Semi-discrete Valuation Rings

• Semi-discrete valuation rings are valuation rings with a discrete, step-wise ordered value group that need not be cyclic.
• They bridge the gap between discrete valuation rings and arbitrary rank-one valuations by preserving a linear ordering of ideals without full Noetherianity.
• They are pivotal in applications ranging from dimension-two local geometry to algebraic K-theory, enabling approximations by complete intersections in positive characteristic.

Semi-discrete valuation rings are valuation rings whose value group retains a discrete step-by-step order without necessarily being cyclic. A common definition, traced in the literature to Zariski–Samuel, is that a valuation ring $V$ with value group $\Gamma$ is semi-discrete if for every $\gamma\in \Gamma$ the set $\{\delta\in\Gamma\mid \delta>\gamma\}$ has a least element; equivalently, $\Gamma$ is order-isomorphic to a sub-ordered group of $\mathbb{Q}$ that is discrete from above at each element, or to a subgroup of $(\mathbb{R},+)$ such that for every $\gamma\in\Gamma$ there is $\varepsilon>0$ with no element of $\Gamma$ in $\Gamma$0. In this sense, semi-discrete valuation rings include discrete valuation rings and also rank-one valuation rings with successor-like jumps but without uniform cyclic behavior (Popescu, 2020).

1. Definition, rank, and terminological scope

Let $\Gamma$1 be a field and $\Gamma$2 a valuation into a totally ordered abelian group, extended by $\Gamma$3. The associated valuation ring is

$\Gamma$4

with maximal ideal

$\Gamma$5

The rank of the valuation is the Krull rank of $\Gamma$6 as an ordered abelian group, and the height of $\Gamma$7 equals the rank of its value group. Rank $\Gamma$8 means that $\Gamma$9 has no nontrivial proper convex subgroup (Popescu, 2020).

Within this hierarchy, a discrete valuation ring is the special case in which $\gamma\in \Gamma$0 is a Noetherian local domain of dimension $\gamma\in \Gamma$1 and $\gamma\in \Gamma$2; equivalently, the maximal ideal is principal. More generally, a discrete rank-one valuation has value group a discrete subgroup of $\gamma\in \Gamma$3, hence again isomorphic to $\gamma\in \Gamma$4. Semi-discrete valuation rings occupy the intermediate position between these cyclic discrete cases and arbitrary rank-one valuations whose value groups may be dense in $\gamma\in \Gamma$5 (Popescu, 2020).

The terminology is not entirely uniform. Some sources reserve “semi-discrete” for value groups that are discrete from above at each element but not necessarily cyclic, while other discussions of rank-one valuation phenomena use adjacent language for broader countable or step-like groups. This variability is itself part of the subject’s history and explains why many papers state the value-group condition explicitly rather than relying solely on the label (Popescu, 2020).

2. Relation to valuation rings, DVRs, Dedekind domains, and Prüfer domains

A valuation ring $\gamma\in \Gamma$6 is characterized by the condition that for every $\gamma\in \Gamma$7, either $\gamma\in \Gamma$8 or $\gamma\in \Gamma$9. Equivalently, its principal ideals are totally ordered by inclusion; in fact every finitely generated ideal of a valuation ring is principal. Every valuation ring is local, and every valuation ring is integrally closed (Aitken, 2021).

Semi-discrete valuation rings share these basic valuation-theoretic features with all valuation rings, but they need not satisfy the extra finiteness conditions that define DVRs. In particular, they inherit localness, the valuation-ring characterization by total order on principal ideals, integrally closedness, and the fact that finitely generated ideals are principal. What generally fails outside the DVR case is the combination of principal maximal ideal, Noetherianity, and the property that every nonzero ideal is a power of a single prime ideal (Aitken, 2021).

This contrast is central in the global theory of one-dimensional domains. Dedekind domains are precisely the Noetherian domains whose localizations at nonzero prime ideals are DVRs. Almost Dedekind domains relax Noetherianity while still requiring $\{\delta\in\Gamma\mid \delta>\gamma\}$0 to be a DVR for every nonzero maximal ideal. Prüfer domains are more general: $\{\delta\in\Gamma\mid \delta>\gamma\}$1 is a valuation ring for every prime ideal $\{\delta\in\Gamma\mid \delta>\gamma\}$2. Semi-discrete valuation rings therefore arise naturally as local models on the Prüfer side when one allows value groups more general than $\{\delta\in\Gamma\mid \delta>\gamma\}$3, while DVRs remain the discrete cyclic local models underlying Dedekind and almost Dedekind domains (Aitken, 2021).

From this perspective, semi-discreteness records how much of the discrete one-dimensional picture survives after Noetherianity is dropped. The ideals remain linearly ordered, but the value group can support many intermediate cuts between the powers that exhaust the ideal theory of a DVR. This suggests that semi-discrete valuation rings are best viewed not as weakened DVRs in an ad hoc sense, but as a structured rank-one class within the much larger category of valuation rings.

3. Value semigroups and dimension-two local geometry

For a valuation $\{\delta\in\Gamma\mid \delta>\gamma\}$4 dominating a two-dimensional Noetherian local domain $\{\delta\in\Gamma\mid \delta>\gamma\}$5, the valuation semigroup

$\{\delta\in\Gamma\mid \delta>\gamma\}$6

is countable and well-ordered of ordinal type $\{\delta\in\Gamma\mid \delta>\gamma\}$7. When $\{\delta\in\Gamma\mid \delta>\gamma\}$8 is a two-dimensional regular local ring, Cutkosky and Pham give a necessary and sufficient criterion for a semigroup $\{\delta\in\Gamma\mid \delta>\gamma\}$9 to occur as $\Gamma$0: there must exist generators $\Gamma$1 with

$\Gamma$2

and finite group indices

$\Gamma$3

such that

$\Gamma$4

When residue field extension data are prescribed, generators $\Gamma$5 of the residue field and extension degrees

$\Gamma$6

enter through the stronger condition

$\Gamma$7

These results describe simultaneously the semigroup and residue-field growth of valuations dominating two-dimensional regular local rings (Cutkosky et al., 2011).

In the discrete rank-one case, the generating sequence is finite and $\Gamma$8 is a numerical semigroup. In non-discrete rank-one situations, the generating sequence is typically infinite. One of the striking examples constructs a semigroup $\Gamma$9 with

$\mathbb{Q}$0

for which any valuation dominating a two-dimensional regular local ring and having semigroup $\mathbb{Q}$1 must have trivial residue field extension. Thus the semigroup alone can force residue-field rigidity (Cutkosky et al., 2011).

These dimension-two classification results are not restricted to semi-discrete valuation rings in the narrow rank-one discrete-from-above sense, but they are highly relevant to the subject because they show how valuation growth is encoded combinatorially. In particular, they make precise the difference between finite-generation phenomena characteristic of DVRs and infinite generating sequences characteristic of broader rank-one valuation behavior.

4. Approximation by complete intersections in characteristic $\mathbb{Q}$2

A major structural theorem due to Popescu states that a valuation ring $\mathbb{Q}$3 containing a perfect field $\mathbb{Q}$4 of characteristic $\mathbb{Q}$5 is a filtered direct limit of complete intersection $\mathbb{Q}$6-algebras under broad hypotheses. In its final form, if either $\mathbb{Q}$7 or $\mathbb{Q}$8 is Henselian, then

$\mathbb{Q}$9

where each $(\mathbb{R},+)$0 is a complete intersection $(\mathbb{R},+)$1-algebra. When the value group is finitely generated and the residue field is contained in $(\mathbb{R},+)$2, the result sharpens to a filtered union of complete intersection subalgebras (Popescu, 2020).

The proof proceeds through valuation-theoretic approximation. Immediate extensions are analyzed via pseudo-convergent sequences, first in simple algebraic and transcendental extensions and then in general. Theorem 17 of the paper shows that any immediate extension $(\mathbb{R},+)$3 is a filtered union of complete intersection $(\mathbb{R},+)$4-algebras, and model-theoretic tools are then used to pass from these immediate-extension statements to the global filtered-direct-limit theorem (Popescu, 2020).

For semi-discrete valuation rings, especially rank-one rings in characteristic $(\mathbb{R},+)$5, this has a direct interpretation. Under the dimension-$(\mathbb{R},+)$6 or Henselian hypotheses, a semi-discrete valuation ring can be approximated by complete intersections even when it is non-Noetherian. The paper notes that, because the rank is $(\mathbb{R},+)$7, the approximating complete intersections are often one-dimensional complete intersections, frequently close to DVRs or local complete intersections in one variable (Popescu, 2020).

This approximation theorem is important conceptually because it connects valuation theory to the homological and geometric toolkit available for complete intersections. It shows that semi-discrete valuation rings in positive characteristic are not isolated pathologies: they admit systematic approximation by finite-type objects with controlled singularity theory.

5. Semi-valuation rings, $(\mathbb{R},+)$8-theory, and Riemann–Zariski spaces

A recent generalization due to Temkin and its use in algebraic $(\mathbb{R},+)$9-theory replaces valuation rings by semi-valuation rings. A semi-valuation ring is a ring $\gamma\in\Gamma$0 equipped with a valuation

$\gamma\in\Gamma$1

such that every zero divisor of $\gamma\in\Gamma$2 lies in $\gamma\in\Gamma$3, and whenever $\gamma\in\Gamma$4, one has $\gamma\in\Gamma$5 in $\gamma\in\Gamma$6. Writing

$\gamma\in\Gamma$7

one gets a Milnor square

$\gamma\in\Gamma$8

in which $\gamma\in\Gamma$9 is the semi-fraction ring and $\varepsilon>0$0 is a valuation ring. If $\varepsilon>0$1, then the semi-valuation ring is literally a valuation ring. Semi-discrete valuation rings therefore appear as the $\varepsilon>0$2 special case of this broader local formalism (Dahlhausen, 2024).

The main theorem in this context is a homotopy invariance statement for algebraic $\varepsilon>0$3-theory. If $\varepsilon>0$4 is a semi-valuation ring whose semi-fraction ring $\varepsilon>0$5 is stably coherent and regular, then the canonical maps

$\varepsilon>0$6

are equivalences, and $\varepsilon>0$7 is also an equivalence. Moreover, if the valuation is non-trivial, then $\varepsilon>0$8 is regular. For a semi-discrete valuation ring $\varepsilon>0$9, this specializes to the known homotopy invariance statements for valuation rings (Dahlhausen, 2024).

The same paper identifies semi-valuation rings as the local models of Temkin’s relative Riemann–Zariski spaces: for every point $\Gamma$0, the stalk $\Gamma$1 is a semi-valuation ring. Under finiteness and regularity assumptions on $\Gamma$2, the resulting $\Gamma$3-theory sheaf on $\Gamma$4 is homotopy invariant. This places semi-discrete valuation rings inside a broader birational-geometric framework in which valuation-type local rings control the stalkwise behavior of $\Gamma$5-theory (Dahlhausen, 2024).

6. Semiring and characteristic-one analogues

Several papers extend valuation-theoretic ideas beyond rings. In the semiring setting, a valuation semiring is defined using a surjective valuation on a semifield of fractions, and a multiplicatively cancellative semiring is a valuation semiring if and only if its ideals are totally ordered by inclusion. A discrete valuation semiring is characterized equivalently as a multiplicatively cancellative principal ideal semiring with a nonzero unique maximal ideal. If the unique maximal ideal is subtractive, then the valuation semiring is integrally closed; in the discrete case, subtractivity is also equivalent to the Gaussian property. The same framework suggests a natural notion of semi-discrete valuation semiring, namely a valuation semiring whose value group is an arbitrary discrete ordered abelian group rather than specifically $\Gamma$6, although that extension is not developed in full (Nasehpour, 2015).

In idempotent and characteristic-one algebra, valuations acquire an especially transparent form. For a unitgenerated idempotent semiring $\Gamma$7, a valuation into $\Gamma$8 is exactly a surjective semiring homomorphism

$\Gamma$9

and valuation subsemirings $\Gamma$00 are in bijection with such valuations. The associated value group is $\Gamma$01, and the ideals of a valuation subsemiring are totally ordered. The paper does not explicitly define discrete or semi-discrete valuation semirings, but it states that the order type of $\Gamma$02 is where discrete, semi-discrete, and higher-rank behavior is encoded, and that the framework encompasses $\Gamma$03, $\Gamma$04, and arbitrary totally ordered abelian groups (Tolliver, 2016).

A complementary tropical viewpoint comes from hyperfield valuations on semirings. Jun’s analysis of $\Gamma$05 shows that the nontrivial geometric valuations used to construct tropical abstract curves all have value group isomorphic to $\Gamma$06, hence are discrete valuations in the semiring sense. These valuations play the role of tropical DVR-valuations in the construction of the abstract curve associated with $\Gamma$07, its relation to $\Gamma$08, and its hyperfield-valued enhancement to the tropical projective line (Jun, 2015).

Taken together, these generalizations show that semi-discrete phenomena are not confined to classical commutative algebra. The ordered structure of the value group, the total ordering of ideals, and the passage from valuation rings to valuation-like local objects persist in semiring, tropical, and characteristic-one settings, even though the precise vocabulary of “semi-discreteness” is not always used explicitly.