nLab
recollement

Idea

The recollement situation is a diagram of six additive functors

$$\mathcal{A}' \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}} \mathcal{A}\stackrel{\overset{j_!}{\longleftarrow}}{\stackrel{\overset{j^*}{\longrightarrow}}{\underset{j_*}{\longleftarrow}}}\mathcal{A}''$$

among three abelian or triangulated categories satisfying a strong list of exactness and adjointness axioms. The paradigmatic situation is about the categories of abelian sheaves $\mathcal{A}' = Sh(C)$, $\mathcal{A} = Sh(X)$, $\mathcal{A}'' = Sh(U)$, where $U\subset X$ is an open subset of a topological space, $C = X\backslash U$, and the functors among the sheaf categories are induced by the open embedding $j:U\hookrightarrow X$ and closed embedding $i:C\hookrightarrow X$.

Definition

A modern treatment for the recollement of abelian categories is in (Franjou-Pirashvili 04), where the following axioms are listed:

(i) $j_!\dashv j^*\dashv j_*$

(ii) the unit $Id_{\mathcal{A}''}\to j^* j_!$ and the counit $j^* j_*\to Id_{\mathcal{A}''}$ are iso (hence $j_\ast$ and $j_!$ are fully faithful)

(iii) $i^*\dashv i_*\dashv i^!$

(iv) the unit $Id_{\mathcal{A}'}\to i^! i_*$ and the counit $i^* i_*\to Id_{\mathcal{A}'}$ are iso

(v) the functor $i_*:\mathcal{A}'\to Ker(j^*)$ is an equivalence of categories.

In fact (i) and (ii) for $j^*:\mathcal{A}\to\mathcal{A}''$ enable one to define $\mathcal{A}'$ as the full subcategory of $\mathcal{A}$ whose objects $a$ satisfy $j^* a = 0$ such that one satisfies the recollement situation.

A standard treatment for the sequence of triangulated functors

$$\mathcal{D}' \overset{i_*}{\to} \mathcal{D}\overset{j^*}{\to}\mathcal{D}''$$

is in (Beilinson-Bernstein-Deligne 82) where in 1.4.3 the following axioms are listed

(a) $i_* = i_!$ admits a triangulated left adjoint $i^*$ and triangulated right adjoint $i^!$

(b) $j^* = j^!$ admits a triangulated left adjoint $j_*$ and triangulated right adjoint $j_!$

(c) $j^* i_* = 0$ (hence by adjointness, also $i^*j_! = 0$ and $i^! j_*=0$)

(d) given $d\in Ob{\mathcal{D}}$, there exist (necessarily unique) distinguished triangles

$$i_! i^! d \to d\to j_* j^* d\to (i_! i^! d) [1]$$

$$j_! j^! d \to d\to i_* i^* d\to (j_! j^! d) [1]$$

(e) $i_*, j_*, j_!$ are full embeddings.

Again in good situations, less data is needed to provide the recollement.

Examples

• The forgetful functor from global equivariant stable homotopy theory to plain stable homotopy theory exhibits a recollement,

See at global equivariant stable homotopy theory – Relation to plain stable homotopy theory.

Related concepts

• adjoint modality

References

• V. Franjou, T. Pirashvili, Comparison of abelian categories recollements, Doc. Math. 9 (2004), 41–56, MR2005c:18008, pdf

• A.A. Beilinson, J. Bernstein, Pierre Deligne, Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Aste’risque 100, Soc. Math. France, Paris 1982.

In references

• E. Cline, B. Parshall, L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math, 1988, 391: 85—99, MR90d:18005, goettingen
• E. Cline, B. Parshall, L. Scott, Algebraic stratification in representative categories, J. of Algebra 117, 1988, 504—521.

one studies the following kind of sources of recollement situations for triangulated categories: $k$ is a commutative field, $A$ a finite dimensional unital associative $k$-algebra, $e$ an idempotent, and $D^b(A)$ the bounded derived category of right $A$-modules. Suppose $eA(1-e) = 0$ and the global dimension of $A$ is finite. Then there is a recollement of triangulated categories involving $D^b(eAe)$, $D^b(A)$ and $D^b((1-e)A(1-e))$.

• S. Koenig, Tilting complexes, perpendicular categories and recollements of derived module categories of rings., MR92k:18009, doi
• Qinghua Chen,Yanan Lin, Recollements of extension algebras, Science in China 46, 4, 2003 pdf

Another source of examples is due MacPherson and Vilonen

• Kari Vilonen, Perverse sheaves and finite dimensional algebras, Trans. A.M.S. 341 (1994), 665–676, MR94d:16012, doi

• Michael Artin, Grothendieck Topologies. Harvard University, 1962.

• Yuri Berest, Oleg Chalykh, Farkhod Eshmatov, Recollement of deformed preprojective algebras and the Calogero-Moser correspondence, Mosc. Math. J. 8 (2008), no. 1, 21–37, 183, arxiv/0706.3006, MR2009h:16030

• Roy Joshua, pdf

• Yang Han, Recollements and Hochschild theory, arxiv/1101.5697

For treatment in the setting of $\infty$-categories:

• Jacob Lurie, section A.8 of Higher Algebra,

• Clark Barwick, Saul Glasman, A note on stable recollements (arXiv:1607.02064)