Krull Dimension of Tame Generalized Multicoil Algebras

We determine the Krull dimension of the module category of finite dimensional tame generalized multicoil algebras over an algebraically closed field, which are domestic.

F leads to a hierarchy of exact sequences in mod A, where the Auslander-Reiten sequences form the lowest level (see [10]). It is expected that the existence of K-dim(mod A) implies that A is domestic, that is, there is a common bound for the numbers of one-parameter families of indecomposable A-modules of any fixed dimension.
We would like to mention that the generalized multicoil algebras (respectively, tame generalized multicoil algebras) form a prominent class of algebras of global dimension at most 3, containing the class of quasitilted algebras of canonical type [14,30] (respectively, tame quasitilted algebras of canonical type), and are obtained by sophisticated gluings of concealed canonical algebras (respectively, tame concealed algebras) using admissible algebra operations (see Section 3 for details). Moreover, recently the tame generalized multicoil algebras showed to be important in describing the structure of the module category ind of an arbitrary cycle-finite algebra (see [18,Theorems 7.1 and 7.2] or [19,Theorem 1.8]). We also refer to the article [24] for the Hochschild cohomology of generalized multicoil algebras.
The following theorem is the main result of the paper.

Theorem 1.1 Let
A be a tame generalized multicoil algebra. The following statements are equivalent: In the representation theory of algebras a prominent role is played by the algebras with a separating family of components in the following sense. A family C = (C i ) i∈I of components of the Auslander-Reiten quiver A of an algebra A is called separating in mod A if the modules in indA split into three disjoint classes P A , C A = C and Q A such that: (S1) C A is a sincere generalized standard family of components; (S2) Hom A (Q A , P A ) = 0, Hom A (Q A , C A ) = 0, Hom A (C A , P A ) = 0; (S3) any morphism from P A to Q A factors through the additive category add C A of C A . We then say that C A separates P A from Q A and write A = P A ∪ C A ∪ Q A . We note that then P A and Q A are uniquely determined by C A (see [4, (2.1)] or [26, (3.1)]). Moreover, C A is called sincere if any simple A-module occurs as a composition factor of a module in C A , and generalized standard if rad ∞ (X, Y ) = 0 for all modules X and Y from C A . We refer also to the survey article [23] for the structure of arbitrary algebras with separating families of Auslander-Reiten components.
Frequently, we may recover A completely from the shape and categorical behaviour of the separating family C A of components of A . For example, the tilted algebras [12,26], or more generally double tilted algebras [25], are determined by their (separating) connecting components. Further, it was proved in [13] that the class of algebras with a separating family of stable tubes coincides with the class of concealed canonical algebras. This was extended in [21] to a characterization of algebras with a separating family of almost cyclic coherent Auslander-Reiten components. Recall that a component of an Auslander-Reiten quiver A is called almost cyclic if all but finitely many modules in lie on oriented cycles contained entirely in . Moreover, a component of A is said to be coherent if the following two conditions are satisfied: (C1) For each projective module P in there is an infinite sectional path P = X 1 → X 2 → · · · → X i → X i+1 → X i+2 → · · · (that is, X i = τ A X i+2 for any i ≥ 1) in .
It has been proved in [21, Theorem A] that the Auslander-Reiten quiver A of an algebra A admits a separating family of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a (possibly decomposable) concealed canonical algebra C. Moreover, for such an algebra A, we have that A is triangular, gl.dim A ≤ 3, and pd A X ≤ 2 or id A X ≤ 2 for any module X in ind A (see [21,Corollary B and Theorem E]).
As an immediate consequence of Theorem 1.1, Theorem 3.1, the definition of separating family of components of the Auslander-Reiten quiver A of an algebra A, and [17, Theorem 1.1] we obtain the following fact.

Corollary 1.2 Let
A be a tame algebra with a separating family of almost cyclic coherent Auslander-Reiten components. The following statements are equivalent:

Preliminaries
Throughout this paper, k will denote a fixed algebraically closed field. An algebra A will always mean a basic, connected (unless otherwise specified), associative finite dimensional k-algebra with an identity. Thus there exists a connected bound quiver (Q A , I A ) and an isomorphism A ∼ = kQ A /I A . Equivalently, A ∼ = kQ A /I A may be considered as a k-linear category, of which the object class A 0 is the set of points of Q A , and the set of morphisms A(x, y) from x to y is the quotient of the k-vector space kQ A (x, y) of all formal linear combinations of paths in Q A from x to y by the subspace I A (x, y) = kQ A (x, y) ∩ I A (see [7]). An algebra A with Q A acyclic (without oriented cycles) is said to be triangular. A full subcategory C of A is said to be convex if any path in Q A with source and target in Q C lies entirely in Q C .
By an A-module is meant a finitely generated right A-module. We denote by mod A the category of A-modules, by ind A the full subcategory consisting of a complete set of representatives of the isomorphism classes of indecomposable A-modules, by A the Auslander-Reiten quiver of A and by τ A the Auslander-Reiten translation in A . We shall agree to identify the vertices of A with the corresponding modules in ind A, and the components of A with the corresponding full subcategories of ind A. A component P of A is called postprojective if P is acyclic and every module in P lies in the τ A -orbit of a projective module. Dually, a component Q of A is called preinjective if Q is acyclic and every module in Q lies in the τ A -orbit of an injective module. Recall also that the Jacobson radical rad(mod A) of the module category mod A is the ideal of mod A generated by all noninvertible morphisms in ind A. Then the infinite radical rad ∞ (mod A) of mod A is the intersection of all powers rad i (mod A), i ≥ 1, of rad(mod A).
Let A be an algebra and Q be an infinite preinjective component of A . Let S be a set of indecomposable representatives of each infinite τ A -orbit of modules from Q. Moreover, assume that for any indecomposable module M from S there exist an indecomposable N from S and an irreducible morphism M → N or N → M. Then we say that S is left stable quasi-section of Q. Let C be an abelian category. A full subcategory C ⊆ C is called a Serre subcategory if it is closed under subobjects, quotients and extensions. If C ⊆ C is a Serre subcategory, then one defines the quotient category C/C as follows. The objects of C/C coincide with the objects of C, and if X and Y are objects of C, then Hom C/C (X, Y ) := lim − → Hom C (X , Y/Y ), where X and Y run through all subobjects of X and Y , respectively, such that X/X and Y belong to C . Again C/C is an abelian category and the quotient functor T : C → C/C is exact.
Let C be a small abelian category. The Krull-Gabriel filtration (C α ) α of C is defined as follows: C −1 = 0, C 0 is the Serre subcategory of all objects of finite length in C. In the case when α is an ordinal number of the form β + 1 then C α is defined to be the Serre subcategory of all objects in C which become of finite length in C/C β . If α is a limit ordinal, then C α is the union of all C β with β < α. If there exists an ordinal α with C α = C, then the smallest ordinal with this property is called the Krull dimension of C, denoted by K-dim C. We shall also denote by T 0 and T 1 the quotient functors T 0 : C → C/C 0 and T 1 : C → C/C 1 , respectively.
Let D be a subcategory of mod A for some algebra A. Denote by F(D) the category of finitely presented contravariant functors from D to the category Ab of abelian groups. Assume that F(D) is abelian. Then K-dim D is by definition the Krull dimension of F(D).
The following result from [32, Lemma 2.1] will be applied.
In the proof of our main result we need also the following fact.

Lemma 2.2 Let
For basic background on the representation theory of algebras applied in the paper, we refer to the books [1,[26][27][28].

Tame Generalized Multicoil Algebras
In this section we introduce and exhibit basic properties of the class of tame generalized multicoil algebras, playing the fundamental role in our proof of Theorem 1.1. This is the class of tame algebras among the class of all algebras having a separating family of almost cyclic coherent components investigated in [21,22] It has been proved in [20, Theorem A] that a connected component of an Auslander-Reiten quiver A is almost cyclic and coherent if and only if is a generalized multicoil, obtained from a family of stable tubes by a sequence of operations called admissible. We recall the latter and simultaneously define the corresponding enlargements of algebras.
We start with the one-point extensions and one-point coextensions of algebras. Let A be an algebra and M be a module in mod A. Then the one-point extension of A by M is the matrix algebra with the usual addition and multiplication.
Dually, one defines also the one-point coextension of A by M as the matrix algebra For r ≥ 1, we denote by T r (k) the r × r-lower triangular matrix algebra ⎡ Given a generalized standard component of A , and an indecomposable module X in , the support S(X) of the functor Hom A (X, −) | is the R-linear category defined as follows [3]. Let H X denote the full subcategory of consisting of the indecomposable modules M in such that Hom A (X, M) = 0, and I X denote the ideal of H X consisting of the morphisms f : M → N (with M, N in H X ) such that Hom A (X, f ) = 0. We define S(X) to be the quotient category H X /I X . Following the above convention, we usually identify the R-linear category S(X) with its quiver.
From now on let A be an algebra and be a family of generalized standard infinite components of A . For an indecomposable brick X in , called the pivot, one defines five admissible operations (ad 1)-(ad 5) and their duals (ad 1 * )-(ad 5 * ) modifying the translation quiver = ( , τ ) to a new translation quiver ( , τ ) and the algebra A to a new algebra A , depending on the shape of the support S(X) (see [20,Section 2] for the figures illustrating the modified translation quivers ).
(ad 1) Let t ∈ N and assume S(X) consists of an infinite sectional path starting at X: and the modified translation quiver of to be obtained by inserting in the rectangle For the remaining vertices of , τ coincides with the translation of , or D , respectively.
Finally, if t = 0 we define the modified algebra A to be the one-point extension A = A[X] and the modified translation quiver to be the translation quiver obtained from by inserting only the sectional path consisting of the vertices X i , i ≥ 0.
The non-negative integer t is such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangle equals t + 1. We call t the parameter of the operation.
Since is a generalized standard family of components of A , we then have that is a generalized standard family of components of A .
In case is a stable tube, it is clear that any module on the mouth of satisfies the condition for being a pivot for the above operation. Actually, the above operation is, in this case, the tube insertion as considered in [8].
(ad 2) Suppose that S(X) admits two sectional paths starting at X, one infinite and the other finite with at least one arrow: where t ≥ 1. In particular, X is necessarily injective. We define the modified algebra A of A to be the one-point extension A = A[X] and the modified translation quiver of to be obtained by inserting in the rectangle consisting of the modules For the remaining vertices of , τ coincides with the translation τ of .
The integer t ≥ 1 is such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangle equals t + 1. We call t the parameter of the operation.
Since is a generalized standard family of components of A , we then have that is a generalized standard family of components of A .
(ad 3) Assume S(X) is the mesh-category of two parallel sectional paths: with the upper sectional path finite and t ≥ 2. In particular, X t−1 is necessarily injective. Moreover, we consider the translation quiver of obtained by deleting the arrows Y i → τ −1 A Y i−1 . We assume that the union of connected components of containing the vertices is a finite translation quiver. Then is a disjoint union of and a cofinite full translation subquiver * , containing the pivot X. We define the modified algebra A of A to be the one-point extension A = A[X] and the modified translation quiver of to be obtained from * by inserting the rectangle consisting of the modules The translation τ of is defined as follows: For the remaining vertices of , τ coincides with the translation τ of * . We note that X t−1 is injective.
The integer t ≥ 2 is such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangle equals t + 1. We call t the parameter of the operation.
Since is a generalized standard family of components of A , we then have that is a generalized standard family of components of A .
(ad 4) Suppose that S(X) consists an infinite sectional path, starting at X with t ≥ 1, be a finite sectional path in A . Let r ∈ N. Moreover, we consider the translation quiver of obtained by deleting the arrows Y i → τ −1 A Y i−1 . We assume that the union of connected components of containing the vertices τ −1 is a finite translation quiver. Then is a disjoint union of and a cofinite full translation subquiver * , containing the pivot X. For r = 0 we define the modified algebra A of A to be the onepoint extension A = A[X ⊕ Y ] and the modified translation quiver of to be obtained from * by inserting the rectangle consisting of the modules Z ij = k, X i ⊕ Y j , 1 1 for For the remaining vertices of , τ coincides with the translation of * .
For r ≥ 1, let G = T r (k), U 1,t+1 , U 2,t+1 , . . ., U r,t+1 denote the indecomposable projective G-modules, U r,t+1 , U r,t+2 , . . ., U r,t+r denote the indecomposable injective Gmodules, with U r,t+1 the unique indecomposable projective-injective G-module. We define the modified algebra A of A to be the triangular matrix algebra of the form: with r + 2 columns and rows and the modified translation quiver of to be obtained from * by inserting the rectangles consisting of the modules U sl = k, Y l ⊕ U s,t+1 , 1 1 for 1 ≤ s ≤ r, 1 ≤ l ≤ t, and Z ij = k, X i ⊕ U rj , 1 1 for i ≥ 0, 1 ≤ j ≤ t + r, and For the remaining vertices of , τ coincides with the translation of * , or G , respectively.
We note that the quiver Q A of A is obtained from the quiver of the double one-point The integers t ≥ 1 and r ≥ 0 are such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangles equals t + r + 1. We call t + r the parameter of the operation.
Since is a generalized standard family of components of A , we then have that is a generalized standard family of components of A .
For the definition of the next admissible operation we need also the finite versions of the admissible operations (ad 1), (ad 2), (ad 3), (ad 4), which we denote by (fad 1), (fad 2), (fad 3) and (fad 4), respectively. In order to obtain these operations we replace all infinite sectional paths of the form X 0 → X 1 → X 2 → · · · (in the definitions of (ad 1), (ad 2), (ad 3), (ad 4)) by the finite sectional paths of the form X 0 → X 1 → X 2 → · · · → X s . For the operation (fad 1) s ≥ 0, for (fad 2) and (fad 4) s ≥ 1, and for (fad 3) s ≥ t − 1. In all above operations X s is injective (see [20] or [21] for the details). Since is a generalized standard family of components of A , we then have that is a generalized standard family of components of A .
Observe that any stable tube is trivially a generalized coil. A tube is a generalized coil having the property that each admissible operation in the sequence defining it is of the form (ad 1) or (ad 1 * ). Moreover, if we apply only operations of type (ad 1) (respectively, of type (ad 1 * )) then such a generalized coil is a ray tube (respectively, a coray tube). Observe that a generalized coil without injective (respectively, projective) vertices is a ray tube (respectively, a coray tube). A quasi-tube is a generalized coil having the property that each of the admissible operations in the sequence defining it is of type (ad 1), (ad 1 * ), (ad 2) or (ad 2 * ). Finally, following [3] a coil is a generalized coil having the property that each of the admissible operations in the sequence defining it is one of the forms (ad 1), (ad 1 * ), (ad 2), (ad 2 * ), (ad 3) or (ad 3 * ). We note that any generalized multicoil is a coherent translation quiver with trivial valuations and its cyclic part c (the translation subquiver of obtained by removing from all acyclic vertices and the arrows attached to them) is infinite, connected and cofinite in , and so is almost cyclic.
Let C be the product C 1 × . . . × C m of a family C 1 , . . . , C m of tame concealed algebras and T C the disjoint union T C 1 ∪ . . . ∪ T C m of P 1 (k)-families T C 1 , . . . , T C m of pairwise orthogonal generalized standard stable tubes of C 1 , . . . , C m , respectively. Following [21], we say that an algebra A is a generalized multicoil enlargement of C 1 , . . . , C m if A is obtained from C by an iteration of admissible operations of types (ad 1)-(ad 5) and (ad 1 * )-(ad 5 * ) performed either on stable tubes of T C or on generalized multicoils obtained from stable tubes of T C by means of the operations done so far. It follows from [21, Corollary B] that then A is a triangular algebra. In fact, in [21] generalized multicoil enlargements of finite families of arbitrary concealed canonical algebras (generalized multicoil algebras) have been introduced and investigated. But in the tame case we may restrict to the generalized multicoil enlargements of tame concealed algebras. Namely, we have the following consequence of [21, Theorems A and F]. From now on, by a tame generalized multicoil algebra we mean a connected tame generalized multicoil enlargement of a finite family of tame concealed algebras. As a consequence of [21, Theorems C and F] and the proof of [21, Theorem C] we obtain the following fact.

(iv) The Auslander-Reiten quiver A of A is of the form
where C A is a family of generalized multicoils separating P A from Q A such that: (a) C A is obtained from the P 1 (k)-families T C 1 , . . . , T C m of stable tubes of C 1 , . . . , C m by admissible operations corresponding to the admissible operations leading from C 1 , . . . , C m to A; i is a representation-infinite tilted algebra of Euclidean type, or P A (l) i is a representation-infinite tilted algebra of Euclidean type, or Q A (r) i is a tubular algebra.

Remark 3.3 From the proof of [21, Theorem C] we know that
It follows from [29, Theorem 4.1] and Theorem 3.2 that, if A is tame generalized multicoil algebra, then A is cycle-finite (see Section 5 for the definition). Applying now [29, Theorem 5.1], we obtain the following fact.  M) is of finite length. Assume that M belongs to a generalized multicoil C. Since different generalized multicoils in mod are pairwise orthogonal, it follows from [11], that if C is a coray tube of A (l) , then T 1 Hom (−, M) = 0. Thus we may assume that M is a module from the generalized multicoil of different from the coray tubes of A (l) . If M is a directing -module which is not an A (l) -module then again Hom (−, M) is of finite length. If M is a non-directing -module which is not an A (l) -module then we have the following three cases to consider.
(a) If is a modified algebra of A (l) obtained by applying the admissible operation of type (ad 1), (ad 2), (ad 3) or (ad 4) with r = 0 then M is isomorphic to Z ij or X i (see Section 3). Assume first that M ∼ = Z ij . Then we have an obvious monomorphism X i ⊕ Y j → Z ij , which induces a monomorphism of functors α : Hom (−, X i ⊕ Y j ) → Hom (−, Z ij ). It follows from the description of generalized multicoils that the set S α of all indecomposable modules N such that coker α(N) = 0 is finite. Indeed, we have: • For (ad 1) and (ad 4) with r = 0, Moreover, coker α is finitely generated. Therefore, we get that coker α is of finite length and T 0 Hom (−, X i ⊕ Y j ) ∼ = T 0 Hom (−, Z ij ). Assume now that M ∼ = X i . Again, we have an obvious monomorphism X i → X i , which induces a monomorphism of functors β : Hom (−, X i ) → Hom (−, X i ) and the set S β of all indecomposable modules N such that coker β(N) = 0 is finite. In this subcase we get: • For (ad 1) and (ad 4) with r = 0, Note that in the above two subcases t denotes the parameter of the suitable admissible operation.
• For (ad 3), Hence coker β is of finite length since, moreover, it is finitely generated. Thus T 0 Hom (−, X i ) ∼ = T 0 Hom (−, X i ). (b) If is a modified algebra of A (l) obtained by applying the admissible operation of type (ad 4) with r ≥ 1 then M is isomorphic to U kl for 1 ≤ k ≤ r, 1 ≤ l ≤ t, Z ij for i ≥ 0, 1 ≤ j ≤ t + r, or X i for i ≥ 0 (see Section 3). Assume first that M ∼ = U kl , 1 ≤ k ≤ r, 1 ≤ l ≤ t, where t + r is the parameter of (ad 4). Then we have a monomorphism Y l → U kl , which induces a monomorphism of functors γ : Hom (−, Y l ) → Hom (−, U kl ) and the set S γ of all indecomposable modules N such that coker γ (N) = 0 is finite.
• For i ≥ 0, 1 ≤ j ≤ t we have monomorphisms X i ⊕ U rj → Z ij and Y j → U rj . Hence, by Lemma 2.2, we infer that X i ⊕ Y j → Z ij is a monomorphism. Again, we get the induced monomorphism of functors δ : Finally, assume that M ∼ = X i , i ≥ 0. Again, we have an obvious monomorphism X i → X i , the induced monomorphism of functors ζ : is a modified algebra of A (l) obtained by applying the admissible operation of type (ad 5). Since in the definition of (ad 5) we use the finite versions (fad 1), (fad 2), (fad 3), (fad 4) of the admissible operations (ad 1), (ad 2), (ad 3), (ad 4) and the admissible operation (ad 4), we conclude that the required statements follows from the above considerations. Therefore, we may assume that M is in fact an A (l) -module. Let F = Hom (−, M) and G = Hom (−, M)| mod A (l) . Let I be the simple -module corresponding to the extension vertex of A (l) [X], where X is the pivot of the suitable admissible operation. Since Hom (I, M) = 0 then for any A-module Z we get F (Z) = G(Z ), where Z is the restriction of Z to A (l) . Moreover, the category mod A (l) is contained in the obvious way into the category mod . From this we conclude that, if T 1 G = 0, then T 1 F = 0, where T 1 : F(mod A (l) ) → F(mod A (l) )/F 1 (mod A (l) ) and T 1 : F(mod ) → F(mod )/F 1 (mod ) are the canonical quotient functors. By [11] we have T 1 G = 0, and hence T 1 F = 0. Since F is not of finite length, we get that T 0 F = 0.
Let X be the full subcategory of mod generated by all indecomposable modules from the generalized multicoils and the postprojective components, and let Y be the full subcategory of mod generated by all indecomposable modules from the preinjective components. Note that, since the projective cover of any finitely presented functor is a functor Hom (−, N) for some module N , it is enough to check only Hom-functors. Therefore, by the above arguments we have K-dim X = 1. Moreover, using Lemma 2.1 we get that K-dim Y = 2. Hence, applying [11, Theorem 2.6] we obtain K-dim(mod ) = max(K-dim X + 1, K-dim Y) = 2.
Finally, we can complete the proof by an obvious induction on the number of admissible operations leading from A (l) to A.

Concluding Remarks
Since the tame quasitilted algebras of canonical type form a distinguished special class of tame generalized multicoil algebras, we obtain the following fact.
Corollary 5.1 Let A be a tame quasitilted algebra of canonical type. The following statements are equivalent: Let A be an algebra. Recall that a cycle in a module category mod A is a sequence X 0 f 1 − − → X 1 → · · · → X r−1 f r − − → X r = X 0 of nonzero nonisomorphisms in ind A, and the cycle is said to be finite if f i ∈ rad ∞ (mod A) for any 1 ≤ i ≤ r. If every cycle in mod A is finite then A is said to be cycle-finite. Recall also that a component C of A is called semiregular if C does not contain both a projective and an injective module. It has been proved in [15] that a semiregular component C of A contains an oriented cycle if and only if C is a ray tube or coray tube (see remarks after definitions of admissible operations).
As an immediate consequence of Corollary 5.1 and [31, Theorem 5.1] we obtain the following fact.

Corollary 5.2 Let
A be a cycle-finite algebra such that every component of A is semiregular, and pd A X ≤ 1 or id A X ≤ 1 for all but finitely many isomorphism classes of modules X in ind A. Then the following statements are equivalent: Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.