nLab
recollement

The recollement situation is a diagram of six additive functors

$$𝒜\prime \stackrel{\stackrel{i^{*}}{\leftarrow }}{\stackrel{\stackrel{i_{*}}{\to }}{\underset{i^{!}}{\leftarrow }}}𝒜\stackrel{\stackrel{j_{!}}{\leftarrow }}{\stackrel{\stackrel{j^{*}}{\to }}{\underset{j_{*}}{\leftarrow }}}𝒜″$$\mathcal{A}' \stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\to}}{\underset{i^!}{\leftarrow}}} \mathcal{A}\stackrel{\overset{j_!}{\leftarrow}}{\stackrel{\overset{j^*}{\to}}{\underset{j_*}{\leftarrow}}}\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 $𝒜\prime =\mathrm{Sh}(C)$, $𝒜=\mathrm{Sh}(X)$, $𝒜″=\mathrm{Sh}(U)$, 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

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

where the following axioms are listed:

(i) $j_{!}⊣j^{*}⊣j_{*}$

(ii) the unit ${\mathrm{Id}}_{𝒜″}\to j^{*}{j}_{!}$ and the counit $j^{*}{j}_{*}\to {\mathrm{Id}}_{𝒜″}$ are iso

(iii) $i^{*}⊣i_{*}⊣i^{!}$

(iv) the unit ${\mathrm{Id}}_{𝒜″\to i^{!}{i}_{*}}$ and the counit $i^{*}{i}_{*}\to {\mathrm{Id}}_{𝒜\prime }$ are iso

(v) the functor $i_{*}:𝒜\prime \to \mathrm{Ker}(j_{*})$ is an equivalence of categories.

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

A standard treatment for the sequence of triangulated functors

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

is in

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

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

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

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

References

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 $\mathrm{eA}(1-e)=0$ and the global dimension of $A$ is finite. Then there is a recollement of triangulated categories involving $D^b(\mathrm{eAe})$, $D^b(A)$ 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

• 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, 21–37, 183, arxiv/0706.3006, MR2009h:16030

• Roy Joshua, pdf

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