Generalized Matrix Artin Algebras Edward L. Green Department of Mathematics Virginia Tech Blacksburg, VA, USA Chrysostomos Psaroudakis Department of Mathematics University of Ioannina 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 N where A and B are artin algebras, M a B-A-bimodule, and N an A-B-bimodule, we say that Λ is a generalized matrix artin algebra. ), Need bimodule homomorphisms ϕ: M A N B and ψ : 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 ( A N Λ (ϕ,ψ) = ).
Examples EXAMPLE 1: Let R be an artin algebra and let U and V be finitely generated R-modules. Then ( ) Λ (ϕ,ψ) = End R (U V ) EndR (U) Hom R (V, U) = Hom R (U, V ) End R (V ) is a generalized matrix artin algebra (where ϕ and ψ are given by composition). EXAMPLE 2: Let Λ be an artin algebra and let {ɛ 1,..., ɛ n } be a full set of primitive orthogonal idempotents. Suppose n 2 and 1 r < n. Set e 1 = r i=1 ɛ i and e 2 = n i=r+1 ɛ i. Then ( ) e1 Λe Λ (ϕ,ψ) = 1 e 1 Λe 2 e 2 Λe 1 e 2 Λe 2 is a generalized matrix artin algebra (where ϕ and ψ are given by multiplication).
Example con t and special cases EXAMPLE 3: Let A and B be artin algebras, M a finitely generated B-A-bimodule, and N a finitely generated A-B-bimodule. Then ( ) A N Λ (0,0) = is a generalized matrix artin algebra. Special Cases: ( ) A 0 If M = N = (0) then Λ (0,0) = and the category of left 0 B 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, Y and Y are left B-modules, and α Hom A (X, X ), and β Hom B (Y, Y ).
Definitions continued ( ) A 0 If N = (0) then Λ (0,0) = 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 Y is a homomorphism of B-modules. The morphisms are given by pairs (α, β): (X, Y, f ) (X, Y, f ), where α: X X is a A-module homorphism and β : Y Y is a B-module homorphism satisfying M A X f y commutes. 1 α M A X f Y β Note that Λ (0,0) = (A B) M.
Representations ( ) A N Let Λ (ϕ,ψ) = 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 and g : N B Y X is an A-module homorphism, such the following diagrams commute: N B M A X 1 N f N B Y M A N B Y 1 M g M A X ψ 1 X A A X g ϕ 1 Y mult X B B Y mult f Y
morphisms The morphisms of M are pairs (α, β): (X, Y, f, g) (X, Y, f, g ), where α: X X is an A-module homomorphism, and β : Y Y is a B-module homomorphism such that the following diagrams commute: M A X f y N B Y g X 1 α β 1 β M A X f Y N B Y g X Theorem ( ) A N Let Λ (ϕ,ψ) = be a generalized matrix artin algebra. The categories Mod(Λ (ϕ,ψ) ) and M are equivalent. α
Matrix interpretation ( ) A N If Λ (ϕ,ψ) = be a generalized matrix artin algebra and (X, Y, f, g) M, then if a A, n N, m M, b B, x X, and y Y, then ( ) ( ) ( ) a n x ax + g(n y) =. m b y by + f (m x)
adjoint isomorphism and some functors Recalling the adjoint isomorphisms Hom B (M A X, Y ) = Hom A (X, Hom B (M, Y )) and Hom A (N B Y, X ) = Hom B (Y, Hom A (N, X )), we see there is a category M, equivalent to M, whose objects are of the form (X, Y, h, k), where X Mod(A), Y Mod(B), h : X Hom B (M, Y ) and k : Y Hom A (N, X ) satisfying.... We now consider two functors. The first functor T A : Mod(A) M is given by T A (X ) = (X, M A X, 1 M X, µ(ψ 1 X )) where µ: A A X X is given by multiplication. Similarly, there is a functor T B : Mod(B) M given by T B (Y ) = (N B Y, Y, µ(ϕ 1 Y ), 1 N Y ). (N B M) A X ψ 1 X A A X µ X
Projective modules and other subcategories Proposition ( ) A N Let Λ (ϕ,ψ) = be a generalized matrix artin algebra. Every indecomposable projective Λ (ϕ,ψ) -module is isomorphic to either T A (P) for some indecomposable projective A-module P or T B (Q), for some indecomposable projective B-module Q. Furthermore, P is an indecomposable projective A-module and Q is an indecomposable projective B-module, then both T A (P) and T B (Q) are projective Λ (ϕ,ψ) -modules. Subcategories ( ) A N Since Λ (ϕ,ψ) =, Mod(A) and Mod(B) are NOT necessarily isomorphic to full subcategories of Mod(Λ (ϕ,ψ) ) in the obvious 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, which need not be true.
Subcategories continued We now restrict our attention ( to generalized ) matrix artin algebras A N with ϕ = ψ = 0, i.e., Λ (0,0) =. One reason for this restriction is that we have the following categorical full embeddings. Let L denote the full subcategory of M whose objects are of the form (X, Y, f, 0), with f an epimorphism. from ( A 0 Mod(A) L Mod( ) ) Mod(Λ (0,0) ) {(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 of the form (X, Y, 0, g), with g an epimorphism, we have the following full embeddings: from ( Mod(B) L A N Mod( 0 B ) ) Mod(Λ (0,0) ) {(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 category A, we say that B is covariantly finite if, for each A A, there is some B A B and a map A B A such that, for all B B, the induced map Hom A (B, B ) Hom A (A, B ) is surjective. We say that B is contravariantly finite if, for each A A, there is some C A B and a map C A A such that, for all B B, the induced map Hom A (B, C A ) Hom A (B, A) is surjective. Finally we say that B is functorially finite if it is both covariantly and contravariantly finite.
Categorical finiteness of subcategories. We now state our main result on finiteness of subcategories. Theorem ( ) A N Let Λ (0,0) = be a generalized matrix artin algebra. Then 1. The category L = {(X, Y, f, 0) f epi} is contravariantly finite in mod(λ (0,0) ), closed under extensions and quotients, and T A (Mod(A)) L. 2. The category L = {(X, Y, 0, g) g epi} is contravariantly finite in mod(λ (0,0) ), closed under extensions and quotients, and T B (Mod(B)) L.
Theorem continued Theorem (con t) 1. The category L = {(X, Y, f, 0) f mono} is covariantly finite in mod(λ (0,0) ), closed under extensions and quotients, and T A (Mod(A)) L. 2. The category L = {(X, Y, 0, g) g mono} is covariantly finite in mod(λ (0,0) ), closed under extensions and quotients, and T B (Mod(B)) L.
Theorem continued Theorem (con t) 3. The categories mod(a) and mod(b) are functorially finite in mod(λ (0,0) ). ( ) ( ) A 0 A N 4. The categories mod( ) and mod( ) are 0 B 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 the canonical surjection. Since ϕ: M A N B is 0 and since M A N B Y 1 M g M A X ϕ 1 Y mult f commutes, we see that there B B Y Y is a factorization M A X 1M π M A coker(g) 0 f h Y Check that (coker(g), Y, h, 0) M. We claim (X, Y, f, g) (π,1 Y ) ( ) (coker(g), Y, h, 0) is a left A 0 mod( )-approximation.
Proof continued Let (c, d): (X, Y, f, g) (U, V, j, 0). Then g c = 0 and there is an induced map e : coker(g) U such that (X, Y, f, g) (π,1 Y ) (coker(g), Y, h, 0) (c,d) (e,d) (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 Hom B (M, Y ) and k : Y Hom A (N, X ) satisfying the appropriate ( conditions. ) A right A 0 Mod( )-approximation of (X, Y, h, k) is (X, ker(k), l, 0), where l : X Hom B (M, Y ) is defined is a similar fashion to h above.
An example Example 4: Let K be a field and Q be the quiver v a w. b Let Λ = KQ/ ab, ba. Let P v (respectively P w ) be the projective Λ-cover of the simple module at v (resp. w). Then Λ = P v P w. Hence Λ is isomorphic to the generalized matrix algebra ( Λ (0,0) EndΛ (P) op Hom Λ (P, Q) = Hom Λ (Q, P) End Λ (Q) op Each entry in this 2 2-matrix ( is K) but the multiplication of two 0 0 elements, one of the form and the other of the form α 0 ( 0 β 0 0 ), in any order, is 0. ).
Example continued ( ) A N Thus, as a generalized matrix algebra, A = B = M = N = K and ϕ = ψ = 0. Hence A and B have global dimension 0 and M and N have projective dimension 0 over both A and B and as bimodules. But Λ has infinite global dimension. From ( this ) example, we see that, unlike a ring of the form A 0, finite global dimension of A and B and hence M and N is not ( sufficient for ) finite global dimension of A N Λ (0,0) =
Tight projective modules and resolutions ( ) A N Let Λ (0,0) =. If P = (P A, 0, 0, 0) is a projective Λ-module for some left A-module P A, then we say that P is an A-tight projective Λ-module. Note that if (P A, 0, 0, 0) is an A-tight projective Λ-module then P A is a projective A-module and M A P A = 0. Conversely, if P A is a projective A-module and M A P A = 0, then (P A, 0, 0, 0) is an A-tight projective Λ-module. We say a left A-module (X, 0, 0, 0) has an A-tight projective Λ-resolution if X has a projective Λ-resolution in which each projective Λ-module is A-tight.
Basic facts about tight projective resolutions 1. A direct sum of modules having A-tight projective Λ-resolutions also has an A-tight projective Λ-resolution. 2. A summand of an A-tight projective Λ-module is again an A-tight projective Λ-module. 3. If X is an A-module such that (X, 0, 0, 0) has an A-tight projective Λ-resolution, then pd Λ (X, 0, 0, 0) = pd A (X ). 4. The subcategory of modules with A-tight projective resolutions is closed under extensions. Proposition A Λ (0,0) -module of the form (X, 0, 0, 0) has an A-tight projective Λ (0,0) -resolution if and only if M A P = 0, where P is the direct sum of projective A-modules in a minimal projective A-resolution of M.
B-tightness and first bound result We define B-tight projective Λ-modules, (0, Q, 0, 0) is a similar fashion as A-tight projective Λ-modules and also Λ-modules (0, Y, 0, 0) having B-tight projective Λ-resolutions. Proposition ( ) A N Suppose that Λ (0,0) = 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 projective Λ-resolution, then pd Λ ((X, 0, 0, 0)) pd A (X ) + 1 + pd B (M). If (N, 0, 0, 0), as a left Λ-module, has an A-tight projective Λ-resolution, then pd Λ ((0, Y, 0, 0)) pd B (Y ) + 1 + pd A (N).
Global dimension result Theorem ( ) A N Suppose that Λ (0,0) = is a generalized matrix artin algebra, M has a B-tight projective Λ (0,0) -resolution, and N has an A-tight projective resolution. Then gl.dim(λ (0,0) ) gl.dim(a) + gl.dim(b) + 1.
Examples and sharpness of result We provide two examples, the first of which shows that the inequality is sharp and the second shows that the inequality can be proper. Example 5 Let Q be the quiver v 1 a w 1 v 2 d w 2 b v 3 c v 4 g w 3 e f w 4 Let I =< arrows > 2 and let Λ = KQ/I. gldim(λ) = 4 Set ɛ 1 = v 1 + v 2 + v 3 + v 4 and ɛ 2 = w 1 + w 2 + w 3 + w 4. View Λ as the generalized matrix artin algebra ( ɛ1 Λɛ 1 ɛ 1 Λɛ 2 ɛ 2 Λɛ 1 ɛ 2 Λɛ 2 ).
Example 5 continued The global dimension of ɛ 1 Λɛ 1 is 2 and the global dimension of ɛ 2 Λɛ 2 is 1. Thus gl.dim(λ) = gl.dim(ɛ 1 Λɛ 1 ) + gl.dim(ɛ 2 Λɛ 2 ) + 1. The following can be checked 1. M = ɛ 2 Λɛ 1, which, as a left ɛ 2 Λɛ 2 -module is isomorphic the simple module at w 1. 2. N = ɛ 1 Λɛ 2, which, as a left ɛ 1 Λɛ 1 -module is isomorphic to the simple module at v 2. 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 v 1 a v 2 b v 3 c d w 1 e w 2 We again take I to be the ideal generated by all paths of length 2 and set Λ = KQ/I. Now set ɛ 1 = v 1 + v 2 + v 3 and ɛ 2 = w 1 + w 2. View Λ as the generalized matrix artin algebra ( ) ɛ1 Λɛ 1 ɛ 1 Λɛ 2. ɛ 2 Λɛ 1 ɛ 2 Λɛ 2 It is easy to check that the hypotheses of Theorem above are satisfied. But the global dimension of Λ is 2 while the global dimension of ɛ 1 Λɛ 1 is 2 and the global dimension of ɛ 2 Λɛ 2 is 1.
The non-tight case We consider the case where either (0, M, 0, 0) or (N, 0, 0, 0) does not 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, 1 N 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 we have little control. The next bound results will require that M, as a left B-module, is projective and N, as a left A-module is projective.
Second bounding result Theorem ( ) A N Suppose that Λ (0,0) = is a generalized matrix artin algebra such that, the global dimensions of A and B are finite, M is 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 above theorem are sharp.
Sharpness of the result. Example 7 LetQ be the quiver v 1 v 3 v 2 v 4 Let Λ be the quotient KQ/I, where I is the ideal generated by all paths of length 2. Let ɛ 1 = v 1 + v 3 + v 5 and ɛ 2 = v 2 + v 4. View Λ as a the generalized matrix artin algebra ( ɛ1 Λɛ 1 ɛ 1 Λɛ 2 ɛ 2 Λɛ 1 ɛ 2 Λɛ 2 ik j denotes the simple Λ-module, which on the left is isomorphic to S i, and on the right is isomorphic to S j. v 5 ).
Example 7 continued Then M = ɛ 2 Λɛ 1 = 4 K 3 2 K 1, and N = ɛ 1 Λɛ 2 = 5 K 4 3 K 2. gl.dima = gl.dimɛ 1 Λɛ 1 = gl.dimb = gl.dimɛ 2 Λɛ 2 = 0. M is a projective left B-module and N is a projective left A-module and we see that N B (M A N) is isomorphic to 5 K 2. Moreover, (N B (M A N), 0, 0, 0) is an A-tight projective Λ-module. Thus, we can apply part (3) of Theorem 2.9 with s = 1 to get gl.dim(λ) 4. But the global dimension of Λ is 4. This example can be adjusted to get that all the inequalities are sharp.
Lower bounds Theorem ( ) A N Let Λ (ϕ,ψ) = be a generalized matrix artin algebra. Suppose that M A is a flat A-module and N B is a flat B-module. Then gl.dim(λ ϕ,ψ ) max{gl.dim(a), gl.dim(b)}