Representations of (D,O)-species and flat mixed matrix problems



REPRESENTATIONS OF (D,O)-SPECIES and FLAT MIXED MATRIX PROBLEMS

Nadiya Gubareni

Institute of Mathematics, Czestochowa University of Technology
Częstochowa, Poland
nadiya.gubareni@yahoo.com

Abstract. The problem of describing representations of (D,O)-species is reduced to flat mixed matrix problems over discrete valuation rings and their common skew field of fractions.

Keywords: O-species, (D,O)-species, representations of (D,O)-species, (D,O)-species of bounded representation type, flat mixed matrix problem, discrete valuation ring

1. Introduction

We continue the study of (D,O)-species that was started in [1]. These species generalize the notion of species introduced by Gabriel [2] and are the special kind of species considered in [3].

Let {O_i} be a family of discrete valuation rings (not necessarily commutative) with a common skew field of fractions D. Consider a (D,O)-species W = (F_i, _iM_j)_{i,j Î I} , where F_i = for i = 1, 2, ..., k, and F_j = D for j = k +1, ..., n, moreover _iM_j is an (,)-bimodule that is finite dimensional both as the left D-vector space and as the right D-vector space, where is a classical ring of fractions of F_i for i = 1, 2, ..., n.

A (D,O)-species W is called weak if F_i = O_i for all i = 1, 2, ..., k, and moreover, _iM_j = 0 if F_j = O_j, and _iM_j = _jM_i = 0 for i,j ÎI and i ¹ j.

For (D,O)-species the representations of O-species were defined in [1]. A representation V = (M_i, V_r, _jj_i, _jy_r) of a weak (D,O)-species W = {F_i, _iM_j}_i,jÎI is a family of right F_i-modules M_i (i = 1, 2, ..., k), a set of right D-vector spaces V_r (r = k +1, k +1, ..., n) and D-linear maps:

for each i = 1, 2, ..., k; j = k +1, k +2, ..., n; and

for each r, j = k +1, k +2, ..., n.

A representation V is said to be finite dimensional if all M_i are finitely generated F_i-modules and all V_r are finite dimensional D-vector spaces. A (D,O)-species is of bounded representation type if the dimensions (see (3.13) in [1]) of its indecomposable finite dimensional representations have an upper bound.

In this paper, we show that the description of representations of (D,O)-species can be reduced to some flat mixed matrix problems over discrete valuation rings and their common skew field of fractions. The definition of such matrix problems is given in Section 2. These matrix problems are some sort of generalization of a flat matrix problem considered by Zavadskii and Revitskaya [4]. Earlier such matrix problems were considered by Gubareni [5, 6], and Zavadskii and Kirichenko [7, 8]. Some examples of such flat matrix problems were also considered in [9]. The reduction of the problem of description of (D,O)-species of bounded representation type to some flat mixed matrix problems is given in Section 3.

With each weak (D,O)-species W = (F_i, _iM_j)_i,jÎI we can associate a D-species , where In Section 4, we prove that if W is a simply con- nected weak (D,O)-species of bounded representation type, then is a D-species of finite representation type.

2. Flat mixed matrix problems

Let O be a discrete valuation ring (DVR) with a classical division ring of fractions D. By left O-elementary transformations of rows of a matrix T with entries in D we mean transformations of two types:

a) multiplying a row on the left by an invertible element of O;

b) adding a row multiplied on the left by an element of O to another row.

In a similar way we can define left D-elementary transformations of rows and, by symmetry, right O-elementary and right D-elementary transformations of columns.

Elementary transformations (a) and (b) can be given by invertible elementary matrices. The automorphism of a finitely generated module P corresponding to an elementary transformation is an elementary automorphism. Multiplications on the left (right) side of a matrix T by elementary matrices correspond to elementary row (column) transformations.

By [10, Proposition 13.1.3], any invertible matrix B over a local ring O can be reduced by O-elementary row (column) transformations on B to the identity matrix. By [10, Corollary 13.1.4], the matrix B can be decomposed into a product of elemen- tary matrices. Moreover, by [10, Theorem 13.1.6] any automorphism of a finitely generated projective module P over a semiperfect ring A can be decomposed into a product of elementary automorphisms.

Let D = {O_i}_{{i =1, …, k}} be a family of discrete valuation rings O_i with a common skew field of fractions D. We define the general flat matrix problem over D and D in the following way.

Let

T11	…	T1j	…	T1m
…	…	…	…	…
Ti1	…	Tij	…	T1m
…	…	…	…	…
Tn1	…	Tnj	…	Tnm

be a block rectangular matrix T with entries in D partitioned into n horizontal strips T₁, …, T_n and m vertical strips T¹, …, T^m so that each block T_ij is the intersection of the j-th vertical strip and the i-th horizontal strip; some of these blocks may be empty.

Assume that the ring Î D È D corresponds to the i-th horizontal strip T_i and the ring Î D È D corresponds to the j-th vertical strip T^j.

The following transformations with the matrix T are admissible:

1. Left -elementary transformations of rows within the strip T_i.

2. Right -elementary transformations of rows within the strip T^j.

3. Additions of rows in the strip T_j multiplied on the left by elements of F_rÎDÈD to rows in the strip T_i.

4. Additions of columns in the strip Tⁱ multiplied on the right by elements of F_pÎDÈD to columns in the strip T^j.

Indecomposable matrices and equivalent matrices are defined in a natural way.

A flat matrix problem is said to be of finite type if the number non-equivalent indecomposable matrices is finite.

Definition 2.1. The vector

d = d(T) = (d1,d2, … , dn; d1, d2, …, dm),

(2.2)

where d_i is the number of rows of the i-th horizontal strip of T for i = 1, ..., n and d ^j is the number of columns of the j-th vertical strip of T for j = 1, ..., m, is called the dimension vector of the partition matrix T. Also set

dim(T) =

(2.3)

Definition 2.4.

A flat matrix problem is said to be of bounded representation type if there is a constant C such that dim(X) < C for all indecomposable matrices X. Otherwise it is of unbounded representation type.

3. The main matrix problem

Let W = (F_i, _iM_j)_{i,j Î I}, where F_i = O_i for i = 1,2, …, k and F_j = D for j = k+1, …, n, be a weak (D,O)-species of bounded representation type.

Suppose that V = (M_i, V_r, _jj_i, _jy_r) is an indecomposable finite dimensional representation of W. Then M_i is a finitely generated F_i-module for i =1,2, …, k and V_r is a finite dimensional D-vector space for r = k +1, …, n. Since F_i = O_i is a discrete valuation ring, by [3, Proposition 5.4.18], any O_i-module M_i is torsion-free and faithful. Therefore any indecomposable representation of W has the following form:

V = (Mi, Vr, jji, jyr)

(3.1)

where M_i is a free F_i-module.

Consider the category R(W) whose objects are representations V = (M_i, V_r, _jj_i, _jy_r), and a morphism from an object V to an object is a set of homo- morphisms (a_i, b_r), in which is a homomorphism of F_i -modules, is a homomorphism of D-vector spaces (r = k +1, …, n), and the following equalities hold:

(3.2)

(3.3)

Let V be an indecomposable finite dimensional representation of the (D,O)-species W. Thus, each M_i is a finitely generated free O_i -module with basis (i = 1,2, …, k); and V_r is a finite dimensional D-space with basis (r = k +1, …, n).

Suppose

(3.4)

(3.5)

Then the matrices A_ij = (), B_ij = () define the representation V uniquely up to equivalence.

Let U_i Î be the matrix corresponding to the homomorphism a_i, and let W_i Î be the matrices corresponding to the homomorphisms b_i , i Î I. If , are the matrices corresponding to a representation V ¢ then the equalities (3.2) and (3.3) have the following matrix form:

Wi Bij = Uj (i =1, …, k; j = k+1, …, n)

(3.6)

Wj Ajr = Wr (j, r = k+1, …, n)

(3.7)

If representations V and V ¢ are equivalent, then a_i, b_r are isomorphisms. Therefore, the matrices U_i and W_r are invertible and the equalities (3.2) and (3.3) are equivalent to the following equalities:

Wi Bij = (i = 1, …, k; j = k+1, …, n)

(3.8)

Wj Ajr = (j, r = k+1, …, n)

(3.9)

Thus we obtain the following matrix problem for description of indecomposable finite dimensional representations of a (D,O)-species W.

Main mixed matrix problem

Let D = {O_i}_i=1,2,…,k be a family of discrete valuation rings O_i with a common skew field of fractions D.

Let T be a block matrix with entries in D partitioned into n horizontal strips {T_i}_{i=1,…,n} and m vertical strips {T^j}_{j=1,…,m} so that each block T_ij is the intersection of j-th vertical strip and i-th horizontal strip, some of these matrices may be empty.

The following transformations with the matrix T are admissible:

1. Left -elementary transformations of rows within the strip T_i, where

2. Right -elementary transformations of rows within the strip T^j, where

The admissible transformations with the matrix T can be given in the form T ® XTY, where X = diag (X₁, …, X_n) and Y = diag (Y₁, …, Y_m), and all X_i and Y_j are square invertible matrices. Moreover, X_i Ì , and Y_j Ì , where

Clearly, the matrix T is indecomposable if and only if the corresponding representation of W is indecomposable. It is easy to prove the following statement.

Lemma 3.10. A (D,O)-species W is of bounded representation type if and only if the corresponding main matrix problem is of bounded representation type.

4. Weak (D,O)-species of bounded representation type

Let W = (F_i, _iM_j)_{i,j Î I} be O-species. The quiver G(W) of an O-species W is defined as the directed graph whose vertices are 1, …, n, and there is an arrow from the vertex i to the vertex j if and only if _iM_j ¹ 0.

An O-species W is called acyclic if its quiver has no oriented cycles, i.e. the indices can be chosen so that _iM_i = 0 for all i, and _iM_j = 0 for j £ i.

A vertex i Î I is called marked if F_i = . Let I₁ = {1, 2, ..., k} be the set of marked vertices of an O-species W. A marked vertex i Î I₁ is called minimal if _iM_j = _jM_i = 0 for all j Î I₁. An O-species W is called min-marked if all its marked vertices are minimal.

An O-species W is simply connected if the underlying graph of G(W) is a tree.

A (D,O)-species W = (F_i, _iM_j)_{i,j Î I} is said to be weak if W is min-marked and all F_i, are O_i or D.

For each O-species W = (F_i, _iM_j)_{i,j Î I} in [1], the tensor algebra T(W) = , where T₀ = , T_i+1 = T_i Ä_B M and M = , was constructed.

Lemma 4.1. Let W = (F_i, _iM_j)_{i,j Î I}, where all F_i = D, be a simply connected D-species of finite representation type. Then the tensor algebra T(W) is a hereditary Artinian semidistributive ring.

Proof. Since W is a simply connected species, the tensor algebra T(W) is Morita equivalent to the algebra

where

Aij =

(4.2)

Since all _iM_j are finitely dimensional right and left D-spaces, A is an Artinian ring. From [11, Corollary 2.2.13] it follows that A is a hereditary ring.

Note that the ring

(4.3)

where V₁₂ is a (D, D)-bimodule, is of finite representation type if and only if V₁₂ has dimension 1 both as right and as left D-vector space. Since W is a D-species of finite representation type, the tensor algebra T(W) is of finite representation type as well, and so it does not contain a minor that is isomorphic to the ring (4.3). Therefore, A is a semidistributive ring.

Besides a weak (D,O)-species W = (F_i, _iM_j)_{i,j Î I} we can also consider a D-species , where since each _iM_j is an (,)-bimodule. Let T() be a tensor algebra of D-species . Since T() is an Artinian ring, by [12, 13] it is of bounded representation type if and only if it is of finite representation type.

Proposition 4.4. If W is a weak simply connected (D,O)-species of bounded representation type, then is a D-species of finite representation type.

Proof. Let W be a weak simply connected (D,O)-species with set of marked vertices J = {1,2, ..., k}. Then the tensor algebra A = T(W) is a basic primely triangular ring whose two-sided Peirce decomposition has the following form

(4.4)

where each U_i is a (D,T)-bimodule. Moreover, the ring T is the tensor algebra of a species W₁ = (F_i, _iM_j)_{i,j Î I \ J}, where F_i = D for all i Î I \ J.

Since W is a (D,O)-species of bounded representation type, then the tensor algebra T(W) is also is of bounded representation type by [1, Corollary 3.15]. Then by [1, Corollary 3.16], T is also of bounded representation type. Since W₁ is a D-species, T is an Artinian ring and so it is of finite representation type. Since W is simply connected, W₁ is also simply connected. By Lemma 4.1, T is an Artinian hereditary semidistributive ring.

Let be a right classical ring of fractions of A. We will use the following notation: if M is a right A-module, then M¢ = M Ä_A ; and if M is a right -module, then M¢ is the module M considered as an A-module. The length of a composition series of a right -module X is denoted by l(X).

Let us prove that for any right -module M there is a right -module X such that M¢¢ = M Å X.

We have

M¢¢ = M¢¢ ÄA = (M ÄA ) ÄA

Taking into account (4.5), we have that , where M_i is an O_i-module and M₀ is a T-module. Then

M ÄA = ÄA =

M¢¢ = (M ÄA ) ÄA =

By [14, Lemma 2], there is an injective torsion-free O_i-module for each i = 1, ..., k. Therefore, the mapping with for each d Î D is a monomorphism, i.e. exact sequences of O_i -modules exist:

0 ® ® Coker (ji) ® 0

Since D is injective, these sequences split, i.e. for i = 1, ..., k. Therefore,

M¢¢ = = = M Å X.

Now suppose that the ring A is of bounded representation type and the ring is of infinite representation type. Then for any N > 0 there is an indecomposable finitely generated -module M such that l(M) > N.

Consider the A-module M¢. It is finitely generated and, by [15, Proposition 1], it decomposes into a direct sum of finitely generated indecomposable A-modules:

M¢ = N1 Å … Å Nt.

Then

Since M¢¢ = M Å X, and M¢¢ is a finitely generated module over an Artinian ring , it follows from the uniqueness of the decomposition that there is a number i such that M is a direct summand of , i.e. there is an -module P such that = M Å P. We have the chain of inequalities

mA(Ni) = ³ l() = l(M) + l(P) ³ l(M) > N,

which contradicts the assumption that A is of bounded representation type.

5. Conclusions

The problem of describing representations of (D,O)-species has been reduced to some flat matrix problems over discrete valuation rings with common skew field of fractions. The main matrix problem for description of (D,O)-species of bounded representation type is given. We establish the connection of (D,O)-species of bounded representation type with D-species of finite representation type. We prove that if W is a weak simply connected (D,O)-species of bounded representation type, then the corresponding D-species is of finite representation type.

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