Tensor Telescope Conjecture

• Tensor Telescope Conjecture is a framework that classifies definable ⊗-ideals via Thomason subsets in rigidly-compactly generated tt-categories.
• It integrates homological residue fields and model-theoretic formulations to systematically study support theories and localizing subcategories.
• The conjecture spans various settings, from derived categories of schemes to noncommutative and stack-theoretic contexts, offering structural and geometric insights.

The Tensor Telescope Conjecture concerns the interplay between smashing localizations, compact or dualizable generators, and tensor-triangular structures within a broad array of triangulated and derived categories. Its resolution yields categorical and geometric classifications for localizing subcategories and reveals deep synergies with support theory, residue fields, and the structure of tensor ideals.

1. Definition and Fundamental Formulations

Let $T$ be a big tt-category: a rigidly-compactly generated tensor-triangulated category with a small tensor-triangulated subcategory $T^c$ of compact objects. A definable $\otimes$-ideal $D \subseteq T$ is a full subcategory closed under products, coproducts, pure subobjects, and tensoring by arbitrary objects; equivalently, $$D = \{ X \in T \mid \operatorname{Hom}_T(f,X) = 0\;\forall f \in \Phi\}$$ for some set of maps $\Phi \subset T^c$.

A Thomason subset of $\operatorname{Spc}(T^c)$ is a union of closed subsets with quasi-compact complements. There is a bijection between Thomason subsets and compactly generated definable $\otimes$-ideals, assigning $V \subseteq \operatorname{Spc}(T^c)$ to the ideal $T_V$ such that $$T_V^c = \{ x \in T^c \mid \operatorname{supp}(x) \subseteq V \}$$. The central statement is:

Tensor Telescope Conjecture (TC):

$T$ satisfies (TC) if every definable $\otimes$-ideal is compactly generated; equivalently, every $D \subseteq T$ as above is $T_V$ for some Thomason $V$ (Hrbek, 2023).

2. TC via Homological Residue Fields and Model-Theoretic Characterization

The homological spectrum $\operatorname{Spc}_h(T^c)$ consists of maximal Serre $\otimes$-ideals in $\operatorname{mod}\text{-}T^c$. For each $B \in \operatorname{Spc}_h(T^c)$, one defines a homological residue field $E_B \in T$ via Gabriel localization. $E_B$ is pure-injective, and support theories at this level recover Zariski points and, in the stable homotopy case, Morava primes.

The conjecture can be restated: If $$\varphi\colon \operatorname{Spc}_h(T^c) \to \operatorname{Spc}(T^c)$$ is a homeomorphism (as posited by the Nerves of Steel Conjecture), $T$ satisfies (TC) if and only if, for every Thomason $V$,

$$D_V = \langle E_p \mid p \in \mathrm{clp}(V^c) \rangle$$

is compactly generated, with $\mathrm{clp}(V^c)$ the closed points of $V^c$ (Hrbek, 2023). This model-theoretic formulation bridges tt-geometry with pure-injective module theory and provides a mechanism to track the generation of localizing subcategories.

3. Locality Principles and Schemes

For derived categories of quasi-compact, quasi-separated schemes $X$, i.e., $T = D(X)$, a strong Stalk-Locality Principle (SLP) holds: If a definable $\otimes$-ideal $D \subset T$ restricts to a compactly generated ideal in each stalk $T_p$ (the triangulated subcategory for the Thomason set of primes above $p$), then $D$ is compactly generated.

This allows a reduction to geometric points:

$D(X)$ satisfies (TC) if and only if for each $x \in X$, the residue field $k(x)$ generates $D(\mathcal{O}_{X,x})$ as a definable $\otimes$-ideal: $$D(X)\text{ satisfies (TC)} \iff \forall x \in X:\;\; \langle k(x)\rangle = D(\mathcal{O}_{X,x})$$

(Hrbek, 2023).

This stalk-local property strengthens the well-known affine-locality (local checks on Zariski opens) and enables descent techniques (Antieau, 2013).

4. TC in Noncommutative, Stack-Theoretic, and Path Algebra Settings

• Von Neumann Regular Rings:

For $R$ von Neumann regular, every homological epimorphism is a universal localization, so all smashing tensor-ideal localizing subcategories in $D(R)$ are compactly generated by compacts; hence, the tensor telescope conjecture holds (Zhang, 2021).

• Algebraic Stacks:

If an algebraic stack $\mathcal{X}$ is noetherian and satisfies the Thomason condition (compact generation of $D_{\mathrm{Qcoh}}(\mathcal{X})$ and support detection for closed subsets with quasi-compact complement), then the thick tensor ideals of $\operatorname{Perf}(\mathcal{X})$ are classified by Thomason subsets of $|\mathcal{X}|$, and inflation to smashing tensor ideals in $D_{\mathrm{Qcoh}}(\mathcal{X})$ is a bijection (Hall et al., 2016).

• Path Algebras:

For $R$ commutative noetherian and $Q$ a finite acyclic quiver, the derived category $D(RQ)$, with the vertexwise tensor product, is compactly generated but not rigid. The tensor telescope conjecture still holds, and all homotopically smashing tensor-t-structures are compactly generated, classified by filtrations of specialization-closed subsets of $$\operatorname{Spc}(D^c(RQ)) \cong \operatorname{Spec}(R) \times Q_0$$ (Sabatini, 25 Nov 2025).

5. Counterexamples, Wild Scenarios, and Spectra

In stable module categories for infinite groups beyond the type $\mathrm{FP}_\infty$ case, the telescope conjecture can fail. For infinite free products of finite $p$-groups, compact objects may not be a tensor subcategory, and the category of dualisable objects splits into infinitely many blocks (Stone-Čech compactification phenomena). In these cases, the Balmer spectrum can have cardinality $2^{2^{\aleph_0}}$ and stratification or control by dualisables fails, resulting in the non-surjectivity or non-injectivity of the correspondence between thick tensor ideals and smashing ideals (Kendall, 23 Apr 2025).

Conversely, under strong finiteness conditions (e.g., for $\mathrm{H}_1\mathfrak{F}$-groups of type $\mathrm{FP}_\infty$), the spectrum is Proj $H^*(G;k)$, and (TC) holds (Kendall, 23 Apr 2025).

6. Connections with Adic Topology, Local Rings, and Separation

The telescope conjecture for $D(R)$, with $R$ local, is closely linked to the adic separation of $R$: $$\langle k \rangle = D(R) \iff \widehat{R} \text{ builds all of } D(R)$$ where $\widehat{R} = \varprojlim R/m^n$ is the $m$-adic completion. Necessary and sufficient conditions:

• If $D(R)$ satisfies (TC), then $R$ must be transfinitely separated (descending chains of powers of $m$ stabilize at zero).
• If every localization $R_p$ is purely (transfinitely) separated, $D(R)$ satisfies (TC).
• Existence of a nonzero idempotent ideal in $R$ implies failure of (TC), with Keller’s example as a classical case (Hrbek, 2023).

Explicit constructions produce separated local rings not purely separated where (TC) fails, and zero-dimensional separated local rings where (TC) holds despite the lack of pure separation.

7. T-Structures, Classifications, and Structural Bijections

The telescope conjecture for tensor t-structures asserts that every homotopically smashing tensor-t-structure is compactly generated (Sahoo et al., 2022, Sabatini, 25 Nov 2025). On (separated) noetherian schemes, the classification is bijective: $$\left\{ \begin{array}{c} \text{Thomason filtrations } \phi: \mathbb{Z} \to 2^X \end{array} \right\} \longleftrightarrow \left\{ \begin{array}{c} \text{aisles of compactly generated tensor t-structures on } D(\mathrm{Qcoh}(X)) \end{array} \right\}$$ with $$U_\phi = \{ E \mid \operatorname{Supp} H^i(E) \subseteq \phi(i)\ \forall i \}$$ (Sahoo et al., 2022, Sabatini, 25 Nov 2025). This generalizes to non-rigid settings (e.g., path algebras) and is compatible with classification of thick tensor ideals by specialization-closed subsets of the Balmer spectrum.

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Summary Table: Notions and Classification Correspondences

Setting	TC Holds When	Classification
Big tt-category $T$	All definable $\otimes$-ideals compactly generated	Thomason subsets $\leftrightarrow$ compactly generated definable $\otimes$-ideals (Hrbek, 2023)
Derived category $D(X)$ of scheme $X$	Residue fields generate $D(\mathcal{O}_{X,x})$ for all $x$	Stalk-local criterion (Hrbek, 2023)
Stacks (Perf, $D_{\mathrm{Qcoh}}$)	Thomason condition holds	Thick $\otimes$-ideals $\leftrightarrow$ Thomason subsets (Hall et al., 2016)
Stable module/stable categories	Finiteness/stratification (e.g. $\mathrm{FP}_\infty$ groups)	Balmer spectrum $\leftrightarrow$ Proj $H^*(G;k)$ (Kendall, 23 Apr 2025)
Path algebra $D(RQ)$	Always for noetherian $R$, acyclic $Q$	Filtrations of specialization-closed subsets (Sabatini, 25 Nov 2025)
T-structures	Homotopically smashing $\implies$ comp. generated	Filtrations of Thomason subsets (Sahoo et al., 2022)

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The tensor telescope conjecture thus unifies a broad array of phenomena in tensor-triangular geometry, derived categories, and tt-geometry, linking support theory, model-theoretic residue structures, stratification by Balmer spectra, and structural classification of localizations across both commutative and noncommutative paradigms. Its precise formulation and consequences continue to drive both the structure theory of triangulated categories and applications to algebraic and arithmetic geometry (Hrbek, 2023, Kendall, 23 Apr 2025, Antieau, 2013, Hall et al., 2016, Sahoo et al., 2022, Zhang, 2021, Sabatini, 25 Nov 2025).