Representations of (D,O)-species and flat mixed matrix problems
Nadiya Gubareni
Journal of Applied Mathematics and Computational Mechanics |
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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}, _{i}M_{j})_{i,j }_{Î}_{ I}_{ }, where F_{i} = for i = 1, 2, ..., k, and F_{j} = D for j = k +1, ..., n, moreover _{i}M_{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, _{i}M_{j} = 0 if F_{j} = O_{j}, and _{i}M_{j} = _{j}M_{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}, _{j}j_{i}, _{j}y_{r}) of a weak (D,O)-species W = {F_{i}, _{i}M_{j}}_{i,j}_{ÎI} 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}, _{i}M_{j})_{i,j}_{ÎI} 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
T_{11} |
… |
T_{1j} |
… |
T_{1m} |
… |
… |
… |
… |
… |
T_{i}_{1} |
… |
T_{ij} |
… |
T_{1m} |
… |
… |
… |
… |
… |
T_{n}_{1} |
… |
T_{nj} |
… |
T_{nm} |
be a block rectangular matrix T with entries in D partitioned into n horizontal strips T_{1}, …, T_{n} and m vertical strips T^{1}, …, 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_{r}ÎDÈD to rows in the strip T_{i}.
4. Additions of columns in the strip T^{i} multiplied on the right by elements of F_{p}ÎDÈD 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) = (d_{1},d_{2}, … , d_{n}; d^{1}, d^{2}, …, d^{m}), | (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}, _{i}M_{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}, _{j}j_{i}, _{j}y_{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 = (M_{i}, V_{r}, _{j}j_{i}, _{j}y_{r}) | (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},_{ j}j_{i},_{ j}y_{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:
W_{i} B_{ij} =_{ }U_{j} (i =1, …, k; j = k+1, …, n) | (3.6) |
W_{j} A_{jr} =_{ }W_{r} (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:
W_{i} B_{ij}_{ }_{ }=_{ } (i = 1, …, k; j = k+1, …, n) | (3.8) |
W_{j} A_{jr}_{ }=_{ } (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 (X_{1}, …, X_{n}) and Y = diag (Y_{1}, …, 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},_{ i}M_{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 _{i}M_{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 _{i}M_{i} = 0 for all i, and _{i}M_{j} = 0 for j £ i.
A vertex i_{ }Î_{ }I is called marked if F_{i} = . Let I_{1} = {1, 2, ..., k} be the set of marked vertices of an O-species W. A marked vertex i_{ }Î_{ }I_{1} is called minimal if _{i}M_{j} = _{j}M_{i} = 0 for all j_{ }Î_{ }I_{1}. 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},_{ i}M_{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}, _{i}M_{j})_{i,j }_{Î I} in [1], the tensor algebra T(W) =_{ }, where T_{0} =_{ }, T_{i}_{+1 }= T_{i} Ä_{B} M and M =_{ }, was constructed.
Lemma 4.1. Let W = (F_{i}, _{i}M_{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
A_{ij} =_{ } | (4.2) |
Since all _{i}M_{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 V_{12} is a (D, D)-bimodule, is of finite representation type if and only if V_{12} 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},_{ i}M_{j})_{i,j }_{Î I} we can also consider a D-species , where since each _{i}M_{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 W_{1} = (F_{i},_{ i}M_{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 W_{1} is a D-species, T is an Artinian ring and so it is of finite representation type. Since W is simply connected, W_{1} 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¢¢ Ä_{A} = (M Ä_{A}_{ }) Ä_{A}_{ } |
Taking into account (4.5), we have that , where M_{i} is an O_{i}-module and M_{0} is a T-module. Then
M Ä_{A}_{ }=_{ }Ä_{A} =_{ } |
M¢¢ = (M Ä_{A}_{ }) Ä_{A} =_{ } |
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 (j_{i}) ® 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¢ = N_{1 }Å … Å N_{t}. |
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
m_{A}(N_{i}) =_{ } 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.
References
[1] Gubareni N., O-species and tensor algebras, Journal of Applied Mathematics and Computational Mechanics 2016, 2(14).
[2] Gabriel P., Indecomposable representations I, Manuscripta Math. 1972, 6, 71-103.
[3] Drozd Yu.A., The structure of hereditary rings (in Russian), Mat. Sbornik 1980, 113(155), N.1(9), 161-172. English translation: Math. USSR Sbornik 1982, 41(1), 139-148.
[4] Zavadskij A.G., Revitskaya U.S., A matrix problem over a discrete valuation rings, Mat. Sb. 1999, 190(6), 835-858.
[5] Gubareni N.M., Semiperfect right hereditary rings of module representation type, Preprint 78.1, Academy of Sciences of USSR, IM, Kiev 1978 (in Russian).
[6] Gubareni N.M., Right hereditary rings of bounded representation type, Preprint-148 Inst. Electrodynamics Akad. Nauk Ukrain. SSR, Kiev 1977 (in Russian).
[7] Zavadskij A.G., Kirichenko V.V., Torsion-free modules over primary rings, Zap. Nauchn. Sem. LOMI 1976, 57, 100-116 (in Russian). English translation: J. Soviet Math. 1979, 11, 598-612.
[8] Zavadskij A.G., Kirichenko V.V., Semimaximal rings of finite type, Mat. Sb. 1977, 103, 323-345.
[9] Gubareni N., Some mixed matrix problems over several discrete valuation rings, Journal of Applied Mathematics and Computational Mechanics 2013, 4(12), 47-58.
[10] Hazewinkel M., Gubareni N., Kirichenko V.V., Algebras, Rings and Modules. Vol. 1, Mathematics and Its Applications, v.575, Kluwer Academic Publisher, Dordrecht/Boston/London 2004.
[11] Hazewinkel M., Gubareni N., Kirichenko V.V., Algebras, Rings and Modules. Vol. 2, Springer, 2007.
[12] Roiter A.V., Unbounded dimensionality of indecomposable representations of an algebra with an infinite number of indecomposable representations of an algebra with an infinite number of indecomposable representations, Math. USSR Izv. 1968, 2(6), 1223-1230.
[13] Auslander M., Representation theory of Artin algebras II, Comm. Algebra 1974, 1, 269-310.
[14] Gubareni N., Structure of finitely generated modules over right hereditary SPSD-rings, Scientific Research of the Institute of Mathematics and Computer Science 2012, 3(11), 45-55.
[15] Rowen L.H., Finitely presented modules over semiperfect rings, Proc. Amer. Math. Soc. 1986, 97(1), 1-7.