Krull-Gabriel Dimension

• Krull-Gabriel Dimension is a transfinite ordinal invariant that measures the structural complexity of abelian and functor categories through inductively defined Serre filtrations.
• It bridges classical Krull dimension with modern topological insights by aligning Serre subcategory filtrations with the Cantor-Bendixson rank in Ziegler spectra, influencing representation theory.
• Its applications range from classifying module categories and verifying Auslander's theorem to differentiating algebra types from finite to wild, highlighting its significance in both algebra and model theory.

The Krull-Gabriel dimension is a transfinite ordinal-valued invariant for abelian categories—especially functor categories or module categories over rings and finite-dimensional algebras. It generalizes classical Krull dimension from commutative algebra to broader categorical contexts, measuring the complexity of length filtrations by Serre subcategories and encoding layer-by-layer decomposition via Serre quotients, filtration theory, and connections with the topological properties of pure-injectives (Ziegler spectra). It plays a fundamental classification role in representation theory, model theory of modules, and categorical localization.

1. Formal Definition and Filtration

Let $\mathcal{A}$ be a skeletally small abelian category. A Serre subcategory $\mathcal{S} \subseteq \mathcal{A}$ is closed under subobjects, quotients, and extensions. The Krull-Gabriel filtration is constructed inductively:

• $\mathcal{A}_{-1} = 0$.
• For each successor ordinal $\alpha+1$, $\mathcal{A}_{\alpha+1}$ comprises objects whose image in the quotient abelian category $\mathcal{A}/\mathcal{A}_\alpha$ has finite length (i.e., admits a finite composition series of simple objects).
• For a limit ordinal $\lambda$, set $$\mathcal{A}_\lambda = \bigcup_{\beta<\lambda} \mathcal{A}_\beta$$.

The Krull-Gabriel dimension of $\mathcal{A}$ is the least ordinal $\alpha$ such that $\mathcal{A}_\alpha = \mathcal{A}$, if it exists, and $\infty$ otherwise. For individual objects $X \in \mathcal{A}$, their Krull-Gabriel dimension is $\min \{ n \mid X \in \mathcal{A}_n \}$ (López-Aguayo, 2015). Equivalent formulations occur in terms of the maximal chain of proper Serre subcategories, modular lattice dimension (m-dim), and, via model theory, Cantor-Bendixson rank in Ziegler spectral topology (Laking et al., 2015, López-Aguayo, 2015).

In the context of module categories, or more generally functor categories $\mathcal{F}(A)$ (finitely presented contravariant $K$-linear functors from $\mathrm{mod}(A)$ to $\mathrm{mod}(K)$ for a finite-dimensional algebra $A$), one sets $\mathrm{KG}(A)=\mathrm{KGdim}(\mathcal{F}(A))$ (Sardar, 16 Jan 2026, Erdmann et al., 28 Mar 2025).

2. Topological Connections: Ziegler Spectrum and Atom Spectra

For a ring or algebra $R$, the Ziegler spectrum $\mathrm{Zg}\,R$ is the set of isomorphism classes of indecomposable pure-injective $R$-modules, with a spectral topology defined by vanishing sets of coherent functors (López-Aguayo, 2015). The Krull-Gabriel filtration on functor categories aligns with the Cantor-Bendixson filtration and rank in $\mathrm{Zg}\,R$:

• Points of CB-rank 0: isolated (finite-dimensional) modules.
• Higher ranks measure the removal of layers under successive filtration.
• $\mathrm{KG}(R)$ equals the Cantor-Bendixson rank of $\mathrm{Zg}\,R$.

For Grothendieck categories, atom spectra $\mathrm{ASpec}\,\mathcal{A}$ generalize prime spectra, equipped with a support topology. Filtration strata correspond to open subsets in this topology, and in semi-noetherian locally coherent categories, $\mathrm{ASpec}\,\mathcal{A}$ and Ziegler spectra are homeomorphic, so stratifications coincide (Alipour et al., 2020).

3. Key Theorems and Classification Results

The Krull-Gabriel dimension provides a robust bridge between representation theory and categorical invariants:

• Auslander's theorem: $\mathrm{KG}(A)=0$ if and only if $A$ is of finite representation type (López-Aguayo, 2015, Hiramatsu, 2021, Sardar, 16 Jan 2026, Erdmann et al., 28 Mar 2025).
• Prest's conjecture: $A$ is domestic (including tame hereditary and certain domestic string algebras) if and only if $\mathrm{KG}(A)$ is finite (Sardar, 16 Jan 2026, Erdmann et al., 28 Mar 2025).
• No Artin algebra of dimension 1: $0<\mathrm{KG}(A)<2$ does not occur (López-Aguayo, 2015, Bobinski et al., 2014).
• Classification (typical values):

| Representation type | KG-dim | Example/Reference | |------------------------------------|----------|---------------------------------------| | Finite (representation-finite) | $0$ | (Hiramatsu, 2021, López-Aguayo, 2015, Sardar, 16 Jan 2026) | | Tame hereditary | $2$ | (López-Aguayo, 2015, Laking et al., 2015, Pastuszak, 2018) | | Domestic string algebra (A$_n$) | $n+1$ | (Laking et al., 2015, López-Aguayo, 2015) | | Tubular type (canonical/tubular) | $3$ | (Pastuszak, 2018) | | Wild/algebras of non-domestic type | $\infty$ | (López-Aguayo, 2015, Erdmann et al., 28 Mar 2025, Sardar, 16 Jan 2026) |

Algebras for which the associated filtration never exhausts in finite steps are "wild," with undefined (infinite) Krull-Gabriel dimension (Erdmann et al., 28 Mar 2025).

4. Functor Categories, Covering Theory, and Invariance

Krull-Gabriel dimension is preserved under certain categorical constructions:

• Galois covering functors: If $F:R\to A=R/G$ is a Galois covering between locally bounded $K$-categories (with a torsion-free group $G$ acting freely), then $\mathrm{KG}(R)=\mathrm{KG}(A)$ (Pastuszak, 2018, Pastuszak, 24 Feb 2025). This result extends to various settings, including skew group algebras, repetitive algebras, and cluster-tilted algebras (Sardar, 16 Jan 2026, Jaworska-Pastuszak et al., 2022).
• Matrix subcategories and surjective functors: If $A\to B$ is a surjection or embedding, then $\mathrm{KG}(B)\le\mathrm{KG}(A)$ (Erdmann et al., 28 Mar 2025, Pastuszak, 24 Feb 2025).
• Skew-group algebras and semi-coverings: The Krull-Gabriel dimension of a skew group algebra $\Lambda\,G$ equals that of $\Lambda$, so the classification applies uniformly to both (Sardar, 16 Jan 2026).
• Functorial push-downs, precoverings: Exact faithful functors (e.g., push-down functors induced by coverings) do not increase KG-dimension; under density hypotheses, they are genuine covering functors between functor categories (Pastuszak, 24 Feb 2025).

5. Explicit Computations and Special Cases

String Algebras: For domestic string algebras, the maximal length $n$ of an oriented path in the bridge quiver $B(R)$ determines the dimension as $n+2$, confirming Schröer's conjecture (Laking et al., 2015). All indecomposable pure-injective modules admit explicit rank formulas aligned with CB-rank and KG-dimension stratification.

Weighted Surface and Hybrid Algebras: Weighted surface algebras, except for certain polynomial-growth exceptions, have undefined KG-dimension (i.e., $\infty$). The same holds for non-domestic idempotent and socle-equivalent variants (Erdmann et al., 28 Mar 2025).

Selfinjective Algebras: For standard representation-infinite self-injective algebras of polynomial growth:

• Domestic type: $\mathrm{KG}=2$;
• Tubular type: $\mathrm{KG}=3$;
• Wild: $\mathrm{KG}=\infty$ (Pastuszak, 2018).

Derived Categories: For derived discrete algebras:

• Piecewise hereditary (Dynkin): $\mathrm{KGdim}=0$;
• Non-piecewise hereditary: $\mathrm{KGdim}=1$ if global dimension is infinite, $2$ if finite;
• Wild (non-derived discrete): $\mathrm{KGdim}\ge2$ (Bobinski et al., 2014).

6. Krull-Gabriel Dimension and Categorical/Topological Stratification

For locally noetherian Grothendieck categories, Krull-Gabriel dimension of an object can be phrased as the deviation of the lattice of subobjects. Critical modules and the associated critical composition series stratify objects in layers corresponding to dimensions. Topologically, indecomposable injectives are stratified by their critical dimensions, and specialization chains in the injective spectrum are strictly dimension-decreasing (Gulliver, 2019).

The Cantor-Bendixson rank of the Ziegler spectrum equals the Krull-Gabriel dimension of the functor category (López-Aguayo, 2015, Laking et al., 2015). Open subsets in atom or Ziegler spectra correspond to localizing subcategories, and the filtration stratifies the full category via combinatorial and topological invariants (Alipour et al., 2020).

7. Conjectures, Stable Rank, and Open Questions

Prest's conjecture links domesticity to finiteness of Krull-Gabriel dimension, now verified for multiple classes, including gentle and skew-gentle algebras (Sardar, 16 Jan 2026, Erdmann et al., 28 Mar 2025). Schröer's conjecture relates stable rank (powers at which the radical stabilizes) to KG-dimension: for KG$\,=n\ge2$, the stable rank is either $\omega(n-1)$ or $\omega n$ (Sardar, 16 Jan 2026).

Open questions remain on detailed computation for intermediate tame non-domestic classes, extension to non-finite-dimensional rings, and model-theoretic implications of definable subcategories and their impact on spectral rank and functor categories (López-Aguayo, 2015, Alipour et al., 2020, Erdmann et al., 28 Mar 2025).

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Summary Table: Main Values of Krull-Gabriel Dimension (Selected Classes)

Class	KG-dim	Reference
Finite representation type	$0$	(López-Aguayo, 2015)
Domestic string algebra ($A_n$)	$n+2$	(Laking et al., 2015)
Tame hereditary (Euclidean type)	$2$	(López-Aguayo, 2015)
Canonical/tubular	$3$	(Pastuszak, 2018)
Wild/non-domestic	$\infty$	(Erdmann et al., 28 Mar 2025)

The Krull-Gabriel dimension functions as a refined length invariant, is preserved under various categorical coverings, and enables a model-theoretic and topological approach to module and algebra classification (López-Aguayo, 2015, Pastuszak, 24 Feb 2025, Sardar, 16 Jan 2026, Erdmann et al., 28 Mar 2025).