Homological Theory of Recollements of Abelian Categories

Homological Theory of Recollements of Abelian Categories Chrysostomos Psaroudakis University of Ioannina ICRA XV, Bielefeld, Germany August 14, 2012

Table of contents 1 Definitions and Basic Properties Examples 2 3 Rouquier Dimension Triangular Matrix Rings

Definitions and Basic Properties Examples Definition A recollement situation between abelian categories A, B and C is a diagram A q l i e B C R ab (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. Im i = Ker e.

Definitions and Basic Properties Examples Properties of R ab (A, B, C ) 1 The functors e: B C and i: A B are exact. 2 The composition of functors ql = pr = 0. 3 qi Id A, Id A pi, er IdC and Id C el. 4 The functor i induces an equivalence between A and the Serre subcategory Ker e = Im i of B. 5 A is a localizing and colocalizing subcategory of B and there is an equivalence B/A C. 6 For every B B we have exact sequences: 0 Ker µ B le(b) µ B B λ B iq(b) 0 0 ip(b) κ B B ν B re(b) Coker ν B 0

Definitions and Basic Properties Examples Example: (One idempotent) Let R be a ring and e 2 = e R an idempotent. Then we have the recollement: Mod-R/ReR R/ReR R inc Mod-R Re ere e( ) Mod-eRe Hom R (R/ReR, ) Hom ere (er, )

Definitions and Basic Properties Examples Example: (Generalized Matrix Rings) Let R, S be rings, M a S-R-bimodule and N a R-S-bimodule. Let φ: M R N S be a S-S-bimodule homomorphism and let ψ : N S M R be a R-R-bimodule homomorphism. Then the above data allow us to define the generalized matrix ring: ( ) R Λ (φ,ψ) = RN S SM R S where the multiplication is given by ( ) ( r n r n ) ( rr m s m s = + ψ(n m ) rn + ns ) mr + sm ss + φ(m n ) e 1 = ( ) 1 R 0 0 0, e2 = ( ) 0 0 0 1 S idempotents elements of Λ(φ,ψ). Then:

Definitions and Basic Properties Examples Example: (Generalized Matrix Rings) Λ/Λe 1 Λ Λ Λe 1 e1 Λe 1 Mod-Λ/Λe 1 Λ inc e 1 ( ) Mod-Λ (φ,ψ) Mod-e 1 Λe 1 Hom Λ (Λ/Λe 1 Λ, ) Hom e1 Λe 1 (e 1 Λ, ) Λ/Λe 2 Λ Λ Λe 2 e2 Λe 2 Mod-Λ/Λe 2 Λ inc e 2 ( ) Mod-Λ (φ,ψ) Mod-e 2 Λe 2 Hom Λ (Λ/Λe 2 Λ, ) Hom e2 Λe 2 (e 2 Λ, ) Mod-Λ/Λe 1 Λ Mod-S/ Im φ, Mod-e 1 Λe 1 Mod-R Mod-Λ/Λe 2 Λ Mod-R/ Im ψ, Mod-S Mod-e 2 Λe 2

Definitions and Basic Properties Examples Example: (Symmetric Recollement) Let R, S rings and R N S a bimodule. Then we have the triangular matrix ring Λ = ( ) R R N S 0 S and the following recollements: Mod-R q Z R Mod-Λ T S U S Mod-S U R Z S Mod-S U S Z S Mod-Λ p Z R U R Mod-R H R

Let as before R ab (A, B, C ) be a recollement of abelian categories. Since the functors i: A B and e: B C are exact, they induce natural maps: i n X,Y : Extn A (X, Y ) Ext n B (i(x ), i(y )) and e n Z,W : Extn B (Z, W ) Ext n C (e(z), e(w )) Problem Find necessary and sufficient conditions such that the induced homomorphisms i n X,Y and en Z,W are isomorphisms for 0 n k.

For 0 n we define the following full subcategories of B : X n = {B B l(p n ) l(p 0 ) B 0 exact where P i Proj C, 0 i n} Y n = {B B 0 B r(i 0 ) r(i n ) exact where I i Inj C, 0 i n}

For 0 n we define the following full subcategories of B : X n = {B B l(p n ) l(p 0 ) B 0 exact where P i Proj C, 0 i n} Y n = {B B 0 B r(i 0 ) r(i n ) exact where I i Inj C, 0 i n} Note that l(p i ) Proj B and r(i i ) Inj B.

Theorem Let R ab (A, B, C ). Then the following statements are equivalent: 1 The map i n X,Y : Extn A (X, Y ) Extn B (i(x ), i(y )) is an isomorphism, X, Y A and 0 n k. 2 Im µ P X k 1, P Proj B, where µ: le Id B. 3 Im ν I Y k 1, I Inj B, where ν : Id B re. Theorem Let R ab (A, B, C ). Then the following statements are equivalent: 1 The map e n Z,W : Extn B (Z, W ) Extn C (e(z), e(w )) is an isomorphism, W B, (resp. Z B), and 0 n k. 2 Z X k+1 (resp. W Y k+1 ).

Theorem Let R ab (A, B, C ). 1 We have: gl. dim B gl. dim A + gl. dim C 2 If gl. dim B 1 then: + sup{pd B i(p) P Proj A } + 1 gl. dim A 1 and gl. dim C 1 3 If sup{pd B i(p) P Proj A } 1, then the following are equivalent: 1. gl. dim B <. 2. gl. dim A < and gl. dim C <.

Let F: D G be a right exact functor between abelian categories where we assume that D has enough projectives. We say that F has locally bounded homological dimension, if there exists n 0 such that whenever L m F(A) = 0 for m 0 then L m F(A) = 0 for every m n + 1. The minimum such n (if it exists) is called the locally bounded homological dimension of F and is denoted by l.b.hom.dim F.

Theorem Let R ab (A, B, C ). 1 If the functor l: C B has locally bounded homological dimension, then: FPD (C ) FPD (B) + l.b.hom.dim l

Theorem Let R ab (A, B, C ). 1 If the functor l: C B has locally bounded homological dimension, then: FPD (C ) FPD (B) + l.b.hom.dim l 2 If the functors r: C B and p: B A are exact, then: FPD(A ) FPD(B) FPD(A ) + FPD(C ) + 1 Corollary Let R be a ring and e 2 = e R. If the functor Re ere has locally bounded homological dimension then: Fin. dim ere Fin. dim R + l.b.hom.dim Re ere

Corollary Let Λ be an Artin algebra with rep. dim Λ 3 and e an idempotent element of Λ. Then: 1 rep. dim eλe 3. 2 If the Λ-module Λ/ΛeΛ is projective, then: rep. dim Λ/ΛeΛ 3

Corollary Let Λ be an Artin algebra with rep. dim Λ 3 and e an idempotent element of Λ. Then: 1 rep. dim eλe 3. 2 If the Λ-module Λ/ΛeΛ is projective, then: rep. dim Λ/ΛeΛ 3 Corollary Let Λ be an Artin algebra. Then: rep. dim Λ 3 rep. dim End Λ (P) 3 for any finitely generated projective Λ-module P.

Corollary Let Λ be an Artin algebra with rep. dim Λ 3 and e 2 = e Λ. Then: fin. dim eλe <

Corollary Let Λ be an Artin algebra with rep. dim Λ 3 and e 2 = e Λ. Then: fin. dim eλe < Proof. rep. dim Λ 3 rep. dim eλe 3 Igusa Todorov fin. dim eλe <

Corollary Let Λ be an Artin algebra and Γ = End Λ (Λ D Λ). Then: rep. dim Λ gl. dim Λ + gl. dim Γ/ΓeΛ ΓΓ + 1 and gl. dim Γ/ΓeΛ ΓΓ gl. dim Γ/Γe Λ Γ + pd Γ Γ/Γe Λ Γ where gl. dim Γ/ΓeΛ ΓΓ = sup{pd Γ X X mod-γ/γe Λ Γ}. Proof. 1 Γ = End Λ (Λ D Λ) ( ) Λ D Λ Hom Λ (D Λ,Λ) Λ. 2 e Λ = ( ) 1 Λ 0 0 0 idempotent element of Γ. 3 mod-γ/γe Λ Γ mod-γ mod-λ : recollement.

Rouquier Dimension Triangular Matrix Rings Definition A recollement situation between triangulated categories U, T and V is a diagram U q l i e T V R tr (U, T, V) p r of triangulated functors 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. Im i = Ker e.

Rouquier Dimension Triangular Matrix Rings Let T be a triangulated category and X T. We write: X = X 1 = add{x [i] i Z} X n+1 = add{y T M Y N M[1] triangle with M X and N X n }

Rouquier Dimension Triangular Matrix Rings Let T be a triangulated category and X T. We write: X = X 1 = add{x [i] i Z} X n+1 = add{y T M Y N M[1] triangle with M X and N X n } Then the Rouquier dimension of T is defined as follows: dim T = min{n 0 X T such that X n+1 = T}

Rouquier Dimension Triangular Matrix Rings Problem Given a R tr (U, T, V), can we give bounds for the dimension of T in terms of the dimensions of U and V?

Rouquier Dimension Triangular Matrix Rings Problem Given a R tr (U, T, V), can we give bounds for the dimension of T in terms of the dimensions of U and V? Theorem Let (U, T, V) be a recollement of triangulated categories. Then: max {dim U, dim V} dim T dim U + dim V + 1

Rouquier Dimension Triangular Matrix Rings Corollary Let Λ = ( R R N S 0 S ) be a triangular matrix ring. Then: max {dim D b (R), dim D b (S)} dim D b (Λ) dim D b (R) + dim D b (S) + 1