Generalized Matrix Artin Algebras - Northeastern...

Generalized Matrix Artin Algebras

Edward L. Green Chrysostomos Psaroudakis

Department of Mathematics Department of MathematicsVirginia Tech University of Ioannina

Blacksburg, VA, USA Ioannina, Greece

International Conference, Woods Hole, Ma.April 2011

Motivation and definitions

Let Λ be an artin K -algebra.When Λ is written as (

A NM B

),

where A and B are artin algebras, M a B-A-bimodule, and N anA-B-bimodule, we say that Λ is a generalized matrix artin algebra.

Need bimodule homomorphisms ϕ : M ⊗A N → B andψ : N ⊗B M → A satisfying

ϕ(m ⊗ n)m′ = mψ(n ⊗m′) and nϕ(m ⊗ n′) = ψ(n ⊗m)n′,

for all m,m′ ∈ M and n, n′ ∈ N.We use the notation

Λ(ϕ,ψ) =

(A NM B

).

Examples

EXAMPLE 1: Let R be an artin algebra and let U and V befinitely generated R-modules. Then

Λ(ϕ,ψ) = EndR(U ⊕ V ) ∼=(

EndR(U) HomR(V ,U)HomR(U,V ) EndR(V )

)is a generalized matrix artin algebra (where ϕ and ψ are given bycomposition).

EXAMPLE 2: Let Λ be an artin algebra and let {ε1, . . . , εn} be afull set of primitive orthogonal idempotents. Suppose n ≥ 2 and1 ≤ r < n. Set e1 =

∑ri=1 εi and e2 =

∑ni=r+1 εi . Then

Λ(ϕ,ψ) =

(e1Λe1 e1Λe2

e2Λe1 e2Λe2

)is a generalized matrix artin algebra (where ϕ and ψ are given bymultiplication).

Example con’t and special cases

EXAMPLE 3: Let A and B be artin algebras, M a finitelygenerated B-A-bimodule, and N a finitely generatedA-B-bimodule. Then

Λ(0,0) =

(A NM B

)is a generalized matrix artin algebra.

Special Cases:

If M = N = (0) then Λ(0,0) =

(A 00 B

)and the category of left

Λ(0,0)-modules is equivalent to the category whose objects are pairs(X ,Y ), and morphisms are given by pairs(α, β) : (X ,Y )→ (X ′,Y ′), where X and X ′ are left A-modules, Yand Y ′ are left B-modules, and α ∈ HomA(X ,X ′), andβ ∈ HomB(Y ,Y ′).

Definitions continued

If N = (0) then Λ(0,0) =

(A 0M B

)and the category of left

Λ(0,0)-modules is equivalent to the category whose objects are pairs(X ,Y , f ), with X ∈Mod(A),Y ∈Mod(B) and f : M ⊗A X → Yis a homomorphism of B-modules. The morphisms are given bypairs (α, β) : (X ,Y , f )→ (X ′,Y ′, f ′), where α : X → X ′ is aA-module homorphism and β : Y → Y ′ is a B-module homorphism

satisfying M ⊗A Xf //

1⊗α��

y

β

��M ⊗A X ′

f ′ // Y ′

commutes.

Note that Λ(0,0)∼= (A× B) n M.

Representations

Let Λ(ϕ,ψ) =

(A NM B

)be a generalized matrix artin algebra.

Let M denote the category whose objects are tuples (X ,Y , f , g),where f : M ⊗A X → Y is a B-module homomorphism andg : N ⊗B Y → X is an A-module homorphism, such the followingdiagrams commute:

N ⊗B M ⊗A X1N⊗f //

ψ⊗1X

��

N ⊗B Y

g

��

M ⊗A N ⊗B Y1M⊗g //

ϕ⊗1Y

��

M ⊗A X

f

��A⊗A X

mult // X B ⊗B Ymult // Y

morphisms

The morphisms of M are pairs(α, β) : (X ,Y , f , g)→ (X ′,Y ′, f ′, g ′), where α : X → X ′ is anA-module homomorphism, and β : Y → Y ′ is a B-modulehomomorphism such that the following diagrams commute:

M ⊗A Xf //

1⊗α��

y

β

��

N ⊗B Yg //

1⊗β��

X

α

��M ⊗A X ′

f ′ // Y ′ N ⊗B Y ′g ′ // X ′

Theorem

Let Λ(ϕ,ψ) =

(A NM B


The categories Mod(Λ(ϕ,ψ)) and M are equivalent.

Matrix interpretation

If Λ(ϕ,ψ) =

(A NM B

)be a generalized matrix artin algebra and

(X ,Y , f , g) ∈M, then if a ∈ A, n ∈ N,m ∈ M, b ∈ B, x ∈ X , andy ∈ Y , then(

a nm b

)·(

xy

)=

(ax + g(n ⊗ y)by + f (m ⊗ x)

).

adjoint isomorphism and some functors

Recalling the adjoint isomorphismsHomB(M ⊗A X ,Y ) ∼= HomA(X ,HomB(M,Y )) andHomA(N ⊗B Y ,X ) ∼= HomB(Y ,HomA(N,X )), we see there is acategory M′, equivalent to M, whose objects are of the form(X ,Y , h, k), where X ∈Mod(A), Y ∈Mod(B),h : X → HomB(M,Y ) and k : Y → HomA(N,X ) satisfying . . . .

We now consider two functors. The first functorTA : Mod(A)→M is given byTA(X ) = (X ,M ⊗A X , 1M⊗X , µ(ψ ⊗ 1X )) where µ : A⊗A X → Xis given by multiplication. Similarly, there is a functorTB : Mod(B)→M given byTB(Y ) = (N ⊗B Y ,Y , µ(ϕ⊗ 1Y ), 1N⊗Y ).

(N ⊗B M)⊗A Xψ⊗1X // A⊗A X

µ // X

Projective modules and other subcategories

Proposition

Let Λ(ϕ,ψ) =

(A NM B


Every indecomposable projective Λ(ϕ,ψ)-module is isomorphic toeither TA(P) for some indecomposable projective A-module P orTB(Q), for some indecomposable projective B-module Q.Furthermore, P is an indecomposable projective A-module and Qis an indecomposable projective B-module, then both TA(P) andTB(Q) are projective Λ(ϕ,ψ)-modules.

Subcategories

Since Λ(ϕ,ψ) =

(A NM B

), Mod(A) and Mod(B) are NOT

necessarily isomorphic to full subcategories of Mod(Λ(ϕ,ψ)) in theobvious way. For example, for Mod(A), viewing elements as(X , 0, 0, 0), we would need (N ⊗B M)⊗A X → X to be 0, whichneed not be true.

Subcategories continued

We now restrict our attention to generalized matrix artin algebras

with ϕ = ψ = 0, i.e., Λ(0,0) =

(A NM B

).

One reason for this restriction is that we have the followingcategorical full embeddings.Let L denote the full subcategory of M whose objects are of theform (X ,Y , f , 0), with f an epimorphism.

Mod(A)→ L →Mod(

(A 0M B

))→Mod(Λ(0,0))

from

{(X , 0, 0, 0)} ⊂ {(X ,Y , f , 0) | f epi} ⊂ {(X ,Y , f , 0)} ⊂ {(X ,Y , f , g)}

Other subcategories

Letting L′ denote the full subcategory of M whose objects are ofthe form (X ,Y , 0, g), with g an epimorphism, we have thefollowing full embeddings:

Mod(B)→ L′ →Mod(

(A N0 B

))→Mod(Λ(0,0))

from

{(0,Y , 0, 0)} ⊂ {(X ,Y , 0, g) | g epi} ⊂ {(X ,Y , 0, g)} ⊂ {(X ,Y , f , g)}

Covariant, contravariant and functorial finiteness

Recall that if B is a additive full subcategory of an abelian categoryA, we say that B is covariantly finite if, for each A ∈ A, there issome BA ∈ B and a map A→ BA such that, for all B ′ ∈ B, theinduced map HomA(B,B ′)→ HomA(A,B ′) is surjective.

We say that B is contravariantly finite if, for each A ∈ A, there issome CA ∈ B and a map CA → A such that, for all B ′ ∈ B, theinduced map HomA(B ′,CA)→ HomA(B ′,A) is surjective.

Finally we say that B is functorially finite if it is both covariantlyand contravariantly finite.

Categorical finiteness of subcategories.

We now state our main result on finiteness of subcategories.

Theorem

Let Λ(0,0) =

(A NM B


Then

1. The category L = {(X ,Y , f , 0) | f epi} is contravariantlyfinite in mod(Λ(0,0)), closed under extensions and quotients,and TA(Mod(A)) ⊆ L.

2. The category L′ = {(X ,Y , 0, g) | g epi} is contravariantlyfinite in mod(Λ(0,0)), closed under extensions and quotients,and TB(Mod(B)) ⊆ L.

Theorem continued

Theorem (con’t)

1’. The category L = {(X ,Y , f , 0) | f mono} is covariantly finitein mod(Λ(0,0)), closed under extensions and quotients, andTA(Mod(A)) ⊆ L.

2’. The category L′ = {(X ,Y , 0, g) | g mono} is covariantlyfinite in mod(Λ(0,0)), closed under extensions and quotients,and TB(Mod(B)) ⊆ L.

Theorem continued

Theorem (con’t)

3. The categories mod(A) and mod(B) are functorially finite inmod(Λ(0,0)).

4. The categories mod(

(A 0M B

)) and mod(

(A N0 B

)) are

functorially finite in mod(Λ(0,0))

Sketch of proof of (4).

Let (X ,Y , f , g) ∈ mod(Λ(0,0)). Let π : X → coker(g) be thecanonical surjection. Since ϕ : M ⊗A N → B is 0 and since

M ⊗A N ⊗B Y1M⊗g //

ϕ⊗1Y

��

M ⊗A X

f

��B ⊗B Y

mult // Y

commutes, we see that there

is a factorization

M ⊗A X1M⊗π//

f��

M ⊗A coker(g) //

h∗

vvnnnnnnnnnnnnn0

YCheck that (coker(g),Y , h∗, 0) ∈M.

We claim (X ,Y , f , g)(π,1Y )−−−−→ (coker(g),Y , h∗, 0) is a left

mod(

(A 0M B

))-approximation.

Proof continued

Let (c , d) : (X ,Y , f , g)→ (U,V , j , 0). Then g ◦ c = 0 and there isan induced map e : coker(g)→ U such that

(X ,Y , f , g)(π,1Y )//

(c,d)��

(coker(g),Y , h∗, 0)

(e,d)vvlllllllllllll

(U,V , j , 0)

commutes. This proves covariant finiteness.

To prove contravariant finiteness, use the adjoint point of view.That is, that elements of mod(Λ(0,0)) are of the form (X ,Y , h, k),where h : X → HomB(M,Y ) and k : Y → HomA(N,X ) satisfyingthe appropriate conditions. A right

Mod(

(A 0M B

))-approximation of (X ,Y , h, k) is

(X , ker(k), `∗, 0), where `∗ : X → HomB(M,Y ) is defined is asimilar fashion to h∗ above.

An example

Example 4: Let K be a field and Q be the quiver

v◦a // w◦ .b

oo

Let Λ = KQ/〈ab, ba〉. Let Pv (respectively Pw ) be the projectiveΛ-cover of the simple module at v (resp. w). Then Λ = Pv ⊕ Pw .Hence Λ is isomorphic to the generalized matrix algebra

Λ(0,0)∼=(

EndΛ(P)op HomΛ(P,Q)HomΛ(Q,P) EndΛ(Q)op

).

Each entry in this 2× 2-matrix is K but the multiplication of two

elements, one of the form

(0 0α 0

)and the other of the form(

0 β0 0

), in any order, is 0.

Example continued

Thus, as a generalized matrix algebra

(A NM B

),

A = B = M = N = K and ϕ = ψ = 0. Hence A and B have globaldimension 0 and M and N have projective dimension 0 over bothA and B and as bimodules. But Λ has infinite global dimension.

From this example, we see that, unlike a ring of the form(A 0M B

), finite global dimension of A and B and hence M and

N is not sufficient for finite global dimension of

Λ(0,0) =

(A NM B

)

Tight projective modules and resolutions

Let Λ(0,0) =

(A NM B

).

If P = (PA, 0, 0, 0) is a projective Λ-module for some left A-modulePA, then we say that P is an A-tight projective Λ-module.

Note that if (PA, 0, 0, 0) is an A-tight projective Λ-module then PA

is a projective A-module and M ⊗A PA = 0. Conversely, if PA is aprojective A-module and M ⊗A PA = 0, then (PA, 0, 0, 0) is anA-tight projective Λ-module.

We say a left A-module (X , 0, 0, 0) has an A-tight projectiveΛ-resolution if X has a projective Λ-resolution in which eachprojective Λ-module is A-tight.

Basic facts about tight projective resolutions

1. A direct sum of modules having A-tight projectiveΛ-resolutions also has an A-tight projective Λ-resolution.

2. A summand of an A-tight projective Λ-module is again anA-tight projective Λ-module.

3. If X is an A-module such that (X , 0, 0, 0) has an A-tightprojective Λ-resolution, then pdΛ(X , 0, 0, 0) = pdA(X ).

4. The subcategory of modules with A-tight projectiveresolutions is closed under extensions.

Proposition

A Λ(0,0)-module of the form (X , 0, 0, 0) has an A-tight projectiveΛ(0,0)-resolution if and only if M ⊗A P = 0, where P is the directsum of projective A-modules in a minimal projective A-resolutionof M.

B-tightness and first bound result

We define B-tight projective Λ-modules, (0,Q, 0, 0) is a similarfashion as A-tight projective Λ-modules and also Λ-modules(0,Y , 0, 0) having B-tight projective Λ-resolutions.

Proposition

Suppose that Λ(0,0) =

(A NM B

)is an generalized matrix artin

algebra and that X is an A-module and Y is a B-module. If(0,M, 0, 0), as a left Λ-module, has a B-tight projectiveΛ-resolution, then

pdΛ((X , 0, 0, 0)) ≤ pdA(X ) + 1 + pdB(M).

If (N, 0, 0, 0), as a left Λ-module, has an A-tight projectiveΛ-resolution, then

pdΛ((0,Y , 0, 0)) ≤ pdB(Y ) + 1 + pdA(N).

Global dimension result

Theorem


(A NM B

)is a generalized matrix artin

algebra, M has a B-tight projective Λ(0,0)-resolution, and N has anA-tight projective resolution. Then

gl.dim(Λ(0,0)) ≤ gl.dim(A) + gl.dim(B) + 1.

Examples and sharpness of resultWe provide two examples, the first of which shows that theinequality is sharp and the second shows that the inequality can beproper.

Example 5 Let Q be the quiver

v1◦a

��

v2◦b // v3◦

c // v4◦

w1◦d // w2◦ w3◦e

oo f // w4◦

g

ggNNNNNNNNNNNNNNN

Let I =< arrows >2 and let Λ = KQ/I .gldim(Λ) = 4Set ε1 = v1 + v2 + v3 + v4 and ε2 = w1 + w2 + w3 + w4. View Λ asthe generalized matrix artin algebra(

ε1Λε1 ε1Λε2

ε2Λε1 ε2Λε2

).

Example 5 continued

The global dimension of ε1Λε1 is 2 and the global dimension ofε2Λε2 is 1.Thus

gl.dim(Λ) = gl.dim(ε1Λε1) + gl.dim(ε2Λε2) + 1.

The following can be checked

1. M = ε2Λε1, which, as a left ε2Λε2-module is isomorphic thesimple module at w1.

2. N = ε1Λε2, which, as a left ε1Λε1-module is isomorphic to thesimple module at v2.

3. (0,M, 0, 0) and (N, 0, 0, 0) have tight projective Λ-resolutions.

4. ϕ and ψ are both 0 for this example.

Example 6

Example 6 Let Q be the quiver

v1◦a //

c

��========v2◦

b // v3◦

w1◦

d

@@��e // w2◦

We again take I to be the ideal generated by all paths of length 2and set Λ = KQ/I . Now set ε1 = v1 + v2 + v3 and ε2 = w1 + w2.View Λ as the generalized matrix artin algebra(

ε1Λε1 ε1Λε2

ε2Λε1 ε2Λε2

).

It is easy to check that the hypotheses of Theorem above aresatisfied. But the global dimension of Λ is 2 while the globaldimension of ε1Λε1 is 2 and the global dimension of ε2Λε2 is 1.

The non-tight case

We consider the case where either (0,M, 0, 0) or (N, 0, 0, 0) doesnot have a tight projective Λ-resolution.

If M is not a projective B-module then a projective cover of

(0,M, 0, 0) is of the form (N ⊗B Q,Q, 0, 1N⊗Q)(0,β)−→ (0,M, 0, 0),

where β : Q → M is projective B-cover of M.

In particular, N ⊗B Q is a sum of summands of N over which wehave little control. The next bound results will require that M, as aleft B-module, is projective and N, as a left A-module is projective.

Second bounding result

Theorem


(A NM B

)is a generalized matrix artin

algebra such that, the global dimensions of A and B are finite, Mis a projective left B-module, and N is a projective left A-module.

1. If ((N ⊗B M)s⊗A , 0, 0, 0) is an A-tight projective Λ(0,0)-module,

for some s ≥ 1, then

gl.dim(Λ(0,0)) ≤ max{gl.dim(A) + 2s, gl.dim(B) + 2s + 1}.

2. If (0,M ⊗A (N ⊗B M)s⊗A , 0, 0) is a B-tight projective

Λ(0,0)-module, for some s ≥ 0, then

gl.dim(Λ(0,0)) ≤ max{gl.dim(A)+2s +1, gl.dim(B)+2(s +1)}.

Theorem (con’t)

3. If (N ⊗B (M ⊗A N)s⊗B , 0, 0, 0) is an A-tight projective

Λ(0,0)-module, for some s ≥ 0, then

gl.dim(Λ(0,0)) ≤ max{gl.dim(A)+2(s +1), gl.dim(B)+2s +1}.

4. If (0, (M ⊗A N)s⊗B , 0, 0) is an B-tight projective Λ(0,0)-module,

for some s ≥ 1, then

gl.dim(Λ(0,0)) ≤ max{gl.dim(A) + 2s + 1, gl.dim(B) + 2s}.

5. If ((N ⊗B M)s⊗A , 0, 0, 0) is an A-tight projective Λ(0,0)-module

and if (0, (M ⊗A N)s⊗B , 0, 0) is an B-tight projective

Λ(0,0)-module , for some s ≥ 1, then

gl.dim(Λ(0,0)) ≤ max{gl.dim(A) + 2s, gl.dim(B) + 2s}.

Theorem (con’t)

6. If (N ⊗B (M ⊗A N)s⊗B , 0, 0, 0) is an A-tight projective

Λ(0,0)-module, and if (0,M ⊗A (N ⊗B M)s⊗A , 0, 0) is a B-tight

projective Λ(0,0)-module, for some s ≥ 0, then

gl.dim(Λ(0,0)) ≤ max{gl.dim(A) + 2s + 1, gl.dim(B) + 2s + 1}.

We present with an example showing that the bounds in the abovetheorem are sharp.

Sharpness of the result.

Example 7 LetQ be the quiver

v1◦

��

v3◦

��

v5◦

v2◦

AA��v4◦

AA��

Let Λ be the quotient KQ/I , where I is the ideal generated by allpaths of length 2. Let ε1 = v1 + v3 + v5 and ε2 = v2 + v4. View Λas a the generalized matrix artin algebra(

ε1Λε1 ε1Λε2

ε2Λε1 ε2Λε2

).

iKj denotes the simple Λ-module, which on the left is isomorphicto Si , and on the right is isomorphic to Sj .

Example 7 continued

Then

M = ε2Λε1 =4 K3 ⊕2 K1, and N = ε1Λε2 =5 K4 ⊕3 K2.

gl.dimA = gl.dimε1Λε1 = gl.dimB = gl.dimε2Λε2 = 0.M is a projective left B-module and N is a projective leftA-module and we see that N ⊗B (M ⊗A N) is isomorphic to 5K2.Moreover, (N ⊗B (M ⊗A N), 0, 0, 0) is an A-tight projectiveΛ-module. Thus, we can apply part (3) of Theorem 2.9 with s = 1to get gl.dim(Λ) ≤ 4. But the global dimension of Λ is 4.

This example can be adjusted to get that all the inequalities aresharp.

Lower bounds

Theorem

Let Λ(ϕ,ψ) =

(A NM B


Suppose that MA is a flat A-module and NB is a flat B-module.Then

gl.dim(Λϕ,ψ) ≥ max{gl.dim(A), gl.dim(B)}

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