The recollement situation is a diagram of six additive functors
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 $i:C\hookrightarrow X$.
A modern treatment for the recollement of abelian categories is in
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
(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
is in
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
(e) $i_*, j_*, j_!$ are full embeddings.
Again in good situations, less data is needed to provide the recollement.
In references
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 $D^b((1-e)A(1-e))$.
Another source of examples is due MacPherson and Vilonen
Kari Vilonen, Perverse sheaves and finite dimensional algebras, Trans. A.M.S. 341 (1994), 665β676, MR94d:16012, doi
M. 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, 21β37, 183, arxiv/0706.3006, MR2009h:16030
Roy Joshua, pdf
Yang Han, Recollements and Hochschild theory, arxiv/1101.5697