Homological Theory of Recollements of Abelian Categories

32 Pages·2012·0.21 MB·English

PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C)
RecollementsofTriangulatedCategories
Homological Theory of Recollements of Abelian
Categories
Chrysostomos Psaroudakis
University of Ioannina
ICRA XV, Bielefeld, Germany
August 14, 2012
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C)
RecollementsofTriangulatedCategories
Table of contents
1 Preliminaries on Recollements of Abelian Categories
Definitions and Basic Properties
Examples
2 Homological Algebra in Rab(A,B,C)
Extensions, Global and Finitistic Dimension
Representation Dimension
3 Recollements of Triangulated Categories
Rouquier Dimension
Triangular Matrix Rings
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) DExeafimniptlieosnsandBasicProperties
RecollementsofTriangulatedCategories
Definition
A recollement situation between abelian categories A,B and C is
a diagram
q l
(cid:123)(cid:123) (cid:124)(cid:124)
A (cid:98)(cid:98) i (cid:47)(cid:47)B(cid:98)(cid:98) e (cid:47)(cid:47)C Rab(A,B,C)
p r
satisfying the following conditions:
1. (l,e,r) is an adjoint triple.
2. (q,i,p) is an adjoint triple.
3. The functors i, l, and r are fully faithful.
4. Imi = Kere.
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) DExeafimniptlieosnsandBasicProperties
RecollementsofTriangulatedCategories
Properties of R (A,B,C)
ab
1 The functors e: B −→ C and i: A −→ B are exact.
2 The composition of functors ql = pr = 0.
(cid:39) (cid:39) (cid:39) (cid:39)
3 qi −→ IdA, IdA −→ pi, er −→ IdC and IdC −→ el.
4 The functor i induces an equivalence between A and the
Serre subcategory Kere = Imi of B.
5 A is a localizing and colocalizing subcategory of B and there
is an equivalence B/A (cid:39) C.
6 For every B ∈ B we have exact sequences:
0 (cid:47)(cid:47)Kerµ (cid:47)(cid:47)le(B)µB (cid:47)(cid:47)B λB(cid:47)(cid:47)iq(B) (cid:47)(cid:47)0
B
0 (cid:47)(cid:47)ip(B) κB (cid:47)(cid:47)B νB (cid:47)(cid:47)re(B) (cid:47)(cid:47)Cokerν (cid:47)(cid:47)0
B
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) DExeafimniptlieosnsandBasicProperties
RecollementsofTriangulatedCategories
Example: (One idempotent)
Let R be a ring and e2 = e ∈ R an idempotent. Then we have the
recollement:
R/ReR⊗R− Re⊗eRe−
(cid:119)(cid:119) (cid:120)(cid:120)
inc (cid:47)(cid:47) e(−) (cid:47)(cid:47)
Mod-R/R(cid:103)(cid:103)eR Mod-R(cid:102)(cid:102) Mod-eRe
Hom (R/ReR,−) Hom (eR,−)
R eRe
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) DExeafimniptlieosnsandBasicProperties
RecollementsofTriangulatedCategories
Example: (Generalized Matrix Rings)
Let R, S be rings, M a S-R-bimodule and N a R-S-bimodule. Let
φ: M ⊗ N −→ S be a S-S-bimodule homomorphism and let
R
ψ: N ⊗ M −→ R be a R-R-bimodule homomorphism. Then the
S
above data allow us to define the generalized matrix ring:
(cid:18) (cid:19)
R N
Λ = R S
(φ,ψ) M S
S R
where the multiplication is given by
(cid:18)r n(cid:19) (cid:18)r(cid:48) n(cid:48)(cid:19) (cid:18)rr(cid:48)+ψ(n⊗m(cid:48)) rn(cid:48)+ns(cid:48) (cid:19)
· =
m s m(cid:48) s(cid:48) mr(cid:48)+sm(cid:48) ss(cid:48)+φ(m⊗n(cid:48))
e1 = (cid:0)10R 00(cid:1), e2 = (cid:0)00 10S(cid:1) idempotents elements of Λ(φ,ψ). Then:
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) DExeafimniptlieosnsandBasicProperties
RecollementsofTriangulatedCategories
Example: (Generalized Matrix Rings)
Λ/Λe1Λ⊗Λ− Λe1⊗e1Λe1−
(cid:120)(cid:120) (cid:121)(cid:121)
Mod-Λ/Λe Λ inc (cid:47)(cid:47)Mod-Λ e1(−) (cid:47)(cid:47)Mod-e Λe
(cid:102)(cid:102) 1 (φ,(cid:102)(cid:102)ψ) 1 1
HomΛ(Λ/Λe1Λ,−) Home1Λe1(e1Λ,−)
Λ/Λe2Λ⊗Λ− Λe2⊗e2Λe2−
(cid:120)(cid:120) (cid:121)(cid:121)
Mod-Λ/Λe Λ inc (cid:47)(cid:47)Mod-Λ e2(−) (cid:47)(cid:47)Mod-e Λe
(cid:102)(cid:102) 2 (φ,(cid:102)(cid:102)ψ) 2 2
HomΛ(Λ/Λe2Λ,−) Home2Λe2(e2Λ,−)
Mod-Λ/Λe Λ (cid:39) Mod-S/Imφ, Mod-e Λe (cid:39) Mod-R
1 1 1
Mod-Λ/Λe Λ (cid:39) Mod-R/Imψ, Mod-S (cid:39) Mod-e Λe
2 2 2
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) DExeafimniptlieosnsandBasicProperties
RecollementsofTriangulatedCategories
Example: (Symmetric Recollement)
Let R, S rings and N a bimodule. Then we have the triangular
R S
matrix ring Λ = (cid:0)R RNS(cid:1) and the following recollements:
0 S
q T
S
(cid:120)(cid:120) (cid:120)(cid:120)
Mod-(cid:101)(cid:101)R ZR (cid:47)(cid:47)Mod-(cid:101)(cid:101)Λ US (cid:47)(cid:47)Mod-S
U Z
R S
U Z
S R
(cid:120)(cid:120) (cid:120)(cid:120)
Mod-(cid:101)(cid:101)S ZS (cid:47)(cid:47)Mod-(cid:101)(cid:101)Λ UR (cid:47)(cid:47)Mod-R
p H
R
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
Problem
Find necessary and sufficient conditions such that the induced
homomorphisms in and en are isomorphisms for
X,Y Z,W
0 ≤ n ≤ k.
Relate (if possible) the global/finitistic dimension of the
categories involved in R (A,B,C).
ab
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) ERxetpernessieonntsa,tiGolnobDailmaenndsiFoinnitisticDimension
RecollementsofTriangulatedCategories
Let as before R (A,B,C) be a recollement of abelian categories.
ab
Since the functors i: A −→ B and e: B −→ C are exact, they
induce natural maps:
in : Extn (X,Y) (cid:47)(cid:47)Extn (i(X),i(Y))
X,Y A B
and
en : Extn (Z,W) (cid:47)(cid:47)Extn(e(Z),e(W))
Z,W B C
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)
Relate (if possible) the global/finitistic dimension of the
categories involved in R (A,B,C).
ab
PreliminariesonRecollementsofAbelianCategories
HomologicalAlgebrainRab(A,B,C) ERxetpernessieonntsa,tiGolnobDailmaenndsiFoinnitisticDimension
RecollementsofTriangulatedCategories
Let as before R (A,B,C) be a recollement of abelian categories.
ab
Since the functors i: A −→ B and e: B −→ C are exact, they
induce natural maps:
in : Extn (X,Y) (cid:47)(cid:47)Extn (i(X),i(Y))
X,Y A B
and
en : Extn (Z,W) (cid:47)(cid:47)Extn(e(Z),e(W))
Z,W B C
Problem
Find necessary and sufficient conditions such that the induced
homomorphisms in and en are isomorphisms for
X,Y Z,W
0 ≤ n ≤ k.
ChrysostomosPsaroudakis HomologicalTheoryofRab(A,B,C)