Homotopically Smashing t-Structures

• Homotopically smashing t-structures are defined by their coaisle’s closure under directed homotopy colimits, connecting closure properties with definability and purity.
• They underpin classification schemes in derived categories using Thomason filtrations and local-to-global methods, thereby linking geometric supports with algebraic properties.
• Results confirm that for commutative noetherian rings, any homotopically smashing t-structure is compactly generated, reinforcing its connection to the telescope conjecture and cosilting theory.

A homotopically smashing $t$-structure is a $t$-structure whose coaisle is closed under directed homotopy colimits. In the algebraic settings most studied in the literature—especially compactly generated triangulated categories underlying strong stable derivators, and derived categories $D(R)$ of commutative rings—this condition sits between formal closure properties of triangulated subcategories and geometric classifications by supports. It is closely linked to definability, purity, Grothendieck hearts, compact generation, Thomason filtrations, the telescope conjecture, and cosilting theory (Saorín et al., 2017, Laking, 2018, Hrbek, 2018, Hrbek et al., 2019, Hrbek et al., 2021).

1. Definition and categorical setting

A $t$-structure on a triangulated category $T$ is a pair $(\mathcal U,\mathcal V)$ of full subcategories such that $\operatorname{Hom}(\mathcal U,\mathcal V)=0$, $\mathcal U[1]\subseteq \mathcal U$ equivalently $\mathcal V[-1]\subseteq \mathcal V$, and every object $X\in T$ fits into a truncation triangle

$t$0

with $t$1 and $t$2. The class $t$3 is the aisle, $t$4 the coaisle, and the heart is the abelian category $t$5 (Saorín et al., 2017, Hrbek et al., 2019).

The term homotopically smashing is used for those $t$6-structures whose coaisle is closed under directed homotopy colimits. In the derivator framework, if $t$7 is directed and a coherent diagram $t$8 satisfies $t$9 for all $D(R)$0, then

$D(R)$1

For $D(R)$2, directed homotopy colimits agree with direct limits computed in the Grothendieck category of complexes $D(R)$3 (Saorín et al., 2017, Hrbek et al., 2019, Hrbek et al., 2021).

Compact generation is a pervasive sufficient condition. Compactly generated $D(R)$4-structures are homotopically smashing, because orthogonality against compact objects is preserved by directed homotopy colimits (Saorín et al., 2017, Hrbek, 2018). In the stable case, homotopically smashing reduces to the usual notion of smashing for Bousfield localizations; in the non-stable case it is strictly stronger and carries genuinely additional content (Hrbek et al., 2021, Saorín et al., 2017).

2. Definability, purity, and Grothendieck hearts

In compactly generated derivator settings, homotopically smashing $D(R)$5-structures are tied to purity. A morphism is pure if it is tested as such by all compact objects, pure-injective objects are those splitting pure monomorphisms starting from them, and definable subcategories are those characterized by closure under products, pure subobjects or monomorphisms, and directed homotopy colimits. In this framework, a full subcategory is definable if and only if it is closed under products, pure subobjects, and directed homotopy colimits (Laking, 2018).

For $D(R)$6, the coaisle side admits a particularly sharp formulation: a full subcategory $D(R)$7 is the coaisle of a homotopically smashing $D(R)$8-structure if and only if $D(R)$9 is definable and cosuspended, meaning closed under extensions and under $t$0 (Hrbek et al., 2021). In the more general setting of a compactly generated triangulated category underlying a strong stable derivator, a left nondegenerate $t$1-structure $t$2 is homotopically smashing if and only if $t$3 is definable; this is also equivalent to $t$4 being smashing with Grothendieck heart, and to being cogenerated by a pure-injective partial cosilting object (Laking, 2018).

These equivalences explain why homotopically smashing $t$5-structures frequently have robust hearts. Compactly generated $t$6-structures are homotopically smashing, and homotopically smashing $t$7-structures have hearts with exact filtered colimits, hence Ab.5 hearts. When the ambient category is well generated and algebraic or topological, accessibly embedded $t$8-structures have hearts with generators; in particular, the heart of any compactly generated $t$9-structure in a well generated algebraic or topological triangulated category is Grothendieck (Saorín et al., 2017).

A common source of such structures is cosilting theory. If $T$0 is a pure-injective cosilting object, then the associated coaisle is definable and cosuspended, so the induced $T$1-structure is homotopically smashing (Hrbek et al., 2021, Laking, 2018).

3. Compact generation and classification over commutative rings

For a commutative ring $T$2, compactly generated $T$3-structures in $T$4 are classified by Thomason filtrations of $T$5. A Thomason subset is a union of closed subsets $T$6 with $T$7 ranging over finitely generated ideals, and a Thomason filtration is a decreasing sequence $T$8 of Thomason subsets. There is a bijection between Thomason filtrations and compactly generated $T$9-structures, with compact generators given by shifts of Koszul complexes $(\mathcal U,\mathcal V)$0 attached to finitely generated ideals (Hrbek, 2018, Hrbek et al., 2021).

The classification can be written explicitly. For a filtration $(\mathcal U,\mathcal V)$1, the aisle is

$(\mathcal U,\mathcal V)$2

and in the bounded-below case one has the support-theoretic formula

$(\mathcal U,\mathcal V)$3

For noetherian $(\mathcal U,\mathcal V)$4, Thomason subsets are exactly specialization-closed subsets, so the classification reduces to filtrations by specialization-closed supports (Hrbek, 2018).

This classification supplies the geometric model underlying many homotopically smashing phenomena. Every compactly generated coaisle is definable and therefore homotopically smashing. Conversely, over a commutative noetherian ring, a bounded-below homotopically smashing $(\mathcal U,\mathcal V)$5-structure is compactly generated; the proof passes through hereditary torsion pairs of finite type and reconstructs the relevant Thomason filtration from cohomological data and injective stalks (Hrbek, 2018). A stronger noetherian result shows that the bounded-below hypothesis is unnecessary: any homotopically smashing $(\mathcal U,\mathcal V)$6-structure in $(\mathcal U,\mathcal V)$7 is compactly generated (Hrbek et al., 2019).

The resulting compact generators remain concrete. If $(\mathcal U,\mathcal V)$8 is the Koszul complex of a finitely generated ideal $(\mathcal U,\mathcal V)$9, then the compactly generated coaisle associated with $\operatorname{Hom}(\mathcal U,\mathcal V)=0$0 is determined by orthogonality against the family $\operatorname{Hom}(\mathcal U,\mathcal V)=0$1 with $\operatorname{Hom}(\mathcal U,\mathcal V)=0$2 (Hrbek, 2018, Hrbek et al., 2021).

4. Telescope conjectures and compact generation

The telescope problem for $\operatorname{Hom}(\mathcal U,\mathcal V)=0$3-structures asks whether homotopically smashing implies compactly generated. In the stable case this recovers the classical telescope conjecture for smashing localizations; in the non-stable setting it becomes a statement about arbitrary $\operatorname{Hom}(\mathcal U,\mathcal V)=0$4-structures rather than only stable ones (Hrbek et al., 2019, Saorín et al., 2017).

For commutative noetherian rings the answer is affirmative. Any homotopically smashing $\operatorname{Hom}(\mathcal U,\mathcal V)=0$5-structure in $\operatorname{Hom}(\mathcal U,\mathcal V)=0$6 is compactly generated (Hrbek et al., 2019). This extends Neeman’s theorem from stable localizations to general $\operatorname{Hom}(\mathcal U,\mathcal V)=0$7-structures, and it implies that pure-injective cosilting objects over commutative noetherian rings are of cofinite type, meaning that their associated $\operatorname{Hom}(\mathcal U,\mathcal V)=0$8-structures are compactly generated (Hrbek et al., 2019). A weaker but earlier form states that bounded-below homotopically smashing $\operatorname{Hom}(\mathcal U,\mathcal V)=0$9-structures over commutative noetherian rings are compactly generated, which already yields classification results for bounded cosilting complexes (Hrbek, 2018).

The same theme appears in derivator language. In compactly generated strong stable derivators, the coaisle of a compactly generated $\mathcal U[1]\subseteq \mathcal U$0-structure is closed under directed homotopy colimits, and the heart is Grothendieck (Saorín et al., 2017). This suggests a general pattern: compact generation controls both the homotopy-colimit behavior of the coaisle and the exactness properties of the heart.

There are also extensions beyond $\mathcal U[1]\subseteq \mathcal U$1. For derived categories of finite quiver representations over commutative artinian rings, every homotopically smashing $\mathcal U[1]\subseteq \mathcal U$2-structure is compactly generated, and analogous results hold for commutative perfect rings (Hrbek et al., 17 Sep 2025). For path algebras $\mathcal U[1]\subseteq \mathcal U$3 of Dynkin quivers over commutative noetherian rings, any homotopically smashing $\mathcal U[1]\subseteq \mathcal U$4-structure in $\mathcal U[1]\subseteq \mathcal U$5 is compactly generated, with an explicit classification of compactly generated aisles by poset maps $\mathcal U[1]\subseteq \mathcal U$6 (Sabatini, 27 May 2025). These results indicate that the commutative-ring picture is not isolated, although the precise geometric parameter spaces change.

5. Locality and gluing over stalks

A major refinement of the subject is the local-to-global theory over affine schemes. For a commutative ring $\mathcal U[1]\subseteq \mathcal U$7, compactly generated $\mathcal U[1]\subseteq \mathcal U$8-structures in $\mathcal U[1]\subseteq \mathcal U$9 correspond not only to Thomason filtrations on $\mathcal V[-1]\subseteq \mathcal V$0 but also to compatible families of compactly generated $\mathcal V[-1]\subseteq \mathcal V$1-structures over the local rings $\mathcal V[-1]\subseteq \mathcal V$2, where $\mathcal V[-1]\subseteq \mathcal V$3 ranges over maximal ideals (Hrbek et al., 2021).

The compatibility is expressed on Thomason subsets. If $\mathcal V[-1]\subseteq \mathcal V$4 is a family of Thomason subsets and $\mathcal V[-1]\subseteq \mathcal V$5 denotes its image, then compatibility means that for all maximal ideals $\mathcal V[-1]\subseteq \mathcal V$6,

$\mathcal V[-1]\subseteq \mathcal V$7

Under this condition, together with the requirement that $\mathcal V[-1]\subseteq \mathcal V$8 be Thomason, one obtains a bijection between global Thomason subsets and compatible local families; componentwise this yields a bijection between global Thomason filtrations and compatible local filtrations (Hrbek et al., 2021).

The consequence for homotopically smashing $\mathcal V[-1]\subseteq \mathcal V$9-structures is a precise locality criterion. If $X\in T$0 is a homotopically smashing $X\in T$1-structure in $X\in T$2, then it is compactly generated if and only if each localized $X\in T$3-structure $X\in T$4 in $X\in T$5 is compactly generated (Hrbek et al., 2021). The argument uses definability of $X\in T$6, the pure monomorphism $X\in T$7, and the construction of a global Thomason filtration from local data via hereditary torsion pairs of finite type.

This locality transfers to telescope statements. The telescope conjecture and the semistable telescope conjecture in $X\in T$8 are equivalent to their validity on all localizations at maximal ideals, and the $X\in T$9-Telescope Conjecture for a quasi-compact and quasi-separated scheme $t$00 is equivalent to the validity of the classical telescope conjecture on all stalks $t$01, equivalently on all closed-point stalks (Hrbek et al., 2021). In particular, if all stalks are noetherian, then $t$02 satisfies the tensor telescope conjecture (Hrbek et al., 2021).

A technical subtlety arises in the non-noetherian case: the hypothesis that the union $t$03 be Thomason is necessary. An explicit counterexample is provided by the ring $t$04 with a nonprincipal ultrafilter maximal ideal (Hrbek et al., 2021).

6. Cosilting theory, lifting procedures, and limitations

Cosilting theory furnishes one of the main structural interpretations of homotopically smashing $t$05-structures. A cosilting object $t$06 induces a $t$07-structure $t$08, and if $t$09 is pure-injective then the coaisle is definable and cosuspended, hence homotopically smashing (Hrbek et al., 2021). The object $t$10 is of cofinite type precisely when the coaisle is of the form $t$11 for a set $t$12 of compact objects; equivalently, the induced $t$13-structure is compactly generated (Hrbek et al., 2021). Over rings satisfying the semistable telescope conjecture, in particular over noetherian rings, every pure-injective cosilting object is of cofinite type (Hrbek et al., 2019, Hrbek, 2018).

The local theory extends to cosilting objects themselves. There is a bijection between cosilting objects of cofinite type in $t$14, up to equivalence, and compatible families of cosilting objects of cofinite type in the local categories $t$15, up to equivalence. The global object is recovered from the compatible family by taking the product $t$16, while localization is expressed by colocalization $t$17 (Hrbek et al., 2021). For noetherian rings, this specializes further to compatible families of cosilting modules and of $t$18-term cosilting complexes (Hrbek et al., 2021).

Homotopically smashing behavior also appears in comparisons between bounded and unbounded derived categories. For a right coherent ring $t$19, every intermediate $t$20-structure in $t$21 lifts to a compactly generated, hence homotopically smashing, intermediate $t$22-structure in $t$23 by closing the aisle and coaisle under directed homotopy colimits; conversely, an intermediate homotopically smashing $t$24-structure in $t$25 restricts to $t$26 exactly when its coaisle and heart are generated from their bounded finitely presented parts by directed homotopy colimits (Marks et al., 2021).

The notion nevertheless has important limitations. Smashing and homotopically smashing do not coincide in general: there are HRS tilts that are smashing but not homotopically smashing, and their hearts fail to be Ab.5 (Saorín et al., 2017). Conversely, homotopically smashing need not imply compactly generated outside the noetherian and similarly controlled settings: an example from HRS tilting over a ring with a nontrivial idempotent ideal gives a homotopically smashing $t$27-structure that is not compactly generated (Saorín et al., 2017). This shows that the affirmative telescope theorems are not formal consequences of the definition but rely on specific geometric or purity-theoretic input.

These limitations clarify the role of the main positive results. Homotopically smashing $t$28-structures are best understood not as a universal synonym for compact generation, but as a robust closure condition whose force depends strongly on the ambient category: in $t$29 for commutative noetherian $t$30, and more generally in several support-theoretic settings, it recovers compact generation; beyond that range, it remains closely tied to definability, purity, and cosilting, but no longer collapses to a purely compactly generated theory (Hrbek et al., 2019, Hrbek et al., 2021, Saorín et al., 2017).