IRREDUCIBLE MORPHISMS, THE GABRIEL-VALUED QUIVER AND COLOCALIZATIONS FOR COALGEBRAS

Given a basic K-coalgebra C, we study the left Gabriel-valued quiver (CQ,Cd) of C by means of irreducible morphisms between indecomposable injective leftC-comodules and by means of the powers rad of the radical rad of the category C-inj of the socle-finite injective left C-comodules. Connections between the valued quiver (CQ,C d) of C and the valued quiver (CQ,Cd) of a colocalization coalgebra quotient fE : C→ C of C are established.


Introduction
Throughout this paper we fix a field K. Given a K-coalgebra C we denote by C-Comod and C-comod the categories of left C-comodules and left C-comodules of finite K-dimension, respectively. Given a left C-comodule M, we denote by soc C M the socle of M, that is, the sum of all simple C-subcomodules of M. We call M socle-finite (or finitely copresented) if dim K socM is finite. Following [17, page 404], a K-coalgebra C is called basic if the left C-comodules C C and soc C C have direct sum decompositions: where I C is a set, E( j) is an indecomposable injective comodule, S( j) is a simple comodule, and E( j) is the injective envelope of S( j), for each j ∈ I C , E(i) ∼ = E( j), and S(i) ∼ = S( j), for i = j. It was shown in [22] that a K-coalgebra C is basic if and only if dim K S = dim K End C S, for any simple left C-comodule S. Throughout this paper we assume that C is a basic K-coalgebra, the decompositions (1.1) are fixed, and we set 2 Irreducible morphisms and the Gabriel-valued quiver for each j ∈ I C . In this case {S( j)} j∈IC is a complete set of all pairwise nonisomorphic simple left C-comodules. We recall from [10,Definition 4.3] and [19,Definition 8.6] that the left Gabriel-valued quiver of C is the valued quiver where C Q 0 = I C is the set of vertices, C Q 1 is the set of valued arrows, and, given two vertices i, j ∈ C Q 0 , there exists a unique valued arrow In other words, ( C Q, C d) is the opposite to the left valued Ext-quiver of C (see [4,8,14]), which is the valued quiver (Q C Ᏹxt ,d C Ᏹxt ) of the left Ext-species C Ᏹxt of C; see [10]. In practice, it is useful to work with an equivalent form of the valued quiver ( C Q, C d). We define it in Section 2, by applying the well-known concepts of the Auslander-Reiten theory for finite dimensional algebras, see [1], [2,Section 5.5], and [18,Section 11.1]. We introduce the notion of an irreducible morphism between left C-comodules, and we give an equivalent description of the quiver ( C Q, C d) in terms of irreducible morphisms between socle-finite injective C-comodules. Then we study the valued quiver ( C Q, C d) by means of irreducible morphisms between the indecomposable injective C-comodules E( j), by means of the K-species; see (2.10), [10, equation (4.9)], and by means of the powers rad m (E(i),E( j)) of the radical rad of the full subcategory C-inj of C-Comod formed by the socle-finite injective C-comodules. In particular, we show that the existence of a valued arrow (1.4) in the quiver ( C Q, C d) is equivalent to the existence of an irreducible morphism E( j) → E(i) in the category C-inj. One of the main results of this paper is Theorem 2.3 of Section 2. It asserts that, for each pair of indices i, j ∈ I C , we have is nonzero, then there exist an integer m i j ≥ 1 and a path of irreducible morphisms ϕ 1 ,...,ϕ mij in C-inj such that the composition ϕ mij ··· ϕ 1 is nonzero. If, in addition, the vector space Hom C (E( j),E(i)) is of finite K-dimension, then there exists a finite set U i j ⊆ I C such that rad mij (E( j),E(i)) = 0 and rad 1+mij (E( j),E(i)) = 0, and every noninvertible nonzero C-comodule homomorphism f : .., f srs in C-inj, where λ s ∈ K, λ s = 0, r s ≤ m i j , for s = 1,...,t, and j sa ∈ U i j , for all a = 1,...,r s and s = 1,...,t.
Hence we conclude, in Corollary 2.4, that the coalgebra C is a direct sum of two nonzero subcoalgebras if and only if the valued quiver ( C Q, C d) is disconnected. In particular, this implies a new proof of [14,Corollary 2.2].
In Section 3, we study a relationship between the valued quiver ( C Q, C d) of the coalgebra C and the valued quiver ( C Q, C d) of a colocalisation coalgebra quotient of C with respect to an injective comodule where U is a subset of I C ; see Section 3 for details. We show in Theorem 3.2 that if E is as above and, for each j ∈ U, the comodule E( j)/S( j) is E-copresented then the left Gabrielvalued quiver ( C Q, C d) of the coalgebra C = C E has C Q 0 = U and is isomorphic to the restriction of the left Gabriel-valued quiver ( C Q, C d) of C to the subset U ⊆ I C = C Q 0 . Throughout, we use the coalgebra representation theory notation and terminology introduced in [19][20][21]. In particular, given a coalgebra C and a pair of left C-comodules M and N, we denote by Hom C (M,N) the vector space of all C-comodule homomorphisms f : M → N, and by End C M the algebra of all C-comodule endomorphisms g : Given a K-coalgebra C, we denote by C * = Hom K (C,K) the K-dual algebra with respect to the convolution product (see [6,13,23]) viewed as a pseudocompact K-algebra (see [7,19]). The category of pseudocompact left C * -modules is denoted by C * -PC.
Given a ring R with an identity element, we denote by J(R) the Jacobson radical of R, and by mod(R) the category of finitely generated right R-modules.
The reader is referred to [3,6,13,23] for the coalgebra and comodule terminology, and to [1,2,18] for the standard representation theory terminology and notation.

The Gabriel-valued quiver of a coalgebra and irreducible morphisms
Assume that K is an arbitrary field and C is a basic K-coalgebra. We fix the decompositions (1.1), and we set F j = End C S( j), for each j ∈ I C , as in (1.2). Let ( C Q, C d) be the Gabriel-valued quiver ( C Q, C d) of C defined in (1.3). 4 Irreducible morphisms and the Gabriel-valued quiver In this section we present an equivalent form of the valued quiver ( C Q, C d) in terms of irreducible morphisms between injective C-comodules. One of the applications of this new description is to compute the left Gabriel-valued quiver ( CE Q, CE d) of the coalgebra C E = e E Ce E in terms of the left Gabriel-valued quiver ( C Q, C d) of C, given in Section 3.
We denote by C-inj the full subcategory of C-Comod formed by the socle-finite injective C-comodules. Note that a comodule E lies in C-inj if and only if E is isomorphic with a finite direct sum of the comodules E( j), with j ∈ I C .
Irreducible morphisms in any full subcategory of C-Comod are defined analogously.
(b) Given two comodules E and E in C-inj, define the radical of Hom C (E ,E ) to be the K-subspace of Hom C (E ,E ) generated by all nonisomorphisms ϕ : The square of rad(E ,E ) is defined to be the K-subspace and call it the infinite radical of Hom C (E ,E ). Then the chain of vector spaces is defined: Simson 5 The following simple lemma is very useful.
The first statement is an immediate consequence of definition. To prove the second one, we apply the standard Auslander-Reiten theory arguments; see [18, page 174]. For the convenience of the reader we present a proof.
Assume, to the contrary, that are the homomorphisms f = ( f 1 ,..., f t ) and f = ( f 1 ,..., f t ). Since f is irreducible, then f is a section or f is a retraction.
First, assume that f is a section. Then there exists a C-comodule homomorphism Since f s ∈ rad(E(i),Z s ) and Z s is indecomposable, then f s is not an isomorphism and, hence, r s f s is noninvertible, for any s∈{1,...,t}.
Since, by (a), the algebra To prove that f is irreducible, assume that f has a factorization .., f t ) and f = ( f 1 ,..., f t ) are as in ( * ) and the comodules Z 1 ,...,Z t are indecomposable. Since f ∈ rad 2 (E(i),E( j)), then, according to (a), there is an index a ∈ {1, ...,t} such that the map f a is bijective or there is an index b ∈ {1, ...,t} such that the map f b is bijective. It follows that f is a section or f is a retraction. This shows that f is an irreducible morphism in C-inj and finishes the proof of the lemma.
Following the finite-dimensional algebras terminology; see [1], [2, Section 5.5], [10], and [18, Section 11.1], we call the K-vector space the bimodule of irreducible morphisms, for each pair i, j ∈ I C ; see [10,20]. It is easy to see that the K-vector Following [10, Definition 4.9], we consider the K-species One of the main results of this paper is the following useful theorem. (a) There exists a unique valued arrow i

E(i)) is nonzero and the numbers (1.5) have the forms
For each i, j ∈ I C and any noninvertible nonzero homomorphism f ∈ Hom C (E( j), E(i)), there is an integer m ≥ 1 such that (2.12) In this case there is a path is finite and rad(E( j),E(i)) = 0, then there exist an integer m i j ≥ 1 and a finite subset U i j of I C such that

14)
and every noninvertible nonzero homomorphism (b) We recall that the Yoneda map ϕ → ε • ϕ defines an isomorphism Λ C = End C C ∼ = C * of pseudocompact algebras; see also [19,Sections 3 and 4]. Note that, given j ∈ I C , the direct summand projection C C → E( j) of left C-comodules induces a direct summand injection of left pseudocompact Λ C -modules and an isomorphism E( j) * ∼ = C * e j , where e j is the primitive idempotent of Λ C defined by the direct summand injection Hom C (E( j),C) Λ C . Then, in view of the duality C-Comod ∼ = (C * -PC) op given by M → M * (see [19, Theorem 4.5]), for each pair i, j ∈ I C , we get F j -F i -bimodule dualities and for each m ≥ 2, we get F j -F i -bimodule dualities see [11,12] and [9, Section 4]. It then follows that ∞ m=0 e j J(Λ C ) m e i = 0 and consequently rad ∞ (E( j),E(i)) = 0.
Assume that f ∈ Hom C (E( j),E(i)) is a nonzero non-isomorphism. Then f ∈ rad(E( j), E(i)) and there is an integer q = q( f ) ∈ {1, ...,m i j } such that f ∈ rad q E( j),E(i) \ rad q+1 E( j),E(i) . (2.21) 8 Irreducible morphisms and the Gabriel-valued quiver We show, by induction on m i j − q( f ) ≥ 0 that f is a K-linear combination of compositions of irreducible morphisms between indecomposable injective comodules. First, assume that m i j − q( f ) = 0, that is, q( f ) = m i j and f ∈ rad mij C (E( j),E(i)). Then f is a finite sum: of length q = m i j and with λ s ∈ K and f sa ∈ rad(E( j),E(i)), for any s and a = 1,..., q.
Since rad q+1 (E( j),E(i)) = 0, then each f sa is an irreducible morphism, by Lemma 2.2, and we are done.
for all 1 ≤ a ≤ q. It follows that each such a homomorphism f sa is an irreducible morphism. By the choice of f , we get f ∈ rad 1+q( f ) (E( j),E(i)) and therefore q( f ) ≥ q( f ) + 1. Since, by induction hypothesis, f is a K-linear combination of compositions of irreducible morphisms, then so is f = f + f , and we are done. Let ψ 1 ,...,ψ b be a K basis of rad(E( j),E(i)). By applying the above to each of the basis element ψ 1 ,...,ψ b , we find a finite subset U i j of I C such that each f ∈ {ψ 1 ,...,ψ b } is a finite K-linear combination f = t s=1 λ s f srs ··· f s2 f s1 ( * ) of compositions (2.15) of irreducible morphisms f srs ,..., f s2 , f s1 in C-inj, where λ s ∈ K and r s ≤ m i j , for s = 1,...,t, and j sa ∈ U i j , for all a = 1...,r s and s = 1,...,t. It follows that the same holds, for any nonzero element f ∈ rad(E( j),E(i)). This finishes the proof of the theorem.
As a consequence of the properties of irreducible morphisms proved in Lemma 2.2 and Theorem 2.3, we get the following important corollary. We note that the second part of it was proved in [19,Corollary 8.7], by applying the Ext-quiver of C and [14, Corollary 2.2].
where E( j s ) E( j s+1 ) means Hom C (E( j s ),E( j s+1 )) = 0 or Hom C (E( j s+1 ),E( j s )) = 0. In view of (a), the former condition is satisfied, if the valued quiver ( C Q, C d) is connected.
The converse implication follows from the fact that Hom C (E(a),E(b)) = 0 implies the existence of a path E(a) Corollary 2.5. Assume that C is a basic K-coalgebra with fixed decompositions (1.1), and assume that C is left computable, that is, dim K Hom C (E( j),E(i)) is finite, for all i, j ∈ I C ; see [22]. Proof. Since dim K Hom C (E( j),E(i)) is finite, for all i, j ∈ I C , then Theorem 2.3 applies and the corollary follows.
10 Irreducible morphisms and the Gabriel-valued quiver

The Gabriel-valued quiver of a colocalization coalgebra C E
One of the main aims of this section is to compute the left Gabriel-valued quiver ( CE Q, CE d) of the colocalisation coalgebra C E = e E Ce E (defined below) in terms of the left Gabriel-valued quiver ( C Q, C d) of C.
Assume that C is a basic K-coalgebra with fixed decompositions (1.1), and that E is an injective left C-comodule of the form where I E is a subset of I C . In [22], we associate to E the coalgebra surjection where C E is the topological K-dual coalgebra to the pseudocompact K-algebra End C E, called the colocalisation coalgebra quotient of C at E. More precisely, the topological Kdual vector space of End C E is equipped with a natural coalgebra structure induced by the pseudocompact K-algebra structure of End C E; see [19,22] for details, compare with [16]. It is shown in [22] that there is a coalgebra isomorphism C E ∼ = e E Ce E , and the kernel of the coalgebra surjection f E : C → C E is the coideal where e E is the idempotent of C * defined by the direct summand embedding E C. We know from [5,22,24] that the restriction functor (3.5) given by M → Me E , is exact and has a right adjoint and the kernel Ker E of E is a localizing subcategory of C-Comod in the sense of Gabriel [7]. Conversely, every localizing subcategory of C-Comod is of this form; see [15,24]. Now we show that, under a suitable assumption on E, the left Gabriel-valued quiver ( C Q, C d) of the coalgebra C is the restriction to the subset I E = C Q 0 of I C = C Q 0 of the valued quiver ( C Q, C d) of the colocalisation coalgebra quotient homomorphism of C at the injective comodule E = j∈IE E( j).

Daniel Simson 11
To formulate the main result of this section we need some notation. We define a comodule M in C-Comod to be E-copresented, if M admits an E-injective copresentation, that is, that there is a short exact sequence where E 0 and E 1 are direct sums of direct summands of the comodule E.
If, in addition, the comodules E 0 and E 1 are socle-finite, we say that the sequence is a socle-finite E-injective copresentation, and then M is called a finitely E-copresented comodule.
We denote by the full subcategories of C-Comod consisting of the E-copresented comodules and the finitely E-copresented comodules, respectively. We set  (a) If, for each j ∈ I E , the comodule E( j)/S( j) is E-copresented, then the left Gabrielvalued quiver ( C Q, C d) of the coalgebra C = C E has C Q 0 = I E and is isomorphic to the restriction of the valued quiver ( C Q, C d) (1.4) of C to the subset I E ⊆ I C = C Q 0 .
(b) Let (U,d) be a finite full convex valued subquiver of the valued quiver ( C Q, C d) of the coalgebra C. Given j ∈ U, denote by e j ∈ R E the primitive idempotent defined by the left ideal Hom C (E( j),E) ⊆ R E , which is a direct summand of R E . If the algebra and induces F i -F j -bimodule isomorphisms for all i, j ∈ U, and (b2) the right Gabriel-valued quiver (Q RE ,d) of the algebra R E is isomorphic with (U,d).
Proof. (a) Assume that, for each j ∈ I E , the comodule E( j)/S( j) is E-copresented, and consider the pair of adjoint functors (3.16) defined by the formulae res E (−) = (−)e E and E (−) = e E C CE (−). By [22, Proposition 2.7 and Theorem 2.10], the K-coalgebra C E is basic, E is a full and faithful K-linear functor such that res E • E ∼ = id. The functor E is right adjoint to res E , the functor res E is exact, and E is left exact and restricts to the functor E : C E -comod → C-comod. Moreover, for each j ∈ I E , the left C E -comoduleŠ( j) = res E S( j) is simple, socC E ∼ = j∈IEŠ ( j), andŠ(i) ∼ =Š( j), for i = j, i, j ∈ I E . Since, for each j ∈ I E , the comodule E( j)/S( j) is E-copresented, then the left C-comodule socE( j)/S( j) is (up to isomorphism) a direct sum of copies of simple comodules S(i), with i ∈ I E . Then, according to [22,Corollary 2.14], for each j ∈ I C , there are isomorphisms S( j) ∼ = E res E S( j) ∼ =Š( j), the simple C-comodule S( j) lies in C-Comod E , and the minimal injective three-term copresentation of a basic-finite dimensional algebra B is defined as follows. Fix a complete set {e j } j∈U of primitive orthogonal idempotents of B such that B = j∈U e j B. Note that, given j ∈ U, the algebra D j = e j Be j /e j J(B)e j is a division ring. Moreover, given i, j ∈ U, the vector space e i J R E e j /e i J 2 R E e j ∼ = e i J R E /J 2 R E e j