Towers of Recollement (ToR)

• Towers of Recollement (ToR) are hierarchically structured frameworks that decompose large abelian and triangulated categories via iterated recollement diagrams.
• The framework systematically constructs ladders and towers, enabling the transfer of tilting, Gorenstein, and representation-theoretic properties across derived and singularity categories.
• Applications in Brieskorn–Pham singularities and Kadar–Yu algebras demonstrate ToR’s practical impact by organizing categorical decompositions and facilitating explicit homological computations.

A Tower of Recollement (ToR) is an organizational framework for stratifying large abelian or triangulated categories into successively smaller or simpler subcategories by the systematic application and iteration of recollement structures—diagrams encoding fully faithful adjoint functor relationships. Each "rung" in a ToR represents an additional layer of categorical decomposition, producing a multi-layered hierarchy that can be leveraged for the study of derived and singularity categories, tilting theory, and the transfer of homological or representation-theoretic properties. ToR formalizes and extends the classical recollement paradigm, refining the decomposition process by introducing the concept of "height" through ladders of recollement, and finds concrete realization in representation theory (e.g., Brieskorn–Pham singularities, Kadar–Yu, Brauer, and Temperley–Lieb algebras) (Gao et al., 2020, Weng, 10 Dec 2025, Morris et al., 31 Dec 2025).

1. Recollement and Ladder Structures

A recollement of abelian categories is encoded in a diagram of exact functors

$A \xrightarrow[]{i} B \xrightarrow[]{e} C$

with adjoint triples $(q, i, p)$ for $A \leftrightarrows B$ and $(l, e, r)$ for $C \leftrightarrows B$, each functor being exact, with $i, l, r$ fully faithful and $\mathrm{Im}\,i = \ker e$. This exactness and adjoint structure enable splitting $B$ into subcategories controlled respectively by $A$ and $C$. Such recollements can be iterated by forming ladders: sequences of further adjoints extending either to the left or right, resulting in a hierarchy structured by "height" (l-height and r-height), which specifies the maximal length to which these extensions exist. Every adjacent pair in the sequence forms an adjoint relation, and each truncation after an adjoint yields a new recollement (Gao et al., 2020).

The formalism for triangulated (and stable) categories is fully analogous, with recollement diagrams featuring exact triangle functors, preserving the essential axioms (adjoint triples, fully faithful functors, kernel-image relations) (Weng, 10 Dec 2025, Morris et al., 31 Dec 2025).

2. From Ladders to Towers: Construction and Parallelism in Derived Categories

The main innovation of ToR is the passage from individual recollement or ladder configurations to the iterative construction of towers—chains of recollement diagrams obtained by truncating finite ladders after each new adjoint. For a ladder of l-height $n$ and r-height $m$, one obtains $n+m$ distinct recollement diagrams. Each rung in the tower reflects a new short exact sequence (abelian case) or a triangle recollement (triangulated case).

Notably, the theory extends to derived and singularity categories: if a ladder has l-height or r-height at least 3, the total derived functors of the adjoints furnish recollement diagrams in bounded derived categories $\mathrm{D}^b$ and, via quotient constructions, in singularity categories $\mathrm{D}_\mathrm{sg}$ (Gao et al., 2020). This stratifies not only the abelian category $B$ but also its derived and singular features.

Parallel towers in derived categories are constructed by deriving each adjoint triple and then leveraging Verdier quotient arguments; outer terms may swap roles at higher heights, and restrictions to subcategories yield further stratifications.

3. Applications: Tilting, Gorenstein Transfer, and Representation Theory

Towers of recollement support advanced applications in tilting theory and the transfer of homological properties:

• Tilting via ladders: Given a recollement admitting a ladder of l-height 3 and a torsion pair $(\mathcal{T}, \mathcal{F})$ on $B$, tilts can be carried compatibly through the tower. Hearts of t-structures ($\mathcal{H}_{\mathcal{T}}$), which are abelian, admit new recollements. If one heart has enough projectives, this structure is also inherited by its bounded derived category (Gao et al., 2020).
• Gorenstein properties: Adjoint functors in the ToR preserve Gorenstein-projective and -injective objects under explicit ladder-height conditions (e.g., r-height 2 for preservation by $e$, l-height 2 for $l$, etc.). If $B$ is Gorenstein, so is $C$ under sufficient height; the exactness and adjunction yield transfer of total acyclicity and depth inequalities (Gao et al., 2020).

In concrete settings such as the representation theory of algebras, the ToR formalism organizes and explains the structure of standard modules, submodules, and homological invariants, as in the Kadar–Yu, Brauer, and Temperley–Lieb algebras (Morris et al., 31 Dec 2025).

4. Explicit Realizations: Brieskorn–Pham Singularities and Kadar–Yu Algebras

Brieskorn–Pham Singularities

The ToR framework is realized in the study of singularity categories for Brieskorn–Pham (BP) singularities. By constructing recollements associated to graded Cohen–Macaulay modules over BP rings and iterating these via “reduction/insertion” functors, an infinite ladder (tower) structure is built. Each layer in the tower encodes stable category recollements corresponding to ring-reduction steps (Weng, 10 Dec 2025). The tower supports the explicit identification of extended tilting $n$-cuboids, whose endomorphism algebras are specific tensor products of Nakayama algebras. These organize, and make manifest, deep derived equivalences—generalizing classical Happel–Seidel symmetries.

Kadar–Yu Algebras

In the context of Kadar–Yu algebras, which interpolate between Brauer and Temperley–Lieb algebras, the ToR formalism is realized through the successive application of globalisation/localisation functors and idempotent localizations. The recollement diagrams constructed at each level relate module categories of the algebras at adjoining ranks. The Gram determinant bootstrap, specific to the tower, provides a recursive criterion (generalized Chebyshev series) for detecting semisimplicity and constructing the non-semisimple structure, and supports the complete block decomposition in both semisimple and non-semisimple regimes (Morris et al., 31 Dec 2025).

5. Derived Equivalences, Stratifications, and Generalizations

ToR systematically stratifies complex categories by iterated recollement, directly relating singularity categories, derived categories, and module-theoretic features through explicit functorial and homological relationships. In BP singularities, the infinite ToR yields a supply of tilting objects, with their endomorphism algebras tracking both the categorical decomposition and derived equivalence classes (Weng, 10 Dec 2025). In Kadar–Yu, the Chebyshev-type bootstrap recovers classical algebraic and geometric patterns, and the ToR formalism bridges several key paradigms of geometric representation theory (affine $\mathsf{A}_1$, integral, and mixed regimes) (Morris et al., 31 Dec 2025).

A summary of the structural implications and reach of ToR is as follows:

ToR Feature	Categorical Manifestation	Example Context
Ladder/height	Rung count in stratification	BP rings, Gorenstein transfer
Six-functor formalism	Adjoints, exactness, functoriality	Kadar–Yu, abelian cases
Derived/singularity	$\mathrm{D}^b$, $\mathrm{D}_{sg}$ stratifications	BP, module categories
Tilting structure	Carry-over of torsion pairs and tilts	BP singularities, Artin algebras
Gram bootstrap	Chebyshev polynomials, block maps	Kadar–Yu, Temperley–Lieb

6. Alcove Geometry, Role in Homological Algebra, and Open Directions

The ToR methodology carries a geometric interpretation in connection with alcove arrangements and reflection group symmetries. In Temperley–Lieb algebras, the walls of alcoves are located at root-of-unity points, while in Brauer algebras, at integral points—both recovered as special cases of the ToR-induced Chebyshev polynomials in Kadar–Yu. The conjectured arm-property for marginal-vertex functions in the associated Rollet graphs reflects a deep compatibility between combinatorial, algebraic, and geometric decompositions (Morris et al., 31 Dec 2025).

A plausible implication is that the ToR framework, though already unifying several algebraic and homological paradigms, will support the construction of further stratifications and explicit calculations across broad classes of categories, especially as fully general criteria for ladder existence and height are developed. The alignment of algebraic “towers” and geometric combinatorics suggests ongoing applications in modern representation theory and categorical geometry.

---

References:

• (Gao et al., 2020) Gao, Koenig, Psaroudakis, "Ladders of recollements of abelian categories"
• (Weng, 10 Dec 2025) Weng, "A recollement approach to Brieskorn-Pham singularities"
• (Morris et al., 31 Dec 2025) "On semisimplicity criteria and non-semisimple representation theory for the Kadar-Yu algebras"