Geometric Ziegler Spectrum

• Geometric Ziegler spectrum is a topological space of indecomposable g-pure-injective objects in triangulated, exact, and Grothendieck categories, characterized by geometric and model-theoretic data.
• It organizes objects using definable subcategories and functorial vanishing, offering a spectral invariant that aids classification through stratification, Cantor–Bendixson rank, and Krull–Gabriel dimension.
• Applications include support theory in tensor triangulated categories, refining purity concepts and facilitating homological analysis across module, scheme, and singularity frameworks.

The geometric Ziegler spectrum generalizes the classical Ziegler spectrum of module categories to a geometric/topological framework suitable for triangulated, exact, and Grothendieck-categorical contexts. The spectrum encapsulates the isomorphism classes of indecomposable pure-injective objects, classified by geometric and model-theoretic data, and organizes them by a topology arising from definable subcategories or functorial vanishing. Where further structure is available—such as in categories with symmetric monoidal or tensor triangulated structure—one refines purity and the notion of closure to capture localizations (tt-stalks), geometric points, and relations with frames of smashing or tensor ideals. This construction underpins several modern approaches to homological and categorical classification via topological invariants and support theory.

1. Foundational Definitions and Geometric Generalizations

Let $\mathcal{A}$ denote a skeletally small abelian, additive, or rigidly-compactly generated triangulated category. The classical Ziegler spectrum, $\mathrm{Zg}(\mathcal{A})$, consists of isomorphism classes of indecomposable pure-injective objects, equipped with a topology whose basic open sets are determined by either pp-pairs (solutions of systems of coherent or positive-primitive formulas in abelian settings) or, more generally, by the nonvanishing of finitely presented functors arising from compact objects in the category. For schemes $X$, the geometric Ziegler spectrum, $\mathrm{Zg}_X$, is defined for categories such as $\mathrm{Qcoh}(X)$ and their generalizations to enriched or Grothendieck module categories (Garkusha, 2024).

In triangulated settings, one distinguishes between ordinary pure-injectivity and "geometric purity" (g-purity) (Gómez et al., 28 Jan 2026). A triangle in a rigidly-compactly generated tt-category is g-pure if its image in every tt-stalk (localization at a Balmer prime) is pure. The geometric Ziegler spectrum, $\mathrm{GZg}(\mathcal{T})$, is the subspace of indecomposable g-pure-injectives, reflecting local geometric data and support in tt-stalks.

In settings of maximal Cohen–Macaulay module categories, the spectrum $\mathrm{Spc}(R)$ consists of isomorphism classes of non-free, indecomposable MCM modules (Hiramatsu, 2024), and closed sets are defined functorially.

2. Geometric Ziegler Topologies, Localizations, and Spectral Spaces

The topological structure is given by basic opens, typically indexed by functorial data: for a module category, by pp-pairs, and for enrichments, by coherent objects $C$ via $U_{C} = \{ [E] \mid \mathrm{Hom}(C, E) \neq 0 \}$ (Garkusha, 2024). In the setting of schemes or locally coherent Grothendieck categories, basic opens correspond to nonvanishing of coherent functors, and a subbasis is generated by representable functors.

The adjacency and specialization relations reflect underlying categorical or geometric containment: in small abelian categories, point $M$ specializes to $N$ if the Serre-annihilator of $N$ contains that of $M$ (Prest, 2012). Sobriety and spectrality are established via frame-theoretic methods, with the lattice of Serre subcategories anti-isomorphic to the topology's frame of opens and irreducible closed sets corresponding to primes (meet-irreducibles) of the lattice.

For triangulated categories with tensor structure, support theory via Balmer's spectrum and localization at primes provides a stalk-wise realization. Every indecomposable g-pure-injective arises as a pushforward from a pure-injective in a tt-stalk, and the geometric Ziegler spectrum is a quotient of the disjoint union of stalkwise spectra (Gómez et al., 28 Jan 2026). Under suitable local-to-global conditions, closed subsets in the geometric spectrum correspond precisely to definable tt-ideals, providing a spectral, sober space and a bijection with the frame of smashing ideals.

3. Stratification, Cantor–Bendixson Rank, and Krull–Gabriel Dimension

Geometric Ziegler spectra often exhibit stratified topologies governed by Cantor–Bendixson derivatives. In derived-discrete module categories, the spectrum is scattered and of CB-rank 2: the first layer (isolated points) comprises perfect complexes; the second layer (first derivative) comprises one-sided infinite complexes; the third comprises the two-sided infinite (generic) complexes (Arnesen et al., 2016).

Similarly, for Cohen–Macaulay spectra over isolated singularities, points are isolated if and only if corresponding AR-sequences exist; non-isolated points comprise higher layers in the CB stratification, and the rank correlates with representation type (Hiramatsu, 2024, Puninski, 2016). In tube or ray/coray geometries (e.g., Auslander–Reiten quivers), closures involve direct and inverse limits along rays and corays; direct limits yield "Prüfer-like" points, while inverse limits along corays are indecomposable under du-duality or local criteria (Gregory, 2017).

4. Geometric Closure Operators, Functorial Classification, and Exact Structures

Closed subsets are functorially defined: in MCM settings, via vanishing of finitely presented functors; in module or exact categories, via support conditions (Hiramatsu, 2024, Sauter, 2 Jun 2025). In small abelian categories, closed sets correspond bijectively to definable subcategories, reflecting the frame-theoretic anti-isomorphism with the lattice of Serre subcategories (Prest, 2012). For exact categories, every exact structure is classified by a closed subset of a suitable Ziegler subspace, with global dimensions detected by the geometry of closed sets and gaps in the spectrum (Sauter, 2 Jun 2025).

In enriched and Grothendieck categorical contexts, closed embeddings exist for injective or scheme spectra into the geometric Ziegler spectrum, and recollements arise naturally, breaking the spectrum into open and closed pieces corresponding to torsion and torsionfree phenomena (Garkusha, 2024).

5. Geometric Purity, Smashing Ideals, and Support Theory

Geometric purity refines ordinary purity by requiring stalkwise purity at every tt-prime. Geometric pure-injectives form a closed subspace of the Ziegler spectrum under mild conditions (e.g., schemes with finite covers by quasi-compact opens) (Gómez et al., 28 Jan 2026). The interplay between geometric Ziegler spectra and the frame of smashing ideals is mediated by support theory: the frame of smashing tensor ideals becomes spatial when realized by the topology of the geometric spectrum. Counterexamples in model-theoretic approaches are resolved by restricting attention to geometric pure-injectives, ensuring that the spectrum detects all tensor ideals bijectively.

On projective curves, indecomposable geometric pure-injectives correspond to torsion sheaves (supported at closed points), Prüfer and adic sheaves, and the generic field; line bundles, though pure-injective, are not geometric pure-injectives due to possible non-splitting in g-pure triangles (Gómez et al., 28 Jan 2026).

6. Examples, Applications, and Geometric Phenomena

• Tube and Ray/Coray Closures: In module categories over finite-dimensional algebras, the spectrum closes a tube by adjoining colimits along rays, inverse limits along corays, and generic points of finite endolength (Gregory, 2017).
• Cohen–Macaulay Singularities: The Ziegler spectrum for one-dimensional plane singularities stratifies into discrete tiers governed by AR structure and accumulation limits, producing global phenomena such as Ringel's quilt (a Möbius strip realization) (Puninski, 2016, Los et al., 2016).
• Scheme-Theoretic Ziegler Picture: For schemes, the geometric spectrum admits closed embeddings from classical injectives and points of the underlying space, orchestrated via recollement and support theory (Garkusha, 2024).
• Exact Structure Classification: The open support subspace of the Ziegler spectrum is anti-isomorphic to the lattice of exact structures on an idempotent-complete category, with categorical and homological properties detected by the geometry of the spectrum (Sauter, 2 Jun 2025).
• Maximal Cohen–Macaulay Modules: For complete CM local rings, the spectrum is a $T_1$ space, and CB-rank is determined by representation type, with AR sequences carving out the isolated loci (Hiramatsu, 2024).

7. Interrelations, Topological Properties, and Homological Implications

The geometric Ziegler spectrum acts as an invariant for definable subcategories, localizations, and smashing ideals in categorical settings, providing a spectral, sober topological space. The topology encodes geometric, representation-theoretic, and model-theoretic data, harmonizing pp-formulaic, functorial, and support-theoretic perspectives. Classification, stratification, and the interplay with homological dimensions and AR-theory are determined by the geometry of the spectrum and its closure properties. The theory extends from rings and modules to schemes, triangulated, and enriched categories, providing a flexible and powerful geometric apparatus for categorical and homological analysis.