Extensions of Valuation Domains and Going-up

Abstract

Suppose F is a field with valuation v and valuation domain O_v, E/F is a finite-dimensional field extension, and R is an O_v-subalgebra of E such that F ⋅ R = E and R ∩ F = O_v. It is known that R satisfies LO, INC, GD and SGB over O_v; it is also known that under certain conditions R satisfies GU over O_v. In this paper, we present a necessary and sufficient condition for the existence of such R that does not satisfy GU over O_v. We also present an explicit example of such R that does not satisfy GU over O_v.

1 Introduction

Valuation theory has long been a key tool in commutative algebra, with applications in number theory and algebraic geometry. Several generalizations of the notion of valuation were made throughout the last few decades; the main purpose of these generalizations was to utilize them in the study of noncommutative algebra, especially in division rings. See [6] for a comprehensive survey.

In this paper we distinguish between a domain, which refers to an integral domain, and a ring, which may contain zero divisors and might not be commutative. Throughout this paper, for a valuation u on a field K we denote by O_u the valuation domain of K corresponding to u, by I_u the maximal ideal of O_u, and by Γ_u the value group of u. The symbol ⊂ means proper inclusion and the symbol $\subseteq$ means inclusion or equality. Any unexplained terminology is as in [4].

Let S be a commutative ring and let R be an algebra over S. For subsets $I \subseteq R$ and $J \subseteq S$ we say that I is lying over J if J = {s ∈ S∣s ⋅ 1_R ∈ I}. By abuse of notation, we write J = I ∩ S (even when R is not faithful over S). We recall now from [5, Section 1.2] the definitions of the classical lifting conditions of ring extensions. We say that R satisfies LO (lying over) over S if for all P ∈Spec(S) there exists Q ∈Spec(R) lying over P. We say that R satisfies GD (going-down) over S if for any P₁ ⊂ P₂ in Spec(S) and for every Q₂ ∈Spec(R) lying over P₂, there exists Q₁ ⊂ Q₂ in Spec(R) lying over P₁. We say that R satisfies GU (going-up) over S if for any P₁ ⊂ P₂ in Spec(S) and for every Q₁ ∈Spec(R) lying over P₁, there exists Q₁ ⊂ Q₂ in Spec(R) lying over P₂. We say that R satisfies SGB (strong going-between) over S if for any P₁ ⊂ P₂ ⊂ P₃ in Spec(S) and for every Q₁ ⊂ Q₃ in Spec(R) such that Q₁ is lying over P₁ and Q₃ is lying over P₃, there exists Q₂, with Q₁ ⊂ Q₂ ⊂ Q₃ in Spec(R), lying over P₂. We say that R satisfies INC (incomparability) over S if whenever Q₁ ⊂ Q₂ in Spec(R), we have Q₁ ∩ S ⊂ Q₂ ∩ S.

Let K be a field and let L be an algebraic field extension of K. Let T be a valuation domain of K. It is well known (cf. [3, 13.2]) that there exists a valuation domain of L lying over T. Recall (cf. [3, Corollary 13.5]) that T is called indecomposed in L if there exists a unique valuation domain of L lying over T; otherwise, T is called decomposed in L. Moreover, by [3, Corollary 13.7]), whenever the separable degree of K over L is finite, the number of valuation domains of L that are lying over T is less than or equal to the separable degree of K over L.

Recall from [4] that a quasi-valuation on a ring A is a function $w : A \rightarrow ~M \cup \{ \infty \}$, where M is a totally ordered abelian monoid, to which we adjoin an element $\infty$ greater than all elements of M, and w satisfies the following properties:(B1) $w(0) = \infty$;(B2) w(xy) ≥ w(x) + w(y) for all x,y ∈ A;(B3) $w(x+y) \geq {\min \limits } \{ w(x), w(y)\}$ for all x,y ∈ A.

In [4] we studied quasi-valuations that extend a given valuation on a finite-dimensional field extension. We proved that the prime spectrum of the associated quasi-valuation domain and the prime spectrum of the valuation domain are intimately connected. Along these lines, let F be a field with a non-trivial valuation v and a corresponding valuation domain O_v, let E/F be a finite field extension, and let R be an O_v-subalgebra of E such that FR = E and R ∩ F = O_v; we call such R an O_v-nice subalgebra of E. By [4, Theorem 9.38, statements (2) and (3)], there exists a quasi-valuation w on E extending v on F, such that R = O_w (we call O_w = {x ∈ E∣w(x) ≥ 0} the quasi-valuation domain), and thus R satisfies LO, INC and GD over O_v. Note that if FR≠E, then since FR is a finite-dimensional field extension of F, one can apply the results of [4] replacing E by FR. Moreover, by [4, Theorem 9.38, statement (6)], if there exists a quasi-valuation $w^{\prime }$ on E extending v with $R = O_{w^{\prime }}$ such that $w^{\prime }(E \setminus \{ 0 \})$ is torsion over Γ_v, then R satisfies GU over O_v. In [5] we generalized some of the results that had been proved in [4], and proved that every algebra over a commutative valuation ring satisfies SGB over it (see [5, Theorem 3.8]). We also generalized [4, Theorem 9.38, statement (6)] and gave weaker conditions for a quasi-valuation ring to satisfy GU over a valuation domain. So, focusing on field extensions, one can say that if E/F is a finite field extension, and R is an O_v-nice subalgebra of E, then R satisfies LO, INC, GD, and SGB over O_v. In light of the above discussion, there exists a quasi-valuation w on E extending v on F such that O_w = R; one may suspect that such R also satisfies GU over O_v. We shall see that this is not the case.

In fact, in this paper we answer a question asked in [5, discussion after Corollary 4.26], by characterizing the existence of O_v-nice subalgebras of E that do not satisfy GU over O_v, and presenting an explicit example in which an O_v-nice subalgebra of E does not satisfy GU over O_v.

2 Going-up May Not Apply

Let F be a field with valuation v and corresponding valuation domain O_v; let E/F be a finite-dimensional field extension and let R be an O_v-nice subalgebra of E. In this section we characterize the existence of O_v-nice subalgebras of E that do not satisfy GU over O_v in terms of the decomposability in E of proper overrings of O_v (recall that an overring of O_v is a subring of F, the field of fractions of O_v, that contains O_v). Then, we present an example of an O_v-nice subalgebra R₀ of E that does not satisfy GU over O_v; equivalently, there exists a quasi-valuation w on E extending v on F with FO_w = E and for which O_w does not satisfy GU over O_v. At the end of this section, we discuss the filter quasi-valuation induced by (R₀,v).

Theorem 2.1

Let F be a field with valuation v and corresponding valuation domain O_v, and let E/F be a finite-dimensional field extension. There exists an O_v-nice subalgebra of E that does not satisfy GU over O_v iff there exists a proper overring of O_v that is decomposed in E.

Proof

Assume that there exists a proper overring B of O_v that is decomposed in E; then B = (O_v)_P, for some non-maximal prime ideal P of O_v. Let D₁ and D₂ be two valuation domains of E that are lying over B. It is clear that D₁ and D₂ are incomparable with respect to containment; this fact can be easily deduced by [3, Theorem 6.6] and [3, Proposition 13.1]. By [3, Theorem 13.2] there exists a valuation domain C₁ of E that is lying over O_v. By [3, Theorem 6.6], the set of all overrings of C₁ is totally ordered by inclusion; thus, C₁ cannot be contained in both D₁ and D₂. Without loss of generality, we assume that C₁ is not contained in D₂. Denote by I₁ the maximal ideal of C₁, and by I₂ the maximal ideal of D₂. Let R = C₁ ∩ D₂; then, by [1, Ch. 6, § 7, no. 1, Proposition 2], R has two maximal ideals: I₁ ∩ R and I₂ ∩ R. However, since D₂ is lying over B = (O_v)_P, its maximal ideal is lying over P. So, R has a maximal ideal that does not lie over I_v, the maximal ideal of O_v. In particular, R does not satisfy GU over O_v.

In the other direction, assume that there exists an O_v-nice subalgebra R of E that does not satisfy GU over O_v. So, there exist P₁ ⊂ P₂ in Spec(O_v) and Q₁ ∈Spec(R) lying over P₁, such that there exists no Q₁ ⊂ Q₂ in Spec(R) lying over P₂. By [4, Theorem 9.38, statement (3)], R satisfies LO over O_v; thus, there exists ${Q_{2}}^{\prime }$ in Spec(R) lying over P₂. Again, by [4, Theorem 9.38, statement (3)], R satisfies GD over O_v; thus, there exists ${Q_{1}}^{\prime } \subset {Q_{2}}^{\prime }$ in Spec(R) lying over P₁. It is clear that $Q_{1} \neq {Q_{1}}^{\prime }$. By [3, Corollary 9.7] there exists a valuation domain O_u of E, containing R, and having a maximal ideal I_u such that Q₁ = I_u ∩ R; likewise, there exists a valuation domain O_w of E, containing R, and having a maximal ideal I_w such that ${Q_{1}}^{\prime }=I_{w} \cap R$. Since both Q₁ and ${Q_{1}}^{\prime }$ are lying over P₁, we deduce that I_u and I_w are lying over P₁; thus, O_u and O_w are two different valuation domains of E that are lying over $(O_{v})_{P_{1}}$; i.e., $(O_{v})_{P_{1}}$ is a proper overring of O_v that is decomposed in E. □

Remark 2.2

Note that, by [5, Corollary 4.24], an O_v-nice subalgebra R of E that does not satisfy GU over O_v, is not finitely generated as an algebra over O_v. Moreover, by [4, Theorem 9.38, statement (6)], for any quasi-valuation w on E extending v on F with O_w = R, one has w(E ∖{0}) is not torsion over Γ_v.

We present now an explicit example demonstrating the situation discussed in the previous theorem.

Example 2.3

Let C be a field with Char(C)≠ 2 and let F = C(x,y) denote the field of rational functions with two indeterminates x and y. Let Γ_v denote the group $\mathbb {Z} \times \mathbb {Z}$ with the left to right lexicographic order and let v denote the rank 2 valuation on F defined by

$$v(0)=\infty, \ \ v\left( \sum\limits_{0 \leq i,j \leq k} \alpha_{ij} y^{i} x^{j}\right)=\min \{(i,j) \mid \alpha_{ij} \neq 0\}$$

for every nonzero ${\sum }_{0 \leq i,j \leq k} \alpha _{ij} y^{i} x^{j} \in C[x,y]$, and $v(\frac {f}{g})=v(f)-v(g)$ for every f,g ∈ C[x,y], g≠ 0. Let {0}≠P denote the non-maximal prime ideal of O_v, namely P = yO_v; and denote (O_v)_P, the localization of O_v at P, by $O_{\widetilde {v}}$. Note that $O_{\widetilde {v}}$ is a valuation domain of F of rank 1. Let $E=F[\sqrt {1-y}]$ and let u be a valuation on E extending v; it is well known that there exists such u (see [2, Corollary 14.1.2]). By the fundamental inequality of valuation theory (see [2, Theorem 17.1.5]), there exist either one or two extensions of v to E. Note that

$$\sqrt{1-y} \in O_{u}, \ \ (1+\sqrt{1-y})+(1-\sqrt{1-y})=2 \notin I_{v}$$

and

$$u((1+\sqrt{1-y})(1-\sqrt{1-y}))=u(y)=(1,0).$$

Hence, exactly one of the elements $1+\sqrt {1-y}$ or $1-\sqrt {1-y}$ has u-value (1,0). Now, since the map $a+b\sqrt {1-y} \rightarrow a-b\sqrt {1-y}$, a,b ∈ F, is an automorphism of E, there exist two extensions of v to E. We denote them by u₁ and u₂; where $u_{1}(1+\sqrt {1-y})=(1,0)$, $u_{1}(1-\sqrt {1-y})=(0,0)$ and $u_{2}(1-\sqrt {1-y})=(1,0)$, $u_{2}(1+\sqrt {1-y})=(0,0)$. Since $(\mathbb {Z} \times \mathbb {Z}) / (\{0\} \times \mathbb {Z}) \cong \mathbb {Z}$, we may view $\mathbb {Z}$ as the value group of $\widetilde {v}$. Using the same argument as above, we deduce that $\widetilde {v}$ has two extensions to E. We denote them by $\widetilde {u_{1}}$ and $\widetilde {u_{2}}$; where $\widetilde {u_{1}}(1+\sqrt {1-y})=1$, $\widetilde {u_{1}}(1-\sqrt {1-y})=0$ and $\widetilde {u_{2}}(1-\sqrt {1-y})=1$, $\widetilde {u_{2}}(1+\sqrt {1-y})=0$. It is easy to see that $O_{u_{1}} \nsubseteq O_{\widetilde {u_{2}}}$. Let $R_{0}=O_{u_{1}} \cap O_{\widetilde {u_{2}}}$ and note that R₀ is an O_v-nice subalgebra of E. Using the same reasoning as in the previous theorem, we get that R₀ has two maximal ideals: $I_{u_{1}} \cap R_{0}$ and $I_{\widetilde {u_{2}}}\cap R_{0}$. However, since $O_{\widetilde {u_{2}}}$ is lying over $O_{\widetilde {v}}$, its maximal ideal, $I_{\widetilde {u_{2}}}$, is lying over P. So, R₀ has a maximal ideal that does not lie over I_v.

In view of Remark 2.2 and the previous example, let w_f denote the filter quasi-valuation induced by (R₀,v) (cf. [4, Section 9] for the construction of the filter quasi-valuation); then there exists m ∈ w_f(E ∖{0}) such that for all $n \in \mathbb {N}$, $nm \notin \mathbb {Z} \times \mathbb {Z}$. Recall that a cut m = (m^L,m^R) of Γ_v is a partition of Γ_v into two subsets m^L and m^R, such that, for every α ∈ m^L and β ∈ m^R, α < β. Also recall that w_f(E ∖{0}) is contained in ${\mathscr{M}}({{\varGamma }}_{v})$, where ${\mathscr{M}}({{\varGamma }}_{v})$ is the set of all cuts of Γ_v; ${\mathscr{M}}({{\varGamma }}_{v})$ is called the cut monoid of Γ_v. Let $m_{0} \in {\mathscr{M}}({{\varGamma }}_{v})$ be the cut defined by ${m_{0}}^{L}=\{ \alpha \in \mathbb {Z} \times \mathbb {Z} \mid \alpha \leq (0,z) \text { for some } z \in \mathbb {Z} \}$; clearly, m₀ + m₀ = m₀ and thus m₀ is not torsion over Γ_v. Denote $1+\sqrt {1-y}$ by r₀; we show now that w_f(r₀) = m₀. Note that for all $t \in \mathbb {Z}$, u₁(r₀x^t) = (1,t) > (0,0), and $\widetilde {u_{2}}(r_{0} x^{t}) =0$. Thus, for all $t \in \mathbb {Z}$, r₀x^t ∈ R₀. Moreover, $\widetilde {u_{2}}(r_{0} y^{-1})=-1$; thus, r₀y^− 1∉R₀. More generally, r₀a^− 1 ∈ R₀ for every a ∈ O_v ∖ P and r₀a^− 1∉R₀ for every a ∈ P; in particular, the support of r₀, $S_{r_{0} } =\{ a \in O_{v} \mid r_{0} \in a R_{0} \}$, satisfies: for all $a \in S_{r_{0} }$ there exists $b \in S_{r_{0} }$ with v(b) > v(a). Therefore, by the definition of the filter quasi-valuation, w_f(r₀) = m₀. In addition, (Γ_v,∅) and (∅,Γ_v) are not in w_f(E ∖{0}); indeed, by the definition of the filter quasi-valuation, (∅,Γ_v)∉w_f(E ∖{0}), and it is easy to check that in our case (Γ_v,∅)∉w_f(E ∖{0}). Indeed, otherwise there exists r ∈ R₀ such that for all a ∈ O_v we have $ra^{-1} \in R_{0} \subset O_{u_{1}}$, but then r would have an infinite value by u₁; we note that one can also use [4, Theorem 8.14] to deduce that (Γ_v,∅)∉w_f(E ∖{0}). Moreover, by [4, Definition 1.5 and Lemma 1.6], for every $t \in \mathbb {Z}$, w_f(r₀y^t) = (t,0) + m₀; where, of course,

$$((t,0)+ m_{0})^{L}=\{ \alpha \in \mathbb{Z} \times \mathbb{Z} \mid \alpha \leq (t,z) \text{ for some } z \in \mathbb{Z} \}.$$

Thus, $w_{f} (E \setminus \{ 0 \})={\mathscr{M}}({{\varGamma }}_{v}) \setminus \{ ({{\varGamma }}_{v}, \emptyset ),(\emptyset ,{{\varGamma }}_{v})\}$.

Finally, we show that Example 2.3 easily provides us with an example, in a similar setting as in Example 2.3, of a noncommutative algebra lying over a valuation domain and not satisfying GU.

Example 2.4

Let the notation be as in Example 2.3 and let A = M_n(E). Then M_n(R₀) is an O_v-subalgebra of A such that FM_n(R₀) = A and M_n(R₀) ∩ F = O_v; M_n(R₀) has maximal ideals $M_{n}(I_{u_{1}} \cap R_{0})$ and $M_{n}(I_{\widetilde {u_{2}}} \cap R_{0})$, while obviously $M_{n}(I_{\widetilde {u_{2}}} \cap R_{0})$ is not lying over I_v.