Discrete Valuation Ring Criterion

• Discrete Valuation Ring Criterion is a set of valuation-theoretic conditions that uniquely characterize DVRs using uniformizers and nonzero ideals.
• It facilitates integral closure decomposition and singularity detection in both unramified and mixed characteristic algebraic settings.
• The criterion bridges local field theory and algebraic geometry, underpinning techniques in Rees valuations and normalization of rings.

A discrete valuation ring (DVR) is a central object in commutative algebra and algebraic geometry, characterized intrinsically by its valuation-theoretic properties. The discrete valuation ring criterion provides necessary and sufficient conditions for a given domain or subring to be recognized as a DVR, both abstractly and in concrete algebraic constructions. The criterion plays a critical role not only in the structure theory of rings and local fields but also in valuation-theoretic characterizations of integral closures and singularity theory, including in mixed characteristic settings.

1. Discrete Valuations and the Unit Ball Construction

Let $K$ be a field and $v: K \to \mathbb{Z} \cup \{\infty\}$ a discrete valuation (i.e., $v$ is surjective onto $\mathbb{Z} \cup \{\infty\}$ and satisfies $v(xy)=v(x)+v(y)$, $v(x+y)\ge \min\{v(x),v(y)\}$). The subring

$\mathcal{O}_v = \{ x \in K \mid v(x) \ge 0\}$

is called the valuation ring, and its unique maximal ideal is $(\pi) = \{x \in K \mid v(x)>0\}$, where $\pi$ is any element with $v(\pi)=1$, called a uniformizer. Fundamental properties of $\mathcal{O}_v$ are:

• $\mathcal{O}_v$ is a local, Noetherian domain of Krull dimension one,
• its maximal ideal $(\pi)$ is principal,
• every nonzero ideal is generated by a power of $\pi$.

These properties establish that $\mathcal{O}_v$ is a DVR. Conversely, for any DVR $R$ with fraction field $K$, the $\mathfrak{m}$-adic valuation $v_{\mathfrak{m}}$ on $K$ defined via the uniformizer $\pi$ satisfies $R=\{x \in K \mid v_{\mathfrak{m}}(x) \ge 0\}$. This yields the DVR-valuation correspondence: a subring $R$ of $K$ is a DVR if and only if $R$ is the valuation ring for some discrete valuation on $K$, and this valuation is uniquely determined up to scaling the value group (Frutos-Fernández et al., 2023).

2. The Discrete Valuation Ring Criterion

The DVR criterion formalizes this correspondence as an equivalence:

• $R$ is a DVR $\Longleftrightarrow$ there exists a discrete valuation $v: K \to \mathbb{Z} \cup \{\infty\}$ such that

$R = \{x \in K \mid v(x) \ge 0\}$

where $K$ is the field of fractions of $R$. These constructions are mutually inverse: the valuation ring of $v$ is $R$, and the canonical valuation $v_{\mathfrak{m}}$ recovers $v$ up to reindexing. In particular, every DVR admits a unique (up to scaling) discrete valuation for which it is the valuation ring (Frutos-Fernández et al., 2023).

A summary of this equivalence is presented below:

Condition on $R$	Valuation-theoretic property	Consequence
$R$ is a Noetherian local domain, $\dim R = 1$, $\mathfrak{m}$ principal	Existence of discrete valuation $v: K \to \mathbb{Z}\cup\{\infty\}$ such that $R = \mathcal{O}_v$	$R$ is a DVR
Given $v$ discrete on $K$, $R=\mathcal{O}_v$	$R$ is local, Noetherian, integrally closed, $\dim R = 1$	$R$ is a DVR

3. The DVR Criterion in the Valuation Decomposition of Integral Closures

For inclusions $A \subset B$ of integral domains with $A$ Noetherian and $B$ a finitely generated $A$-algebra, the integral closure $A'$ of $A$ in $B$ can be characterized via a minimal intersection of localizations at certain height-one primes. Precisely, the main discrete valuation ring criterion of (Rangachev, 2020) asserts:

Let $A$ be a Noetherian integral domain, $B$ a finitely generated $A$-algebra (also an integral domain), and $A'$ the integral closure of $A$ in $B$. Then, either $A' = B$, or there exists a finite set of uniquely determined discrete valuation rings $V_i = A_{q_i}$, with $q_i$ running over the nonzero associated primes $\operatorname{Ass}_A(B/A)\setminus\{0\}$, such that

$A' = \bigcap_{i=1}^r (V_i \cap B)$

where this intersection is minimal and canonical. Each $V_i$ is a DVR of rank one, corresponding to a divisorial valuation over $A$ in the equidimensional case. This representation is unique: omitting any $V_i$ changes the intersection. The criterion also recovers classical results such as Rees's theorem for the integral closures of ideals, and supplies a valuation-theoretic version of Zariski's Main Theorem (Rangachev, 2020).

4. The Criterion in Mixed Characteristic and Singularity Theory

In the context of singularity detection and the study of hypersurfaces over discrete valuation rings in mixed characteristic (0, $p$), the DVR criterion is foundational for the construction and application of higher-order Jacobian invariants. For $V$ a DVR with uniformizer $\pi$ and residue field $\kappa$, and for $f \in V[[x_1,\ldots,x_n]]$, isolated singularities are detected by:

• constructing suitable Jacobian ideals that incorporate the mixed characteristic structure (using $p$-derivations or, in the ramified case, derivations with respect to $\pi$),
• verifying membership of $p$ or $\pi$ in the radical of the Jacobian ideal, and
• checking that associated length or dimension invariants (Tjurina-type numbers) are finite (Svoray, 20 Dec 2025, KC, 2024).

A key result is that for a ramified DVR $(V,\pi)$, $f$ defines an isolated singularity if and only if $\pi \in \sqrt{J(f)}$ and the dimension

$$\tau^\pi(f) = \dim_{\kappa} \frac{\kappa[[x]]}{J_\pi(f) + \langle \overline{f} \rangle}$$

is finite, where $J_\pi(f)$ is the Jacobian ideal generated by the partials with respect to $x_i$ and $\pi$-derivation (Svoray, 20 Dec 2025, KC, 2024).

5. Applications and Consequences

The DVR criterion underpins several central theoretical results:

• Valuative characterization of integral closures: The integral closure of a Noetherian domain in a finitely generated extension is the canonical intersection of local DVRs corresponding to height-one primes of the module $B/A$ (Rangachev, 2020).
• Rees valuations and blowup algebras: The minimal set of DVRs appearing in the valuation decomposition correspond to Rees valuations controlling integral closures and asymptotic multiplicities of powers of ideals; the formula

$I^n = \bigcap_{i=1}^r (I V_i)^n \cap R$

is a classical example (Rangachev, 2020).

• Singular locus determination: In the mixed characteristic setting, the DVR structure is essential for the formulation and proof of the Jacobian criteria for singular loci, including ramified and unramified cases, unifying and extending standard field-based results (KC, 2024, Svoray, 20 Dec 2025).
• Local field theory: The DVR criterion is formalized in proof assistants and justifies the structure of local fields as completions with respect to a discrete valuation and their classification (Frutos-Fernández et al., 2023).

6. Consequences for Equidimensionality, Uniqueness, and Extensions

If $A$ is locally formally equidimensional, each DVR $V_i$ occurring in the decomposition of $A'$ is divisorial—these are precisely those associated to height-one primes in certain finite covers of $A$ (Rangachev, 2020). The uniqueness aspect is strict: the minimal set of DVRs can neither be enlarged nor omitted without changing the intersection. In the context of algebraic extensions and graded algebra structures, the corresponding analogues of the DVR criterion provide canonical decompositions for homogeneous integral closures and for the integral closures of Rees algebras (Rangachev, 2020).

Finally, the discrete valuation ring criterion serves as a bridge across commutative algebra, algebraic geometry, and valuation theory, allowing the translation of local-to-global principles, singularity theory, and normalization of rings into precise valuation-theoretic statements that admit both computational and conceptual applications.