Recollements of Krause & Neeman–Murfet

• Recollement theory is a categorical framework that glues abelian, triangulated, and stable ∞-categories using adjoint functor pairs and canonical distinguished triangles.
• It integrates localization, duality, tilting, and homological invariants, offering applications in algebraic geometry, modular representation theory, and noncommutative geometry.
• Modern advancements extend recollements to stable ∞-categories with symmetrical adjoint triples, enabling the transfer of invariants such as Hochschild cohomology and algebraic K-theory.

Recollement theory, particularly as developed by Krause and by Neeman-Murfet, formalizes the “gluing” of abelian, triangulated, and more recently stable $\infty$-categories from simpler constituent pieces. This concept provides a categorical framework connecting localization, duality, tilting, and homological invariants, with applications ranging from algebraic geometry to modular representation theory, and stable and stratified $\infty$-categories. The following sections present the core principles, key structural features, and modern generalizations of recollements, focusing on foundational formulations, their compatibility with localization, duality, and tensor product structures, and their impact on homological invariants like Hochschild cohomology and algebraic $K$-theory.

1. Formalism of Recollements

In its classical form, a recollement of triangulated categories $\mathcal{T}_1, \mathcal{T}, \mathcal{T}_2$ is an “exact sequence” of six functors: $\xymatrix{ \mathcal{T}_1 \ar@<-2.5ex>[r]_{i_*} \ar@<-1.0ex>[r]^{i^*} \ar@<+1.0ex>[r]^{i^!} & \mathcal{T} \ar@<-2.5ex>[r]_{j_!} \ar@<-1.0ex>[r]^{j^*} \ar@<+1.0ex>[r]^{j_*} & \mathcal{T}_2 }$ where $(i^*, i_*)$, $(i_*, i^!)$, $(j_!, j^*)$, and $(j^*, j_*)$ are adjoint pairs, and the functors $i_*, j_!$ (or $j_*$) are fully faithful. Axioms require that $\operatorname{Im} i_* = \ker j^*$ and that every object in $\mathcal{T}$ sits in canonical distinguished triangles associated to the images of these functors.

In the context of derived categories of associative $k$-algebras $A$, $A_1$, $A_2$, a standard recollement (see (Han, 2011)) is realized as

$$\left( D(A_1),\, D(A),\, D(A_2); \, i^* = - \otimes_A^{\mathbb{L}} Y,\, i_* = \mathrm{RHom}_{A_1}(Y,-),\, j_! = - \otimes_A^{\mathbb{L}} Y_2,\, j^* = \mathrm{RHom}_A(Y_2, -),\, \dots \right)$$

for appropriate complexes $Y \in D(A^{op} \otimes A_1)$ and $Y_2 \in D(A^{op} \otimes A_2)$.

The recollement formalism extends naturally to stable $\infty$-categories. In this setting (see (Fiorenza et al., 2015, Barwick et al., 2016, Shah, 2021)), six adjoint functors are replaced by two adjoint triples $i_\ell \dashv i \dashv i_r$, $q_\ell \dashv q \dashv q_r$ with properties generalizing the triangulated setup. Every object in $\mathcal{D}$ is determined, up to canonical equivalence, by its images in $\mathcal{D}^0$ and $\mathcal{D}^1$ via these functors and a gluing map reflecting exactness (pullout or pushout) properties.

2. Gluing, Lifting, and Symmetry

A key foundational point is the minimality of the data required to reconstruct a recollement. In stable $\infty$-categories, specifying a full subcategory $U \subset X$ that is both reflective and coreflective (i.e., the inclusion $j_*: U \to X$ admits both a left adjoint $j^*$ and a right adjoint $j^\times$) suffices: the right and left orthogonals

$$Z^\wedge := \{M \in X : \mathrm{Map}_X(N,M) \simeq *\ \forall N\in U\}, \qquad Z^\vee := \{M \in X : \mathrm{Map}_X(M,N) \simeq *\ \forall N\in U\}$$

are themselves reflective and coreflective, and $X$ is “stratified” along these subcategories (see (Barwick et al., 2016)).

Furthermore, this approach reveals a striking symmetry: $Z^\wedge$ and $Z^\vee$ are canonically equivalent via

$$i^\wedge i_\vee : Z^\vee \xrightarrow{\sim} Z^\wedge, \qquad i^\vee i_\wedge : Z^\wedge \xrightarrow{\sim} Z^\vee$$

making the notion of “closed part” of the recollement genuinely self-dual in the stable context.

3. Recollements, Localization, and Model Structures

Recollements are intimately tied to localization theory and the construction of model structures on exact or abelian categories. In exact or Grothendieck categories, every complete hereditary cotorsion pair (e.g., pairs $(\mathcal{A},\mathcal{B})$ in $\mathcal{G}$ with vanishing $\operatorname{Ext}^1$) induces a recollement structure on subcategories of complexes, such as

$\xymatrix{ \text{Acyclics}/\sim \ar[r] & K(\mathcal{B}) \ar[r] & D(\mathcal{B}) }$

where $K(\mathcal{B})$ is the homotopy category of $\mathcal{B}$-complexes, and $D(\mathcal{B})$ is its derived category (Bazzoni et al., 2017, Hu et al., 18 Oct 2025). When three compatible cotorsion pairs satisfy certain inclusion/intersection conditions, the existence of a recollement is automatic (see (Gillespie, 2013), where such constructions explain Krause’s and Neeman–Murfet’s recollements for injective and flat modules in both module and sheaf-theoretic contexts).

In model categories, these recollement diagrams arise as relationships between three homotopy categories corresponding to different model structures, such as projective, injective, and “mock” homotopy model structures. The underlying idea is that recollements can be packaged as localizations and colocalizations at the level of the homotopy categories.

4. Homological Invariants, Duality, and Transfer

Recollements play a deep role in transferring homological properties and dualities. Major consequences include the following:

• Descent/Lifting to Tensor Products and Opposite Algebras: Standard recollements of $D(A)$ (relative to $D(A_1)$, $D(A_2)$) lift to $D(B \otimes A)$ (relative to $D(B \otimes A_i)$), where $B$ is any algebra, via the derived functor $B \otimes^{\mathbb{L}} -$. Recollements also transfer to $D(A^{op})$, using dualizing complexes (see (Han, 2011)).
• Smoothness: An algebra $A$ is smooth (finite projective dimension as an $A^e$-module) if and only if both $A_1$ and $A_2$ are smooth, provided $D(A)$ admits a recollement with respect to $D(A_1)$ and $D(A_2)$ (Han, 2011).
• Hochschild Cohomology: There are explicit long exact sequences (à la Mayer–Vietoris) connecting $HH^n(A)$, $HH^n(A_1)$, $HH^n(A_2)$, arising as the cohomology of triangles in the derived category of $k$-modules induced by the recollement (Han, 2011).
• Gorenstein and Singular Equivalences: For recollements arising from idempotents $a$ in an artin algebra $\Lambda$, the comparison functor $e = (a-)$ yields that $\Lambda$ is Gorenstein if and only if $a\Lambda a$ is Gorenstein, and the singularity categories $D_{sg}(\Lambda)$ and $D_{sg}(a\Lambda a)$ are equivalent, under suitable homological conditions (Psaroudakis et al., 2014, Qin, 2018).
• Algebraic $K$-Theory: Under recollement hypotheses, the higher algebraic $K$-groups decompose additively: $K_n(R) \cong K_n(S) \oplus K_n(T)$ for all $n$, with $R$, $S$, $T$ in recollement; more generally, there is a long Mayer–Vietoris sequence relating $K$-theory via noncommutative tensor products (Chen et al., 2012).

5. Partial Tilting, TTF Triples, and Universal Localizations

Recollements are closely related to the theory of TTF (torsion-torsionfree) triples and partial tilting modules. Any recollement of an abelian or module category corresponds, up to equivalence, to a TTF triple, and in module categories, to an idempotent ideal (Psaroudakis et al., 2013). For derived categories, the presence of a partial tilting complex $P$ induces a recollement decomposing $D(B)$ into $\mathrm{Im}(P \otimes^{\mathbb{L}}_B -)$ and its kernel, corresponding to a quotient by a bireflective subcategory or a generalized universal localization (homological epimorphism) (Bazzoni et al., 2012).

This provides a direct link between recollement theory, the theory of localizations (in the sense of Cohn-Schofield for rings), and the construction of derived or homological invariants.

6. Iterated and Higher Recollement Structures

Beyond the classical three-object setting, recollements have been generalized to higher “ladders” and polygons, most prominently in stable $\infty$-categories. Given a stratified topological space (or a poset stratification), multiple recollements can be iterated: gluing $t$-structures from subcategories yields a global $t$-structure via an associative operation $\uplus$; compatible recollements yield an $n$-fold glueing that is invariant under parenthesization, encoded combinatorially by structures such as the “Urizen compass” (Fiorenza et al., 2015).

In triangulated settings, polygons of recollements (n-gons where every adjacent pair gives a recollement, and the data is cyclically symmetric) encode higher semiorthogonal decompositions relevant in Calabi-Yau contexts and for $N$-complexes (Iyama et al., 2016).

7. Recollements and Stratification, Symmetric Monoidal Refinements

Recently, recollement formalism has been deeply integrated into the theory of stratification in $\infty$-categories, topoi, and monoidal categories. In the symmetric monoidal context, recollements can be structured to respect tensor products, and gluing can be interpreted as right-lax limits over stratifying bases, with the gluing functor (e.g., $\phi = i^* \circ j_*$) given by explicit formulas involving Kan extensions. This underpins the reconstruction theorems for stratified sheaf categories, yielding equivalences between categories of stratified $\infty$-topoi and certain fibrations (Shah, 2021).

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Key Formulas and Summary Table

Structure	Formula or Diagram	Invariant/Implication
Standard recollement	$i^* \cong - \otimes_A^{\mathbb{L}} Y$; $j^* \cong \mathrm{RHom}_A(Y_2, -)$	Decomposes $D(A)$ using $D(A_1)$, $D(A_2)$ (Han, 2011)
Tensor lift	$$I^* \cong - \otimes_A^{\mathbb{L}} (B \otimes^{\mathbb{L}} Y)$$	Transfers recollement to $D(B \otimes A)$
Opposite algebra	$Y^* := \mathrm{RHom}_{A_1}(Y, A_1)$	Recollement for $D(A^{op})$
Additive $K$-theory	$K_n(R) \simeq K_n(S) \oplus K_n(T)$	$K$-group additivity (Chen et al., 2012)
Hochschild triangle	$$R\mathrm{Hom}_{A^e}(A, J(J'A)) \to R\mathrm{Hom}_{A^e}(A, A) \to R\mathrm{Hom}_{A^e}(I(I^*A))$$	Long exact sequence for $HH^n$ (Han, 2011)
Glued $t$-structure	$$D_{\geq 0} = \{ X \mid q(X)\in D^1_{\geq 0},\, i_\ell(X) \in D^0_{\geq 0} \}$$	Global $t$-structure from local data (Fiorenza et al., 2015)

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Significance and Impact

The recollement framework pioneered by Krause and Neeman-Murfet unifies stratification, localization, tilting, and duality in a categorical setting and is foundational for the paper of singularities, noncommutative geometry, and representation theory. It provides a blueprint to “decompose” and “reconstruct” categories (of modules, sheaves, or spectra) from local pieces, to transfer or detect smoothness, Gorenstein and singular behaviors, and to establish long exact sequences and additivity results for invariants such as algebraic $K$-theory and Hochschild cohomology. Its most advanced incarnations in stable $\infty$-category theory or stratified sheaf theory (including symmetric monoidal refinements and higher polygons or ladders) provide powerful tools for addressing new phenomena in geometry, homotopy theory, and beyond.