Transfer Krull Domain: Structure & Invariants

• Transfer Krull domains are commutative integral domains that admit a transfer homomorphism to a Krull monoid, capturing key factorization properties.
• They generalize classical Krull domains by using block monoids and zero-sum sequences over abelian groups to analyze arithmetic invariants.
• Their study has practical applications in orders, subrings, and non-normal domains, providing a unified framework for factorization theory.

A transfer Krull domain is a commutative integral domain $D$ whose multiplicative monoid of nonzero elements $D^\bullet$ admits a transfer homomorphism to a commutative Krull monoid, often realized as a monoid of zero-sum sequences over a subgroup or subset of an abelian group. This generalizes Krull domains by reflecting their factorization-theoretic properties in broader algebraic contexts, with applications to the study of orders in Dedekind domains, subrings of Krull domains, singular or non-normal domains, and beyond. Transfer Krull domains are central in modern factorization theory, as transfer homomorphisms enable the study of arithmetic invariants through combinatorial and structural methods arising from zero-sum theory in finite abelian groups (Geroldinger et al., 2021, Geroldinger et al., 2018, Rago, 2024, Bashir et al., 13 Feb 2025, Bashir et al., 2021).

1. Fundamental Definitions and the Transfer Homomorphism Principle

Let $D$ be a commutative integral domain, written multiplicatively as $D^\bullet = D \setminus \{0\}$. A transfer homomorphism from a commutative, atomic, cancellative monoid $H$ to another such monoid $B$ is a monoid homomorphism $\theta: H \to B$ that satisfies:

• (T1) $B = \theta(H) B^\times$ and $\theta^{-1}(B^\times) = H^\times$,
• (T2) For $u \in H$, $\theta(u) = bc$ in $B$ implies there exist $v,w \in H$, $\varepsilon \in B^\times$ with $u = vw$, $\theta(v) = b\varepsilon^{-1}$, $\theta(w) = \varepsilon c$.

A domain $D$ is a transfer Krull domain if there exists a (weak) transfer homomorphism from $D^\bullet$ to a Krull monoid $B$, typically a block monoid of zero-sum sequences $\mathcal{B}(G_0)$ for some $G_0$ subset of an abelian group $G$ (Geroldinger et al., 2021, Geroldinger et al., 2018, Bashir et al., 2021).

This property, by construction, generalizes the classical setting: every Krull domain is transfer Krull, but transfer Krull domains form a strictly larger class, encompassing various non-Krull and non-normal rings, non-maximal orders, and domains with restricted divisor theory.

2. Block Monoids and Zero-Sum Sequences

Given an additive abelian group $G$ and a subset $G_0 \subseteq G$, the free abelian monoid $\mathcal{F}(G_0)$ consists of unordered finite sequences (words) on $G_0$. The sum-homomorphism is

$$\sigma: \mathcal{F}(G_0) \rightarrow G, \quad \sigma(g_1 \dots g_\ell) = g_1 + \cdots + g_\ell.$$

The block monoid or zero-sum sequence monoid over $G_0$ is: $$\mathcal{B}(G_0) := \{ S \in \mathcal{F}(G_0) \mid \sigma(S) = 0 \}.$$ $\mathcal{B}(G_0)$ is a commutative Krull monoid; the study of its arithmetical invariants (sets of lengths, catenary degrees, elasticity, delta sets, Davenport constant, etc.) is foundational in transfer Krull theory. The class group and distribution of prime divisors in the original domain correlate with the zero-sum structure of $\mathcal{B}(G_0)$ (Geroldinger et al., 2021, Geroldinger et al., 2018).

3. Structural Characterizations and Root Closure Criteria

Characterizations of when $D$ is transfer Krull often proceed through the root closure $\widetilde D = \{x \in \text{Quot}(D) : x^n \in D$ for some $n \in \mathbb{N}\}$, which is the minimal candidate for a Krull overmonoid through which all transfer homomorphisms factor. Three fundamental cases have explicit criteria:

• $\widetilde D$ is a discrete valuation monoid (DVM):

$D$ is transfer Krull if and only if $D \subseteq \widetilde D$ is inert (every $xy \in D$ with $x, y \in \widetilde D$ can be "corrected" by units to both factors in $D$).

• $\widetilde D$ is factorial:

$D$ is transfer Krull if and only if $$\mathcal{A}(\widetilde D) = \{u\varepsilon : u \in \mathcal{A}(D), \varepsilon \in \widetilde D^\times\}$$ (the atoms of $\widetilde D$ are the images of atoms of $D$ up to units).

• $\widetilde D$ is half-factorial:

$D$ is transfer Krull if and only if $\mathcal{A}(D) \subseteq \mathcal{A}(\widetilde D)$ (Bashir et al., 2021).

No analogous atom-matching or inertness criterion characterizes transfer Krullness in the general case where $\widetilde D$ is Krull but not DVM, factorial, or half-factorial; counterexamples exist among reduced affine monoids (Bashir et al., 2021).

4. Explicit Constructions in Orders, Subrings, and Non-Krull Examples

Transfer Krull domains arise naturally in prominent settings:

• Orders in Dedekind domains ($O \subset R$) with torsion class group:

Characterizations rely on the bijectivity of the Spec map and local valuation behavior of atoms. For $|\text{Cl}(R)| \ge 3$, $O$ is transfer Krull iff $O \cdot R = R$ and all $v_p(u) = 1$ for atoms $u$ and $p \in \text{Spec}(O)$. For $|\text{Cl}(R)| = 2$, specific multiplicity constraints apply and an exceptional non-transfer Krull, half-factorial case arises (Rago, 2024).

• Subrings of Krull rings:

For $R \subset D$ with $D$ Krull and the conductor $(R: D)$ being a maximal $v$-ideal of $R$, $R$ is transfer Krull and inherits all arithmetical invariants of $D$ via transfer homomorphism (Bashir et al., 13 Feb 2025).

• Non-normal domains and rings with zero-divisors:

Heritage extends to regular congruence monoids in Dedekind domains, module-theoretic monoids, and certain Bass orders (Geroldinger et al., 2018).

Notably, transfer Krull domains can be atomic but not completely integrally closed, and are not necessarily Krull; for instance, non-maximal orders in quadratic fields that are not integrally closed yet exhibit full transfer Krull structure (Rago, 2024, Bashir et al., 13 Feb 2025).

5. Arithmetic Consequences and Invariant Transfer

The existence of a transfer homomorphism $D^\bullet \to \mathcal{B}(G_0)$ for appropriate $G_0$ ensures:

• The system of sets of lengths is preserved: $$\mathcal{L}_{D^\bullet}(a) = \mathcal{L}_{\mathcal{B}(G_0)}(\theta(a))$$ for all $a \in D^\bullet$.
• All classical factorization-theoretic invariants are inherited:
  • Elasticity: $\rho(D^\bullet) = \rho(\mathcal{B}(G_0))$; transfer Krull monoids are fully elastic—every rational in $[1, \rho]$ is realized (Geroldinger et al., 2018).
  • Catenary and tame degrees: $c(D^\bullet) \sim c(\mathcal{B}(G_0))$, $t(D^\bullet) \sim t(\mathcal{B}(G_0))$; under suitable class group hypotheses, sets of catenary and tame degrees are intervals or realize prescribed finite subsets.
  • Delta sets and Davenport constants: inherit structure from the target block monoid, with the delta-set $\Delta(D^\bullet)$ and the Davenport constant $D(G_0)$ reflecting group-theoretic properties (Geroldinger et al., 2018, Geroldinger et al., 2021).
  • Structure of sets of lengths: arithmetical progressions with prescribed difference and bound, determined by the class group (Bashir et al., 13 Feb 2025).
• Realization results: For finitely generated abelian groups $G_0$ with suitable prime divisor distribution, any finite set of invariants (catenary degrees, delta sets, tame degrees) can occur as arithmetic invariants in a transfer Krull domain (Geroldinger et al., 2018).

A summary table of arithmetic invariants and their transfer is given below:

Invariant	Transfer-preserving property	Governing object
Sets of lengths	$$\mathcal{L}_{D^\bullet} = \mathcal{L}_{\mathcal{B}(G_0)}$$	Block monoid over $G_0$
Elasticity	$\rho(D^\bullet) = \rho(\mathcal{B}(G_0))$	Block monoid over $G_0$
Catenary/Tame degrees	$c(D^\bullet) \sim c(\mathcal{B}(G_0))$	Block monoid over $G_0$
Delta-set	$\Delta(D^\bullet) = \Delta(\mathcal{B}(G_0))$	Block monoid over $G_0$
Davenport constant	$D(D^\bullet) = D(G_0)$	Abelian group structure

All invariants are computed combinatorially in $\mathcal{B}(G_0)$, reducing questions about nonuniqueness of factorization in non-normal or singular rings to zero-sum problems in finite abelian groups (Geroldinger et al., 2021, Geroldinger et al., 2018, Bashir et al., 13 Feb 2025).

6. Examples and Broader Impact

• Krull domains: Classical Dedekind domains, finitely generated normal domains, and Krull subrings are transfer Krull by definition.
• Cluster algebras: Some are Krull and thus transfer Krull.
• Module-theoretic settings: Monoids of isomorphism classes of modules with semilocal endomorphism rings or in maximal orders are transfer Krull (Geroldinger et al., 2018).

Transfer Krull domains facilitate a unified approach to factorization invariants across wide algebraic contexts. Their structure-theoretic results rely on explicit map constructions and valuation-theoretic characterizations. The reduction of arithmetic to block monoids enables the realization of diverse factorization regimes and a systematic understanding of nonuniqueness phenomena (Geroldinger et al., 2021, Geroldinger et al., 2018, Bashir et al., 13 Feb 2025, Rago, 2024, Bashir et al., 2021).