For a commutative ring R, Thomason subsets of the Zariski spectrum Spec R correspond bijectively to compactly generated localizing subcategories of the unbounded derived category D(R), and each such subcategory is always contained in the class of complexes whose cohomological support is contained in the corresponding Thomason set. This inclusion is known to be an equality when R is noetherian (where it recovers Neeman's classification), when the set is Zariski closed with quasi-compact complement (Kock–Pitsch), and when R has weak global dimension at most one (the case proved by Bazzoni and Šťovíček), and Bazzoni and Šťovíček explicitly asked (Remark 6.5 of their Advances in Mathematics paper) whether equality holds for arbitrary commutative rings. Later work maps the boundary of the known — Keller-type and new valuation-domain counterexamples to the telescope conjecture, frame-theoretic approaches to smashing spectra, a full support classification for rings whose module category has Gabriel dimension, and obstructions to generation by residue fields — but neither proves nor refutes the equality in general: the problem remains open.

Background

Let R be a commutative ring and let D(R)\mathbf{D}(R) denote the unbounded derived category of the category of R-modules. A localizing subcategory of D(R)\mathbf{D}(R) is a full triangulated subcategory closed under arbitrary coproducts; a Bousfield localization of D(R)\mathbf{D}(R) is an exact functor L ⁣:D(R)→D(R)L\colon \mathbf{D}(R)\to \mathbf{D}(R) together with a natural morphism id→L\mathrm{id}\to L inducing an isomorphism L→L2L\to L^2. A localizing subcategory S\mathcal{S} is smashing if the corresponding localization functor satisfies L(X)≅L(R)⊗RLXL(X)\cong L(R)\otimes^{\mathbf{L}}_R X for all complexes XX. The telescope conjecture (TC) for D(R)\mathbf{D}(R) asserts that every smashing subcategory is compactly generated, i.e. generated as a localizing subcategory by a set of compact objects; in D(R)\mathbf{D}(R) the compact objects are precisely the perfect complexes, the bounded complexes of finitely generated projective modules. For a complex X∈D(R)X\in \mathbf{D}(R) its cohomological support is Supp⁡X={p∈Spec⁡R∣X⊗RLRp≠0}\operatorname{Supp}X=\{\mathfrak{p}\in \operatorname{Spec}R \mid X\otimes^{\mathbf{L}}_R R_{\mathfrak{p}}\neq 0\}, where RpR_{\mathfrak{p}} is the localization of RR at the prime ideal p\mathfrak{p}. A subset Z⊆Spec⁡RZ\subseteq \operatorname{Spec}R is a Thomason set if it is a union of closed subsets of the form Zi={p∣p⊇Ii}Z_i=\{\mathfrak{p}\mid \mathfrak{p}\supseteq I_i\} for finitely generated ideals IiI_i whose complements Spec⁡R∖Zi\operatorname{Spec}R\setminus Z_i are quasi-compact. Thomason's classification of thick subcategories of perfect complexes, combined with Neeman's work, yields for every commutative ring R a bijection between Thomason subsets of Spec⁡R\operatorname{Spec}R and compactly generated localizing subcategories of D(R)\mathbf{D}(R), given by X↦Supp⁡X\mathcal{X}\mapsto \operatorname{Supp}\mathcal{X} (Proposition 6.4 of Bazzoni and Šťovíček). When R is noetherian, Neeman's 1992 'chromatic tower' theorem gives the far stronger statement that every localizing subcategory of D(R)\mathbf{D}(R), compactly generated or not, has the form {X∣Supp⁡X⊆S}\{X \mid \operatorname{Supp}X\subseteq S\} for a uniquely determined subset S⊆Spec⁡RS\subseteq \operatorname{Spec}R; the compactly generated ones correspond exactly to the Thomason subsets, and TC holds. Beyond the noetherian world this picture breaks down. Bazzoni and Šťovíček (2017) proved, for rings R of weak global dimension at most one (meaning every submodule of a flat module is flat; this class contains all von Neumann regular rings, where every module is flat, and all valuation domains, i.e. integral domains whose finitely generated ideals are totally ordered by inclusion), that smashing subcategories of D(R)\mathbf{D}(R) correspond bijectively to equivalence classes of homological epimorphisms starting in R (ring maps R→SR\to S such that the induced restriction D(S)→D(R)\mathbf{D}(S)\to \mathbf{D}(R) is full and faithful), and for commutative R that compactly generated localizing subcategories correspond precisely to flat epimorphisms; they classified the smashing localizations of the derived category of any valuation domain, gave a criterion for TC for any commutative ring of weak global dimension at most one, and deduced that TC holds for every commutative von Neumann regular ring and, among Prüfer domains (domains whose localizations at maximal ideals are valuation domains), precisely for the strongly discrete ones (those for which no maximal ideal of any such localization is idempotent). Keller had already constructed, before this work, a commutative ring for which TC fails, so the general commutative classification problem genuinely exceeds the compactly generated world. The paper of Bazzoni and Šťovíček isolates a sharper, purely support-theoretic question that is meaningful for every commutative ring. Writing XZ\mathcal{X}_Z for the compactly generated localizing subcategory corresponding to a Thomason set ZZ (equivalently, the smallest localizing subcategory containing all perfect complexes with support in ZZ), the inclusion XZ⊆{X∈D(R)∣Supp⁡X⊆Z}\mathcal{X}_Z\subseteq \{X\in \mathbf{D}(R) \mid \operatorname{Supp}X\subseteq Z\} always holds, and the equality is known when R is noetherian (essentially due to Neeman), when ZZ is Zariski closed with quasi-compact complement (due to Kock and Pitsch), and when the weak global dimension of R is at most one (the case established in the paper). Whether equality holds in full generality, and more broadly what the smashing and Bousfield localizations of D(R)\mathbf{D}(R) look like for commutative rings outside these regimes, is the subject of this problem; the authors note in their introduction that understanding all Bousfield localizations of D(R)\mathbf{D}(R) 'is generally an extremely difficult problem'.

Problem Statement

Let R be an arbitrary commutative ring and D(R)\mathbf{D}(R) its unbounded derived category. For a complex X∈D(R)X\in \mathbf{D}(R) let Supp⁡X={p∈Spec⁡R∣X⊗RLRp≠0}\operatorname{Supp}X=\{\mathfrak{p}\in \operatorname{Spec}R \mid X\otimes^{\mathbf{L}}_R R_{\mathfrak{p}}\neq 0\} be its cohomological support, and call Z⊆Spec⁡RZ\subseteq \operatorname{Spec}R a Thomason subset if Z=⋃i{p∣p⊇Ii}Z=\bigcup_i \{\mathfrak{p}\mid \mathfrak{p}\supseteq I_i\} for finitely generated ideals IiI_i with quasi-compact complements. For a Thomason set ZZ let XZ\mathcal{X}_Z be the smallest localizing subcategory of D(R)\mathbf{D}(R) containing all perfect complexes PP with Supp⁡P⊆Z\operatorname{Supp}P\subseteq Z; by Thomason–Neeman theory XZ\mathcal{X}_Z is compactly generated, Supp⁡XZ=Z\operatorname{Supp}\mathcal{X}_Z=Z, and the inclusion XZ⊆{X∈D(R)∣Supp⁡X⊆Z}\mathcal{X}_Z\subseteq \{X\in \mathbf{D}(R)\mid \operatorname{Supp}X\subseteq Z\} always holds. (1) Support-equality question (Bazzoni–Šťovíček, Remark 6.5, arXiv:1402.7294): does XZ={X∈D(R)∣Supp⁡X⊆Z}\mathcal{X}_Z=\{X\in \mathbf{D}(R)\mid \operatorname{Supp}X\subseteq Z\} hold for every commutative ring R and every Thomason subset Z⊆Spec⁡RZ\subseteq \operatorname{Spec}R? The equality fails precisely when there exist a commutative ring R, a Thomason set Z, and a complex X with Supp⁡X⊆Z\operatorname{Supp}X\subseteq Z but X∉XZX\notin \mathcal{X}_Z. (2) The source's broader program, of which (1) is the precise core: describe the smashing, and more generally the Bousfield, localizations of D(R)\mathbf{D}(R) for commutative rings without the weak-global-dimension-at-most-one hypothesis — in particular determine which smashing subcategories are compactly generated (the telescope conjecture, known to fail in general via Keller-type and valuation-domain examples) and give a structural classification of them. Both parts serve one scientific objective: to extend the geometric, support-theoretic description of the localizations of D(R)\mathbf{D}(R) — complete for noetherian rings and for rings of weak global dimension at most one — to arbitrary commutative rings. Sub-question (1) is the source's precise core question, and sub-question (2) is the broader surrounding research program.

References

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