A December 2024 preprint of Torgeir Aambø (arXiv:2412.09391) proves that, for a stable ∞-category C\mathcal{C} with a suitable t-structure, the weak localizing subcategories of the heart C♡\mathcal{C}^{\heartsuit} biject with the π-stable localizing subcategories of C\mathcal{C}, and it extends the affine Gabriel–Neeman–Krause–Takahashi classification picture to homology-closed subcategories of Dqc(X)\mathrm{D}_{qc}(X) for noetherian schemes XX. The paper explicitly records two open questions that remain untouched by later literature. (a) Conjecture 3.15: for every noetherian scheme XX, Krause's coherent subsets of XX should be in bijection with the weak localizing subcategories of QCoh(X)\mathrm{QCoh}(X), generalizing Krause's affine classification beyond the specialization-closed level of Gabriel. (b) Remark 3.47: for a stable ∞-category C\mathcal{C} with t-structure and a weak localizing subcategory A⊆C♡\mathcal{A}\subseteq\mathcal{C}^{\heartsuit}, is the inclusion D(A)⊆AD(C♡)\mathrm{D}(\mathcal{A})\subseteq\mathcal{A}_{\mathrm{D}(\mathcal{C}^{\heartsuit})} inside D(C♡)\mathrm{D}(\mathcal{C}^{\heartsuit}) always an equivalence; equivalently, is D(A)\mathrm{D}(\mathcal{A}) always a π-stable localizing subcategory of D(C♡)\mathrm{D}(\mathcal{C}^{\heartsuit})? The source also flags, as context, the lack of a direct proof that π-exact localizing subcategories of D(R)\mathrm{D}(R) or Dqc(X)\mathrm{D}_{qc}(X) are exactly the smashing ones, tied to the unresolved telescope conjecture for locally noetherian schemes.

Background

In homological algebra and homotopy theory one studies a triangulated or stable category through its localizing subcategories — full subcategories closed under shifts, cones, and arbitrary coproducts (in a stable ∞-category, the natural home for this story, these are the stable full subcategories closed under colimits). Many such categories carry a t-structure, introduced by Beilinson–Bernstein–Deligne: a pair of subcategories (C≥0,C≤0)(\mathcal{C}_{\geq 0},\mathcal{C}_{\leq 0}) equipping C\mathcal{C} with truncation functors and homology objects πn♡X\pi_n^{\heartsuit}X, in direct analogy with the classical homology of a chain complex. The intersection C♡=C≥0∩C≤0\mathcal{C}^{\heartsuit}=\mathcal{C}_{\geq 0}\cap\mathcal{C}_{\leq 0}, called the heart, is an abelian category (for example, the heart of the natural t-structure on the derived category D(R)\mathrm{D}(R) of a ring is the module category ModR\mathrm{Mod}_R), and one may hope to classify localizing subcategories of C\mathcal{C} in terms of subcategories of C♡\mathcal{C}^{\heartsuit}.

The affine commutative-noetherian case of this program is classical and complete. Gabriel (1962) proved that localizing subcategories of ModR\mathrm{Mod}_R for a commutative noetherian ring RR biject with specialization-closed subsets of the prime spectrum Spec R\mathrm{Spec}\,R (subsets closed under taking closures of points); Bökstedt and Neeman (1992) proved that all localizing subcategories of D(R)\mathrm{D}(R) biject with all subsets of Spec R\mathrm{Spec}\,R, so that the smashing localizations — those whose localization functor preserves coproducts, the subject of the telescope conjecture — correspond exactly to the specialization-closed subsets. Krause (2008) filled the middle level: a thick subcategory of ModR\mathrm{Mod}_R closed under direct sums — equivalently, in the terminology below, a weak localizing subcategory — bijects with the coherent subsets of Spec R\mathrm{Spec}\,R, where a subset Φ⊆Spec R\Phi\subseteq\mathrm{Spec}\,R is called coherent if every morphism I0→I1I_0\to I_1 of injective RR-modules whose associated primes lie in Φ\Phi can be completed to an exact sequence I0→I1→I2I_0\to I_1\to I_2 of injectives (exact at I1I_1) with the associated primes of I2I_2 again in Φ\Phi. Takahashi (2009) unified these results into one commutative diagram: homology-closed localizing subcategories of D(R)\mathrm{D}(R), coherent subsets of Spec R\mathrm{Spec}\,R, and direct-sum-closed thick subcategories of ModR\mathrm{Mod}_R all correspond, as do smashing subcategories of D(R)\mathrm{D}(R), specialization-closed subsets, and localizing subcategories of ModR\mathrm{Mod}_R.

Aambø's preprint (arXiv:2412.09391, v1, the only version) transports this picture to general stable ∞-categories. Call a full subcategory of a Grothendieck abelian category a weak Serre subcategory if, in every exact sequence A1→A2→A3→A4→A5A_1\to A_2\to A_3\to A_4\to A_5, membership of A1,A2,A4,A5A_1,A_2,A_4,A_5 forces membership of A3A_3; a weak localizing subcategory is a weak Serre subcategory also closed under coproducts. Call a localizing subcategory L⊆C\mathcal{L}\subseteq\mathcal{C} π-stable if an object XX lies in L\mathcal{L} exactly when all its homology objects πn♡X\pi_n^{\heartsuit}X lie in L♡=L∩C♡\mathcal{L}^{\heartsuit}=\mathcal{L}\cap\mathcal{C}^{\heartsuit} — that is, membership can be checked on the heart. The paper's Theorem A states that for a right-complete t-structure compatible with filtered colimits, π-stable localizing subcategories of C\mathcal{C} biject with weak localizing subcategories of C♡\mathcal{C}^{\heartsuit} via L↦L♡\mathcal{L}\mapsto\mathcal{L}^{\heartsuit}, generalizing Takahashi's correspondence (the affine case, recovered in Corollary 3.13) and earlier bounded-t-structure results of Zhang–Cai; Theorem B upgrades this to genuine localizing subcategories of the heart, using Lurie's theory of prestable ∞-categories (the connective part C≥0\mathcal{C}_{\geq 0}, an ∞-categorical analogue of the category of non-negative complexes). For a noetherian scheme XX — a scheme covered by spectra of noetherian rings — with category QCoh(X)\mathrm{QCoh}(X) of quasi-coherent sheaves and derived category Dqc(X)\mathrm{D}_{qc}(X), Corollary 3.14 then gives a bijection between homology-closed stable localizing subcategories of Dqc(X)\mathrm{D}_{qc}(X) and weak localizing subcategories of QCoh(X)\mathrm{QCoh}(X), and the paper conjectures (Conjecture 3.15) that the latter are classified by the coherent subsets of XX, Gabriel's specialization-closed bijection being the evident restriction. For context, the telescope conjecture is known for commutative noetherian rings and noetherian schemes (via Bökstedt–Neeman and Alonso Tarrío–Jeremías López–Souto Salorio, with Stevenson's support-theoretic formulation; Hall–Rydh proved it for algebraic stacks), and Kanda classified localizing subcategories of QCoh(X)\mathrm{QCoh}(X) for locally noetherian XX by specialization-closed subsets; but, as the author records, the telescope conjecture remains unresolved for locally noetherian schemes. A realization functor, in the sense of Section 3.3 of the paper, is a functor R:D(C♡)→CR:\mathrm{D}(\mathcal{C}^{\heartsuit})\to\mathcal{C} extending the inclusion of the heart; such functors exist quite generally but are rarely equivalences, and their interplay with the π-stable lifts motivates the second open question.

Problem Statement

Resolve the two open questions that Aambø (arXiv:2412.09391, v1) records inside the program of classifying localizing subcategories of stable categories through the hearts of their t-structures.

(a) For every noetherian scheme XX, is there a bijection between the coherent subsets of XX and the weak localizing subcategories of QCoh(X)\mathrm{QCoh}(X) (Conjecture 3.15 of the source)? The source does not formally define 'coherent subset' for a non-affine scheme; the operative definition here is the direct generalization of the affine definition of Krause that the source cites ([Kra08, 3.1], as restated in Takahashi's Definition 2.10): a subset Φ\Phi of the underlying topological space of XX is coherent if every morphism I0→I1I_0\to I_1 of injective quasi-coherent OX\mathcal{O}_X-modules with all associated points of I0I_0 and I1I_1 inside Φ\Phi can be completed to an exact sequence I0→I1→I2I_0\to I_1\to I_2 of injective quasi-coherent OX\mathcal{O}_X-modules (exact at I1I_1; not necessarily a short exact sequence) with the associated points of I2I_2 also inside Φ\Phi; and a weak localizing subcategory of QCoh(X)\mathrm{QCoh}(X) is a full subcategory that is weak Serre (in any exact sequence A1→A2→A3→A4→A5A_1\to A_2\to A_3\to A_4\to A_5 of quasi-coherent sheaves, if A1,A2,A4,A5A_1,A_2,A_4,A_5 belong to the subcategory then so does A3A_3) and closed under arbitrary coproducts. The expected bijection should be compatible with the source's Corollary 3.14 (weak localizing subcategories of QCoh(X)\mathrm{QCoh}(X) ↔ homology-closed stable localizing subcategories of Dqc(X)\mathrm{D}_{qc}(X)) and should restrict to Gabriel's bijection between specialization-closed subsets of XX and localizing subcategories of QCoh(X)\mathrm{QCoh}(X). For affine X=Spec RX=\mathrm{Spec}\,R this is Krause's theorem, so the question is the non-affine generalization; a negative answer must exhibit a specific noetherian scheme XX together with a rigorous demonstration that no such bijection with the expected compatibilities exists for that XX.

(b) Let C\mathcal{C} be a presentable stable ∞-category (a "stable category" in the sense of the source's conventions) equipped with an accessible t-structure that is right-complete and compatible with filtered colimits (so the heart C♡\mathcal{C}^{\heartsuit} is a Grothendieck abelian category; the primary examples are D(R)\mathrm{D}(R) and Dqc(X)\mathrm{D}_{qc}(X) with their natural t-structures), and let A⊆C♡\mathcal{A}\subseteq\mathcal{C}^{\heartsuit} be a weak localizing subcategory of the heart (a weak Serre subcategory closed under arbitrary coproducts; this includes, but is not restricted to, abelian localizing subcategories, i.e. Serre subcategories closed under coproducts). Writing D(−)\mathrm{D}(-) for the derived ∞-category and, for a weak localizing T⊆C♡\mathcal{T}\subseteq\mathcal{C}^{\heartsuit}, writing TD(C♡)\mathcal{T}_{\mathrm{D}(\mathcal{C}^{\heartsuit})} for the unique π-stable localizing subcategory of D(C♡)\mathrm{D}(\mathcal{C}^{\heartsuit}) with heart T\mathcal{T} (a localizing subcategory L\mathcal{L} is π-stable when X∈LX\in\mathcal{L} holds exactly when all homology objects πn♡X\pi_n^{\heartsuit}X lie in L♡\mathcal{L}^{\heartsuit}), the source records an inclusion D(A)⊆AD(C♡)\mathrm{D}(\mathcal{A})\subseteq\mathcal{A}_{\mathrm{D}(\mathcal{C}^{\heartsuit})} inside D(C♡)\mathrm{D}(\mathcal{C}^{\heartsuit}) (Remark 3.47). Is this inclusion always an equivalence; equivalently, is D(A)\mathrm{D}(\mathcal{A}), treated as a subcategory of D(C♡)\mathrm{D}(\mathcal{C}^{\heartsuit}), always a π-stable localizing subcategory? A negative answer must exhibit an explicit C\mathcal{C}, t-structure, and A\mathcal{A}, together with an object of AD(C♡)\mathcal{A}_{\mathrm{D}(\mathcal{C}^{\heartsuit})} not in the image of D(A)\mathrm{D}(\mathcal{A}) and a proof of its non-membership.

Both questions are components of one objective: extending the affine Gabriel–Neeman–Krause–Takahashi classification diagram, and the heart-based classification of Aambø's Theorems A and B, from noetherian affine schemes to all noetherian schemes (question (a)) and from derived categories of rings to general stable ∞-categories with t-structure (question (b)). Progress that additionally connects π-exact localizing subcategories of D(R)\mathrm{D}(R) or Dqc(X)\mathrm{D}_{qc}(X) directly to smashing localizations, or sheds light on the telescope conjecture for locally noetherian schemes, would carry extra significance but is not required for completion.

References

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