recollement in nLab

Contents

Contents

Idea

The recollement situation is a diagram of six additive functors

𝒜′⟵i !⟶i *⟵i *𝒜⟵j *⟶j *⟵j !𝒜″ \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 𝒜′=Sh(C)\mathcal{A}' = Sh(C), 𝒜=Sh(X)\mathcal{A} = Sh(X), 𝒜″=Sh(U)\mathcal{A}'' = Sh(U), where U⊂XU\subset X is an open subset of a topological space, C=X\UC = X\backslash U the complement, and the functors among the sheaf categories are induced by the open embedding j:U↪Xj \colon U\hookrightarrow X and closed embedding i:C↪Xi \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 !⊣j *⊣j *j_!\dashv j^*\dashv j_*

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

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

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

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

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

A standard treatment for the sequence of triangulated functors

𝒟′→i *𝒟→j *𝒟″ \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 !i_* = i_! admits a triangulated left adjoint i *i^* and triangulated right adjoint i !i^!

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

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

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

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

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

(e) i *,j *,j !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.

• 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: kk is a commutative field, AA a finite dimensional unital associative kk-algebra, ee an idempotent, and D b(A)D^b(A) the bounded derived category of right AA-modules. Suppose eA(1−e)=0eA(1-e) = 0 and the global dimension of AA is finite. Then there is a recollement of triangulated categories involving D b(eAe)D^b(eAe), D b(A)D^b(A) and D b((1−e)A(1−e))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)