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 , , , where is an open subset of a topological space, , and the functors among the sheaf categories are induced by the open embedding and closed embedding .
A modern treatment for the recollement of abelian categories is in (Franjou-Pirashvili 04), where the following axioms are listed:
(ii) the unit and the counit are iso (hence and are fully faithful)
(iv) the unit and the counit are iso
(v) the functor is an equivalence of categories.
In fact (i) and (ii) for enable one to define as the full subcategory of whose objects satisfy such that one satisfies the recollement situation.
A standard treatment for the sequence of triangulated functors
is in (Beilinson-Bernstein-Deligne 82) where in 1.4.3 the following axioms are listed
(a) admits a triangulated left adjoint and triangulated right adjoint
(b) admits a triangulated left adjoint and triangulated right adjoint
(c) (hence by adjointness, also and )
(d) given , there exist (necessarily unique) distinguished triangles
(e) are full embeddings.
Again in good situations, less data is needed to provide the recollement.
See at global equivariant stable homotopy theory – Relation to plain stable homotopy theory.
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.
- 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: is a commutative field, a finite dimensional unital associative -algebra, an idempotent, and the bounded derived category of right -modules. Suppose and the global dimension of is finite. Then there is a recollement of triangulated categories involving , and .
- 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 a treatment in the setting of -categories