Silting-theoretic parametrisation of smashing subcategories of compactly generated triangulated categories beyond the dg setting

For a compactly generated triangulated category $\mathcal{T}$, the smashing localising subcategories — the kernels of coproduct-preserving localisation functors — govern the decompositions of $\mathcal{T}$ into recollements, and their classification contains the Telescope Conjecture as the sub-question whether all of them are generated by compact objects. Angeleri Hügel, Marks and Vitória (Math. Z. 296 (2020) 887–900) proved that for the derived category $D(A)$ of differential graded (dg) modules over a non-positive dg ring $A$, every compactly generated localising subcategory is generated by a single, possibly large, partial silting object, proposing silting theory as the replacement parametrisation in the regime where the Telescope Conjecture fails. The general problem remains open: the non-compactly-generated (Keller-type) smashing subcategories are classified only for special ring classes; the parametrisation by partial silting objects is established neither for them nor in any non-algebraic setting; and the 2023 disproof of Ravenel's Telescope Conjecture creates new, unclassified smashing localisations of the stable homotopy category to which the silting approach has not been exported.

Background

A triangulated category $\mathcal{T}$ with set-indexed coproducts is the natural home of homological algebra and stable homotopy theory. An object $X$ of $\mathcal{T}$ is compact if the functor $\mathrm{Hom}_{\mathcal{T}}(X,-)$ commutes with coproducts; a localising subcategory is a triangulated subcategory closed under coproducts. A localising subcategory $\mathcal{L}$ is smashing if it is coreflective (the inclusion admits a right adjoint) and the right adjoint of the inclusion preserves coproducts; equivalently, $\mathcal{L}$ is the kernel of a localisation functor whose fully faithful right adjoint also preserves coproducts. Smashing subcategories are exactly the data of recollement decompositions of $\mathcal{T}$ — a recollement being a diagram of three triangulated categories tied by six functors that exhibits the middle one as glued together from the outer two — so classifying them means classifying the coproduct-preserving localisations of $\mathcal{T}$. Localising subcategories generated by a set of compact objects are automatically smashing, and the claim that in a compactly generated $\mathcal{T}$ all smashing subcategories are of this form is the Telescope Conjecture (TC), stated by Ravenel in 1984 for the stable homotopy category (the category of spectra, the basic 'topological' triangulated category). TC holds in some settings — Neeman proved it for derived categories $D(R)$ of modules over commutative noetherian rings, and Krause–Šťovíček for hereditary rings — but Keller constructed, for a non-noetherian commutative ring, a smashing subcategory of $D(R)$ containing no nonzero compact object, so TC fails in general and any classification must handle such 'Keller-type' subcategories. In the topological world, Burklund, Hahn, Levy and Schlank proved in 2023 that the telescopic localisations of spectra (localisation with respect to a periodic self-map of a finite complex) and the chromatic localisations (localisation with respect to Morava K-theory, the periodic homology theories underlying the chromatic filtration) differ at all heights $\ge 2$, disproving Ravenel's conjecture and producing non-compactly-generated smashing localisations of the sphere spectrum. A parallel programme replaces compact generators by silting-type objects. An object $T$ of a well generated triangulated category (a weakening of compact generation: the category is generated by a set of objects whose represented functors commute only with coproducts indexed by some fixed regular cardinal) is partial silting if it lies in its own positive orthogonal $T^{\perp_{>0}}$ and this orthogonal is cocomplete; it is silting if additionally $T^{\perp_{\mathbb{Z}}}=0$. A partial silting object generates a smashing subcategory $\mathrm{Loc}(T)$, and silting objects induce nondegenerate t-structures, where a t-structure is a pair of subcategories $(\mathcal{U},\mathcal{V})$ with no maps from $\mathcal{U}$ to $\mathcal{V}[-1]$ such that every object of $\mathcal{T}$ fits into a distinguished triangle with entries in $\mathcal{U}$ and $\mathcal{V}$ — the triangulated analogue of a torsion pair — and nondegenerate means the intersection of all shifts of $\mathcal{U}$ and of all shifts of $\mathcal{V}$ vanish. In the silting literature, classifications of smashing subcategories are expressed through explicit silting-type data — equivalence classes of silting objects, TTF triples (torsion–torsion–free triples, i.e. t-structures whose coaisle is itself the aisle of a further t-structure), and comparable homological invariants. Building on Nicolás–Saorín–Zvonareva's silting theory for categories with coproducts and on Marks–Šťovíček's result that universal localisations of rings (the initial ring homomorphisms making a prescribed set of matrices invertible) are controlled by large partial silting modules, Angeleri Hügel–Marks–Vitória proved that for $\mathcal{T}=D(A)$, the derived category of differential graded (dg) modules over a non-positive (cohomologically concentrated in non-positive degrees) dg ring $A$, every compactly generated localising subcategory $\mathcal{L}$ equals $\mathrm{Loc}(T)$ for a partial silting object $T$. A compact silting generator forces an algebraic triangulated category (a stable category of a Frobenius exact category) to be equivalent to such a $D(A)$, so the hypothesis is exactly 'algebraic with compact silting generator'. Their Remark 3.6(2) shows the parametrisation cannot hold for arbitrary categories with coproducts: for example, the stable category of all modules over a representation-finite self-injective finite-dimensional algebra is periodic and admits no nondegenerate t-structures, so its smashing subcategories cannot all be silting-generated. The precise scope of the silting parametrisation — for non-compactly-generated smashing subcategories, and outside the algebraic setting — is the open target.

Problem Statement

Classify the smashing localising subcategories of a compactly generated triangulated category $\mathcal{T}$, and determine the extent to which they can be parametrised by possibly large (non-compact) partial silting objects, in the generality where the Telescope Conjecture fails — that is, where smashing subcategories need not be generated by compact objects. Concretely: (1) for $\mathcal{T}=D(A)$ the derived category of dg modules over a non-positive dg ring (or an ordinary ring), Angeleri Hügel–Marks–Vitória prove that every compactly generated localising subcategory is $\mathrm{Loc}(T)$ for a partial silting object $T$; determine whether the smashing subcategories that are not compactly generated — Keller-type examples and the valuation-domain families of Bazzoni–Šťovíček and Balchin–Tecklenburg — are likewise generated by a possibly large partial silting object, and otherwise classify them. (2) Beyond the algebraic setting — for well generated or compactly generated triangulated categories without a compact silting generator, including topological categories such as the stable homotopy category, where the disproof of Ravenel's Telescope Conjecture guarantees the existence of non-compactly-generated smashing localisations — establish which smashing subcategories admit a silting-type parametrisation, and characterise the structural obstructions that bound any such parametrisation, given that it cannot hold for all categories with coproducts.

References

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