n}).  (48)

By Proposition 3.1 (5), (44), the assumed scalewise ℚ-linear independence of a_{p+1}, …, a_{n} and the Remark following Definition 3.1, ν_{δ}(x_{p+1}), …, ν_{δ}(x_{n}) are also scalewise ℚ-linearly independent. Now (48) follows from Proposition 3.1 (7). The inclusion (38) is proved.

Finally, the homeomorphism (33) follows from (32), (30) and (31) by letting I, F, G run over all the triples of disjoint subsets, satisfying (21), (22) and (23), such that I is empty whenever G is empty. □

Of course, Theorem 3.1 is true with {1, …, p} replaced by any other subset of {1, …, n}. In the next Corollary we drop the assumption (21) and let I, F, G run over all the triples of disjoint subsets of {1, …, n} such that I = ∅ whenever G = ∅. Similarly, H, T will run over all the pairs of disjoint subsets of {1, …, n}.

Corollary 3.1 We have finite coverings

Sper A = ∐_{I,F,G} U_{I,F,G},

and

Sper A = ⋃_{H,T} U_{H,T}

For each I, F, G as above, the set U_{I,F,G} is homeomorphic to the subset U_{I,F,G}^{∗} of the set Sper^{∗}B_{G} of finite points of Sper B_{G}. For each H, T as above, the set U_{H,T} is homeomorphic to the subset U_{H,T}^{∗} of the set Sper^{∗}B_{T} of finite points of Sper B_{T}.

Remark 3.3 The assumption of scalewise ℚ-linear independence of a_{p+1}, …, a_{n}, is needed in Theorem 3.1 only for the inclusion (38). The usual ℚ-linear independence is needed for the inclusion (36) and for Proposition 3.1. Although at first glance these assumptions seem rather restrictive, we remark that any point δ ∈ Sper A can be transformed into one for which these assumptions hold by a sequence of blowings up. We refer the reader to Corollary 6.2 of [7] for details. Corollary 6.2 of [7] shows how to achieve usual ℚ-linear independence of a_{p+1}, …, a_{n}, but it also works for scalewise ℚ-linear independence after some minor and obvious modifications.

Example. Let n = 5. Let δ ∈ Sper A be the point given by the following semi-curvette. We let Γ = ℤ_{lex}^{2} and k_{δ} = R(z, w), where z and w are independent variables. Let the total order on k_{δ} be given by the following inequalities:

0 <_{δ} w <_{δ} c <_{δ} z for all c ∈ R_{>0}  (49)
\frac{1}{w^{N}} <_{δ} z  for all N ∈ ℕ.  (50)

As usual, we define the total order on k_{δ} ((t^{Γ})) by declaring t to be positive. Define the map δ : A → k_{δ} ((t^{Γ})) by

δ(x_{1}) = w  (51)
δ(x_{2}) = 1 + t^{(0,1)}  (52)
δ(x_{3}) = z  (53)
δ(x_{4}) = t^{(1,0)}  (54)
δ(x_{5}) = zt^{(1,0)}.  (55)

We have ν_{δ}(x_{1}) = ν_{δ}(x_{2}) = ν_{δ}(x_{3}) = 0,

ν_{δ}(x_{4}) = ν_{δ}(x_{5}) = (1, 0) > 0,  (56)

so p = 3. Moreover, I_{δ} = {1}, F_{δ} = {2}, G_{δ} = {3}. Let T = G_{δ} and let δ^{∗} = ψ(δ) ∈ Sper^{∗}B_{T}. We have Γ_{δ^{∗}} = ℤ_{lex}^{4} and k_{δ^{∗}} = R. The semi-curvette δ^{∗} can be defined by the map

δ^{∗}(y_{1}) = t^{(0,0,0,1)}  (57)
δ^{∗}(y_{2}) = 1 + t^{(0,1,0,0)}  (58)
δ^{∗}(y_{3}) = t^{(0,0,1,0)}  (59)
δ^{∗}(y_{4}) = t^{(1,0,0,0)}  (60)
δ^{∗}(y_{5}) = t^{(1,0,1,0)}.  (61)

In this example, ν_{δ}(x_{4}) and ν_{δ}(x_{5}) are not ℚ-linearly independent (56), and the conclusion of Proposition 3.1 does not hold: we do not have the equivalence

(ν_{δ^{∗}}(y_{4}), ν_{δ^{∗}}(y_{5})) ∼_{∘} (ν_{δ}(x_{4}), ν_{δ}(x_{5})).

Let A′ = R[x′_{1}, x′_{2}, x′_{3}, x′_{4}, x′_{5}]. Consider the map π : A → A′ defined by

π(x_{j}) = x′_{j} for j ∈ {1, 2, 3, 4},  (62)
π(x_{5}) = x′_{4}x′_{5}.  (63)

Let δ′ be the unique preimage of δ under the natural map π^{∗} : Sper A′ → Sper A of the real spectra, induced by π (in the terminology of [7], π is an affine monomial blowing up along the ideal (x_{4}, x_{5}) with respect to δ and δ′ is the transform of δ by π). Explicitly, we have Γ_{δ′} = ℤ_{lex}^{2}, k_{δ′} = R(z, w), as above, and δ is given by the semi-curvette

δ(x_{1}) = w  (64)
δ(x_{2}) = 1 + t^{(0,1)}  (65)
δ(x_{3}) = z  (66)
δ(x_{4}) = t^{(1,0)}  (67)
δ(x_{5}) = z.  (68)

This is an example of the fact that every point δ ∈ Sper A can be transformed, after a sequence Sper A′ → Sper A of affine monomial blowings up with respect to δ, into a point δ′ ∈ Sper A′ such that the non-zero elements of the set {ν_{δ′}(x_{1}), …, ν_{δ′}(x_{n})} are (scalewise) ℚ-linearly independent.

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