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, $C = X\backslash U$ the complement, and the functors among the sheaf categories are induced by the open embedding $j \colon U\hookrightarrow X$ and closed embedding $i \colon C\hookrightarrow X$. As suggested by this example, recollement may in fact be regarded as the additive or triangulated version of Artin gluing.

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 prototypical example of recollement is provided by a closed-open decomposition. In this setup (which works equally well for locally compact Hausdorff spaces and schemes) there is the following diagram of inclusions.

$$\begin{array}{ccccc} Z & \xhookrightarrow{i} & X & \xhookleftarrow{j} & U \\ \end{array}$$

Here $i$ is a closed inclusion and $j$ is an open inclusion. Then the categories of sheaves on $X,Z$ and $U$ fit into the recollement structure described above. The recollement structure on the category of sheaves on $X$ enables one to argue inductively about sheaves on $X.$ This idea is usually referred to as Noetherian induction?. This technique allows to deduce results about higher dimensional schemes from information about lower dimensional schemes and analysis of locally standard schemes (e.g. smooth schemes for étale topology). See Achar Exercise 1.3.4, Lemma 2.2.1.

• 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.

• P. Achar. Perverse Sheaves and Applications to Representation Theory. Vol. 258. Mathematical Surveys and Monographs. Providence, Rhode Island: American Mathematical Society, 2021. doi:10.1090/surv/258.

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)