The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

We study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.


Introduction and the main results
Throughout the paper k will denote a fixed algebraically closed field. By an algebra is meant an associative finite-dimensional k-algebra with an identity, which we shall assume (without loss of generality) to be basic. Then such an algebra has a presentation A ∼ = kQ A /I , where Q A = (Q 0 , Q 1 ) is the ordinary quiver of A with the set of vertices Q 0 and the set of arrows Q 1 and I is an admissible ideal in the path algebra kQ A of Q A . If the quiver Q A has no oriented cycles, the algebra A is said to be triangular. For an algebra A, we denote by mod A the category of finitely generated right A-modules, and by ind A a full subcategory of mod A consisting of a complete set of representatives of the isomorphism classes of indecomposable modules. We shall denote by rad A the Jacobson radical of mod A, and by rad ∞ A the intersection of all powers rad i A , i ≥ 1, of rad A . Moreover, we denote by A the Auslander-Reiten quiver of A, and by τ A and τ − A the Auslander-Reiten translations D Tr and Tr D, respectively. We will not distinguish between a module in ind A and the vertex of A corresponding to it. Following [39] a family C of components is said to be generalized standard if rad ∞ A (X, Y ) = 0 for all modules X and Y in C. We note that different components in a generalized standard family C are orthogonal, and all but finitely many τ A -orbits in C are τ A -periodic (see [39, (2.3)]). We refer to [31] for the structure and homological properties of arbitrary generalized standard Auslander-Reiten components of algebras.
Following Assem and Skowroński [8] a triangular algebra A is called simply connected if, for any presentation A ∼ = kQ A /I of A as a bound quiver algebra, the fundamental group π 1 (Q A , I ) of (Q A , I ) is trivial (see Section 2). The importance of these algebras follows from the fact that often we may reduce (using techniques of Galois coverings) study of the module category of an algebra to that for the corresponding simply connected algebras. Recall that A is called strongly simply connected [38] if every convex subcategory of A is simply connected.
From Drozd's Tame and Wild Theorem [16] the class of algebras may be divided into two disjoint classes. One class consists of the tame algebras for which the indecomposable modules occur, in each dimension d, in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory comprises the representation theories of all finite dimensional algebras over k. Hence, a classification of the finite dimensional modules is only feasible for tame algebras. It has been shown by Crawley-Boevey [15] that, if A is a tame algebra, then, for any dimension d ≥ 1, all but finitely many isomorphism classes of indecomposable A-modules of dimension d are invariant on the action of τ A , and hence, by a result due to Hoshino [19], lie in stable tubes of rank one in A . The indecomposable modules over tame algebras which do not lie in stable tubes of rank one are called discrete.
A prominent role in the representation theory of algebras is played by the algebras with separating families of Auslander-Reiten components. A concept of a separating family of tubes has been introduced by Ringel in [33,34] who proved that they occur in the Auslander-Reiten quivers of hereditary algebras of Euclidean type, tubular algebras, and canonical algebras. In order to deal with wider classes of algebras, the following more general concept of a separating family of Auslander-Reiten components was proposed by Assem, Skowroński and Tomé in [11] (see also [27]). 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 components of A split into three disjoint families 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 homomorphism from P A to Q A in mod A factors through the additive category add(C A ) of C A .
Then we 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 [11, (2.1)] or [34, (3.1)]). Moreover, C A is called sincere if any simple A-module occurs as a composition factor of a module in C A . We note that if A is an algebra of finite representation type that C A = A is trivially a unique separating component of A , with P A and Q A being empty. Frequently, we may recover A completely from the shape and categorical behavior of the separating family C A of components of A . For example, the tilted algebras [20,34], or more generally double tilted algebras [32], are determined by their (separating) connecting components. Further, it was proved in [22] that the class of algebras with a separating family of stable tubes coincides with the class of concealed canonical algebras. This was extended in [23] to a characterization of all quasitilted algebras of canonical type, for which the Auslander-Reiten quiver admits a separating family of semiregular tubes. Then, the latter has been extended in [27] 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 → · · · in ; (C2) For each injective module I in there is an infinite sectional path It has been proved in [27, Theorem A] that the Auslander-Reiten quiver A of an algebra A admits a separating family C A of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a finite family C 1 , . . . , C m of concealed canonical algebras by an iterated application of admissible algebra operations of types (ad 1)-(ad 5) and their duals. Note that for such an algebra A, we have that A is triangular, gl. dim A ≤ 3, and pd A M ≤ 2 or id A M ≤ 2 for any module M in ind A (see [27,Corollary B and Theorem E]). Moreover, let A = P A ∪ C A ∪ Q A be the induced decomposition of A . Then, by [27,Theorem C], there are uniquely determined quotient algebras A (l) = A (l) m of A which are the quasitilted algebras of canonical type such that P A = P A (l) and Q A = Q A (r) . The algebras A (l) and A (r) are called the left and right quasitilted algebras of A. We note that A (l) and A (r) are tame if and only if A (l) and A (r) are products of tilted algebras of Euclidean type or tubular algebras. Moreover, by [27,Theorem F], A is tame if and only if A (l) and A (r) are tame.
In order to formulate our main results we need one more definition. Namely, if the sectional paths occurring in the definitions of the operations (ad 4), (fad 4), (ad 4 * ), (fad 4 * ) come from a component or two components of the same connected algebra, then the modified translation quiver is said to be an exceptional configuration of modules (see Section 3 for details).
We mention that applying [30,Theorem 1.1] and [38,Theorem 4.1] we obtain that an algebra whose Auslander-Reiten quiver admits a separating family of almost cyclic coherent components with exceptional configurations of modules is not strongly simply connected (although its left and right quasitilted algebra can be strongly simply connectedsee Example 8.1 for details).
We are now in position to formulate the first main result of the paper. In order to present the first corollary, we need the class of cycle-finite algebras introduced by Assem and Skowroński in [9]. Let A be an algebra. Recall that a cycle in a module category mod A is a sequence of nonzero nonisomorphisms between indecomposable modules in mod A, and the cycle is said to be finite if the homomorphisms f 1 , . . . , f r do not belong to rad ∞ A . Then an algebra A is said to be cycle-finite if all cycles in mod A are finite. It is known (see [9]) that every cycle-finite algebra is tame.
As an application of Theorem 1.2 and results of [27,40] we obtain the following corollary.

Corollary 1.3 Let
A be a cycle-finite algebra with a separating family of almost cyclic coherent components in A without exceptional configurations of modules. The following statements are equivalent: (i) A is strongly simply connected.
(ii) A does not contain a full convex subcategory which is hereditary of type A n .
Following Gerstenhaber [17], a one-parameter deformation of an algebra A is a k Then the algebra A is said to be analytically rigid if any one-parameter deformation of A is isomorphic to the trivial one given by f i = 0 for i ≥ 1. It was shown in [17] that oneparameter deformations of A are related to low Hochschild cohomology spaces, H i (A), 1 ≤ i ≤ 3. In fact, for a one-parameter deformation of A given by f : For a connected concealed canonical algebra C of tubular type p C = (p 1 , . . . , p t ) (see Section 2 for details), we set t C = t. It is known that the class of concealed canonical algebras of type (p 1 , p 2 ) coincides with the class of hereditary algebras Euclidean types A n , n ≥ 1 (see [18]). For the definitions of generalized multicoil enlargement of a (not necessarily connected) concealed canonical algebra and admissible operations see Section 3.
As a direct consequence of Theorem 1.2, [30, Corollary 1.2] and the remark before Theorem 1.1 we obtain the following fact.

Corollary 1.4 Let
A be a strongly simply connected tame algebra with a separating family of almost cyclic coherent components in A . The following statements are equivalent: A is a tame generalized multicoil enlargement of a family C 1 , . . . , C m of connected concealed canonical algebras such that, for any i ∈ {1, . . . , m}, t C i = 3 and there is no admissible operation applied to modules from the mouth of stable tubes of rank one in C i .
This paper is organized as follows. In Section 2 we recall some concepts and facts from representation theory, which are necessary for further considerations. Section 3 is devoted to describing some properties of almost cyclic coherent components of the Auslander-Reiten quivers of algebras, applied in the proofs of the preliminary results and the main theorems. In Section 4 we present and prove several results applied in the proof of the first main result of the paper. Sections 5, 6 and 7 are devoted to the proves of Theorems 1.1, 1.2 and Corollary 1.3, respectively. In Section 8 we present examples of tame algebras whose Auslander-Reiten quivers admit separating families of almost cyclic coherent components, illustrating Theorem 1.2. In Section 9 we present an application of Theorem 1.1. The aim of the final Section 10 is to present concluding remarks.
For basic background on the representation theory of algebras we refer to the books [7,[34][35][36] and for more details on algebras with separating families of Auslander-Reiten components and their representation theory to the survey article [29].

The separation condition
Let A be an algebra and A ∼ = kQ A /I be a presentation of A as a bound quiver algebra. Then the algebra A = kQ A /I can equivalently 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 (x, y) = kQ A (x, y) ∩ I (see [13]). A full subcategory B of A is called convex (in A) if any path in A with source and target in B lies entirely in B. For each vertex v of Q A we denote by S v the corresponding simple A-module, and by P v (respectively, I v ) the projective cover (respectively, the injective envelope) of S v .
Let A be a triangular algebra. Recall that a vertex v of Q A is called separating if the radical of P v is a direct sum of pairwise nonisomorphic indecomposable modules whose supports are contained in different connected components of the subquiver Q(v) of Q A obtained by deleting all those vertices u of Q A being the source of a path with target v (including the trivial path from v to v). Then, generalizing definition proposed in [12], we say that A satisfy the separation condition if each vertex v of Q A is separating.

One-point extensions and coextensions
Frequently an algebra A can be obtained from another algebra B by a sequence of one-point extensions and one-point coextensions. Recall that the one-point extension of an algebra B by a B-module M is the matrix algebra

Tameness
Let A be an algebra and

Simple connectedness
Let (Q, I ) be a connected bound quiver. A relation = m i=1 λ i w i ∈ I (x, y) is minimal if m ≥ 2 and, for any nonempty proper subset J ⊂ {1, . . . , m}, we have j ∈J λ j w j / ∈ I (x, y), where for each i ∈ {1, . . . , m} λ i is a non-zero scalar and w i is a path of length at least two from x to y. We denote by α −1 the formal inverse of an arrow α ∈ Q 1 . A walk in Q from x to y is a formal composition α ε 1 1 α ε 2 2 . . . α ε t t (where α i ∈ Q 1 and ε i ∈ {−1, 1} for all i) with source x and target y. We denote by e x the trivial path at x. Let ∼ be the least equivalence relation on the set of all walks in Q such that: then wuw ∼ wvw whenever these compositions make sense. Let x ∈ Q 0 be arbitrary. The set π 1 (Q, I, x) of equivalence classes u of closed walks u starting and ending at u has a group structure defined by the operation u · v = uv. Since Q is connected, π 1 (Q, I, x) does not depend on the choice of x. We denote it by π 1 (Q, I ) and call it the fundamental group of (Q, I ).
Let A ∼ = kQ A /I be a presentation of a triangular algebra A as a bound quiver algebra. The fundamental group π 1 (Q A , I ) depends essentially on I , so is not an invariant of A. A triangular algebra A is called simply connected if, for any presentation A ∼ = kQ A /I of A as a bound quiver algebra, the fundamental group π 1 (Q A , I ) of (Q A , I ) is trivial [8].

Strong simple connectedness
Let A be a triangular algebra. Following [38] A is called a strongly simply connected if it satisfies one of the following equivalent conditions: In the proof of Theorem 1.1 we will use a characterization of strongly simply connected algebras established by Assem and Liu [4]. We now recall the necessary definitions and the appropriate theorem. Let Q be a locally finite quiver without oriented cycles. A contour (p, q) in Q from x to y consists of a pair of non-trivial paths from x to y. It is interlaced if p, q have a common point besides x and y. A contour (p, q) is irreducible if there exists no sequence of paths p = p 0 , p 1 , . . . , p m = q in Q from x to y such that, for each i ∈ {1, . . . , m}, the contour (p i−1 , p i ) is interlaced. Let C be a simple cycle which is not a contour, and let σ (C) denote the number of sources in C. Then C is reducible if there exist x, y on C and a path p : x → · · · → y in Q such that if w 1 and w 2 denote the subwalks of C from x to y (so that C = w 1 w −1 2 ), then w 1 p −1 and w 2 p −1 are cycles and σ (w 1 p −1 ) < σ(C), σ (w 2 p −1 ) < σ(C). A cycle C is irreducible if either it is an irreducible contour, or it is not a contour, but it is not reducible in the above sense. Let I be an admissible ideal of kQ. Finally, a contour (p, q) from x to y is naturally contractible in a bound quiver (Q, I ) if there exists a sequence of paths p = p 0 , p 1 , . . . , p m = q in Q such that, for each i ∈ {1, . . . , m}, the paths p i−1 and p i have subpaths q i−1 and q i , respectively, which are involved in the same minimal relation in (Q, I ).
We need the following result proved in [

Almost cyclic coherent Auslander-Reiten components
It has been proved in [26, Theorem A] that a connected component of an Auslander-Reiten quiver A of an algebra A is almost cyclic and coherent if and only if is a generalized multicoil, that is, can be obtained, as a translation quiver, from a finite family of stable tubes by a sequence of operations called admissible. We recall briefly the generalized multicoil enlargements of algebras from [27,Section 3].
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 k-linear category defined as follows [10]. 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 k-linear category S(X) with its quiver.
Recall that a module X in mod A is called a brick if End A (X) ∼ = k.
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, five admissible operations are defined, depending on the shape of the support S(X) of the functor Hom A (X, −)| . These admissible operations yield in each case a modified algebra A such that the modified translation quiver is a family of generalized standard infinite components in the Auslander-Reiten quiver A of A (see [26,Section 2] or [29,Section 4] for the figures illustrating the modified translation quiver ).
(ad 1) Assume S(X) consists of an infinite sectional path starting at X: Let t ≥ 1 be a positive integer, D be the full t ×t lower triangular matrix algebra, and Y 1 , . . ., In this case, is obtained by inserting in the rectangle consisting of the modules and the rectangle reduces to the sectional path consisting of the modules X i , i ≥ 0.
(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 set A = A [X]. In this case, is obtained by inserting in the rectangle consisting of the modules (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 modules is a finite translation quiver. Then is a disjoint union of and a cofinite full translation subquiver * , containing the pivot X. We set A = A[X]. In this case, is obtained from * by inserting the rectangle consisting of the modules (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 be a positive integer. Moreover, we consider the translation quiver of obtained by deleting the arrows 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.
In this case, is obtained from * by inserting the rectangle consisting of the modules For r ≥ 1, let G be the full r × r lower triangular matrix algebra, 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 G-modules, with U r,t+1 the unique indecomposable projective-injective G-module. We set define the with r + 2 columns and rows. In this case, is obtained from * by inserting the following modules In the above formulas U sl is treated as a module over the To 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 the figures for (fad 1)-(fad 4) in [26,Section 2] or [29,Section 4]).

(ad 5)
We define the modified algebra A of A to be the iteration of the extensions described in the definitions of the admissible operations (ad 1), (ad 2), (ad 3), (ad 4), and their finite versions corresponding to the operations (fad 1), (fad 2), (fad 3) and (fad 4). In this case, is obtained in the following three steps: first we are doing on one of the operations (fad 1), (fad 2) or (fad 3), next a finite number (possibly zero) of the operation (fad 4) and finally the operation (ad 4), and in such a way that the sectional paths starting from all the new projective modules have a common cofinite (infinite) sectional subpath. By an (ad 5)pivot we mean an indecomposable brick X from the last (ad 4) operation used in the whole process of creating (ad 5).
Finally, let C be a (not necessarily connected) concealed canonical algebra (see [21]) and T C a separating family of stable tubes of C . Following [27] we say that an algebra A is a generalized multicoil enlargement of C using modules from T C if there exists a sequence of algebras C = A 0 , A 1 , . . . , A n = A such that A i+1 is obtained from A i by an admissible operation of one of the types (ad 1)-(ad 5), (ad 1 * )-(ad 5 * ) performed either on stable tubes of T A i , or on generalized multicoils obtained from stable tubes of T A i by means of operations done so far. The sequence C = A 0 , A 1 , . . . , A n = A is then called an admissible sequence for A. Observe that this definition extends the concept of a coil enlargement of a concealed canonical algebra introduced in [11]. We note that a generalized multicoil enlargement A of C invoking only admissible operations of type (ad 1) (respectively, of type (ad 1 * )) is a tubular extension (respectively, tubular coextension) of C in the sense of [34]. An algebra A is said to be a generalized multicoil algebra if A is a connected generalized multicoil enlargement of a product C of connected concealed canonical algebras.
Let A be an algebra. It has been proved in [27, Theorem A] that A admits a separating family of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a concealed canonical algebra C.
Let us note that the class of algebras with generalized standard almost cyclic coherent Auslander-Reiten components is large (see [28,Proposition 2.9] and the following comments).
We also note that the class of tubular extension (respectively, tubular coextension) of concealed canonical algebras coincides with the class of algebras having a separating family of ray tubes (respectively, coray tubes) in their Auslander-Reiten quiver (see [21,23]). Moreover, these algebras are quasitilted algebras of canonical type. See also [27,Theorem C] describing the structure of the module category of an algebra with a separating family of almost cyclic coherent Auslander-Reiten components.

Branch extensions and coextensions
Let A be an algebra and A ∼ = kQ A /I be a presentation of A as a bound quiver algebra. For a given vertex v in Q A , we denote by v → (respectively, by → v) the set of all arrows of the quiver Q A starting at v (respectively, terminating at v). Let now K be a branch at a vertex v ∈ Q A and E ∈ mod A. Recall that the branch extension A[E, K] by the branch K [34, (4.4)] is constructed in the following way: to the one-point extension A[E] with extension vertex w (that is, rad P w = E) we add the branch K by identifying the vertices v and w. If E 1 , . . . , E n ∈ mod A and K 1 , . . . , K n is a set of branches, then the branch The concept of branch coextension is defined dually.

Lemma 4.2 Let
A be a generalized multicoil enlargement of C = C 1 × · · · × C m , where C 1 , . . . , C m is a family of tame concealed algebras. Moreover, let C = A 0 , . . . , A p = A (l) , A p+1 , . . . , A n = A be an admissible sequence for A, j ≥ p, X ∈ ind A j be an (ad 1)pivot, A j +1 be the modified algebra of A j , and v be the corresponding extension point. Then the following statements hold.
Proof Since X is an (ad 1)-pivot, the support S(X) consists of an infinite sectional path X = X 0 → X 1 → X 2 → · · · starting at X. Let t ≥ 1 be a positive integer, D be the full t × t lower triangular matrix algebra, and Y 1 , . . ., Y t be the indecomposable injective Dmodules with Y 1 the unique indecomposable projective-injective D-module (see definition of (ad 1)).
(ii) A case by case inspection shows that X is either simple module or the support of X is a linearly ordered quiver of type A t .

Lemma 4.3 Let A be a generalized multicoil enlargement of
. . , C m is a family of tame concealed algebras. Moreover, let C = A 0 , . . . , A p = A (l) , A p+1 , . . . , A n = A be an admissible sequence for A, j ≥ p, X ∈ ind A j be an (ad 2) or (ad 3)-pivot, and A j +1 be the modified algebra of A j . If v is the corresponding extension point then there is a unique vertex u ∈ A (l) \ A (r) that satisfies: . If X is an (ad 2)-pivot (respectively, (ad 3)-pivot), then in the sequence of earlier admissible operations, there is an operation of type (ad 1 * ) or (ad 5 * ) which contains an operation (fad 1 * ) which gives rise to the pivot X of (ad 2) (respectively, to the pivot X of (ad 3) and to the modules Y 1 , . . . , Y t in the support of Hom A (X, −) restricted to the generalized multicoil containing X -see definition of (ad 3)). The operations done after must not affect the support of Hom A (X, −) restricted to the generalized multicoil containing X. Note that in general, in the sequence of earlier admissible operations can be an operation of type (ad 5) which contains an operation (fad 4) which gives rise to the pivot X of (ad 2) (respectively, to the pivot X of (ad 3)) but from Lemma [27, Lemma 3.10] this case can be reduced to (ad 5 * ) which contains an operation (fad 1 * ).
Let X be an (ad 2)-pivot, A j +1 = A j [X], and u, u 1 , . . . , u t (X = I u ) be the points in the quiver Q A j of A j corresponding to the new indecomposable injective A j -modules obtained after performing the above admissible operation (ad 1 * ) or the operation (fad 1 * ). The bound quiver Q A j +1 of A j +1 is of the form where X 1 is the immediate successor of X on the infinite sectional path in S(X) (see definition of (ad 2)). Let now X be an (ad 3)-pivot, A j +1 = A j [X], and assume that we had r consecutive admissible operations of types (ad 1 * ) or (fad 1 * ), the first of which had X t as a pivot, and these admissible operations built up a branch K in A j with points u, u 1 , . . . , u t in Q A j , so that X t−1 and Y t are the indecomposable injective A j -modules corresponding respectively to u and u 1 , and both Y 1 and τ −1 A j Y 1 are coray modules in the generalized multicoil containing the (ad 3)-pivot X. The bound quiver Q A j +1 of A j +1 is of the form 1 and Y t are as in the definition of (ad 3). Now, the statements follows from the proof of [14,Lemma 2.4].

Lemma 4.4 Let A be a generalized multicoil enlargement of
. . , A n = A be an admissible sequence for A, j ≥ p, X ∈ ind A j be an (ad 4) or (ad 5)-pivot, A j +1 be the modified algebra of A j , and v be the corresponding extension point. If there is a vertex u ∈ A (l) \ A (r) such that for pairwise different arrows α 1 , . . . , α q ∈ v → , q ≥ 2 there are paths ω α 1 , . . . , ω α q ∈ A(v, u), then for arbitrary f, g ∈ {1, . . . , q}, f = g, one of the following cases holds: The paths ω α f , ω α g are non-zero and ω α f − ω α g ∈ I .
Proof It follows from [27, Section 4] that A (l) is a unique maximal convex branch coextension of C = C 1 × · · · × C m inside A, that is, i is a unique maximal convex branch coextension of C i inside A, i ∈ {1, . . . , m}. More precisely, Assume that there is a vertex u ∈ A (l) \ A (r) such that for pairwise different arrows α 1 , . . . , α q ∈ v → , q ≥ 2, there are paths ω α 1 , . . . , ω α q ∈ A(v, u). Then there exists s ∈ {1, . . . , m} such that u ∈ B (l) s . Let X be an (ad 4)-pivot and Y 1 → Y 2 → · · · → Y t with t ≥ 1, be a finite sectional path in A j (as in the definition of (ad 4)). Note that this finite sectional path is the linearly oriented quiver of type A t and its support algebra (given by the vertices corresponding to the simple composition factors of the modules Y 1 , Y 2 , . . . , Y t ) is a tilted algebra of the path algebra D of the linearly oriented quiver of type A t . From [34, (4.4)(2)] we know that is a bound quiver algebra given by a branch in x, where x corresponds to the unique projectiveinjective D-module. Let be a generalized multicoil of A j +1 obtained by applying the admissible operation (ad 4), where X is the pivot contained in the generalized multicoil 1 , and Y 1 is the starting vertex of a finite sectional path contained in the generalized multicoil 1 or 2 . So, is obtained from 1 or from the disjoin union of two generalized multicoils 1 , 2 by the corresponding translation quiver admissible operations. Therefore, the bound quiver Q A j +1 of A j +1 in the first case is of the form for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of . . , w d belong to branch in w generated by the support of Y 1 ⊕ · · · ⊕ Y t , and αβ 1 . . . β h = 0 for some h ∈ {1, . . . , d + 1}. In the second case the bound quiver Q A j +1 of A j +1 is of the form for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of A j [X], w is the extension point of A j [Y 1 ], w 1 , . . . , w d belong to branch in w generated by the support of Y 1 ⊕ · · · ⊕ Y t , αβ 1 . . . β h = 0 for some h ∈ {1, . . . , d + 1}, and y is the coextension point of A j such that y ∈ A (l) \ A (r) . More precisely, y ∈ B (l) s , where s ∈ {1, . . . , m} and s = s. Moreover in both cases, we have P v (u) = X(u) = k or P v (u) = X(u) = 0, and hence all non-zero paths from v to u are congruent modulo I j +1 . So, A j +1 (v, u) is at most one-dimensional. We note that in the first case, the definition of (ad 4) (see the shape of the bound quiver Q A j +1 of A j +1 ) implies that if the paths ω α f , ω α g ∈ A j +1 (v, u) are non-zero and ω α f − ω α g ∈ I , then there is also zero path Let X be an (ad 5)-pivot and be a generalized multicoil of A j +1 obtained by applying this admissible operation with pivot X. Then is obtained from the disjoint union of the finite family of generalized multicoils 1 , 2 , . . . , e by the corresponding translation quiver admissible operations, 1 ≤ e ≤ l, where l is the number of stable tubes of C used in the whole process of creating . Since in the definition of admissible operation (ad 5) we use the finite versions (fad 1)-(fad 4) of the admissible operations (ad 1)-(ad 4) and the admissible operation (ad 4), we conclude that the required statement follows from the above considerations.

The factor space branch
Let A be an algebra, u be a coextension point of A, and K be the branch with root vertex u. We denote the arrows in K by α and β, where αβ = 0. For a given vertex w ∈ K, which is the end point of an arrow β (respectively, the starting point of an arrow α), we denote by is the largest possible. Following [34, (4.4)], we call the maximal subbranch of K of the form u β − − → · · · β − − → y the factor space branch of u. Note that if | → v| = 2 and v does not lie on the factor space branch od u, then τ (σ (v)) is defined. (2) we can apply the operations (ad 4), (fad 4), (that is, also (ad 5)) in two ways. The first way is when the sectional paths occurring in the definitions of these operations come from a component or two components of the same connected algebra. The second one is, when these sectional paths come from two components of two connected algebras. Only in the first case we can obtain a pair (v, u) of vertices in Q A such that v ∈ A (r) \ A (l) , u ∈ A (l) \ A (r) and for all arrows α 1 , . . . , α q ∈ v → , q ≥ 2 there are paths ω α 1 , . . . , ω α q ∈ A(v, u) (see the proof of Lemma 4.4). Now, if we have two such pairs (v 1 , u 1 ) and (v 2 , u 2 ) of vertices obtained by applying operation of type (ad 2), (ad 3), (ad 4) or (ad 5) in the corresponding sequence of admissible operations, where v 1 , v 2 ∈ A (r) \A (l) and u 1 , u 2 ∈ A (l) \A (r) , then in general, the following three cases are possible:

Related pairs
• the pairs (v 1 , u 1 ) and (v 2 , u 2 ) are obtained by applying the corresponding sequence of admissible operations with pivots belonging to the component of A obtained from the same stable tube; • the pairs (v 1 , u 1 ) and (v 2 , u 2 ) are obtained by applying the corresponding sequence of admissible operations with pivots belonging to the component of A obtained from two different stable tubes of the same connected algebra; • the pairs (v 1 , u 1 ) and (v 2 , u 2 ) are obtained by applying the corresponding sequence of admissible operations with pivots belonging to the component of A obtained from two different stable tubes of two connected algebras.
If the first case occurs, then the pairs (v 1 , u 1 ) and (v 2 , u 2 ) we call related. It is easy to see that this is indeed an equivalence relation.

. , A n = A be an admissible sequence for A, and (v 1 , u 1 ) and (v 2 , u 2 ) be two pairs of vertices obtained by applying operation of type (ad 2) or (ad 3) in the corresponding sequence of admissible operations,
where v 1 , v 2 ∈ A (r) \ A (l) and u 1 , u 2 ∈ A (l) \ A (r) . Then the following statements hold.
(i) If (v 1 , u 1 ), (v 2 , u 2 ) are related pairs and u 2 does not lie on the factor space branch of a coextension vertex of C, then there must be a previous operation of type (ad 2) in the sequence for which the associated pair of vertices is (v 1 , τ (σ (u 2 ))). (ii) If there is a path in Q A from v 2 to v 1 , then u 1 and u 2 belong to the same coextension branch K and there is a path u 2

Proof By [27, Section 4], A (l) is a unique maximal convex branch coextension of
i is a unique maximal convex branch extension of C i inside A, i ∈ {1, . . . , m}.
(i) Let be the generalized multicoil of A containing the pivot X of the operation which yields (v 2 , u 2 ). Then contains a ray passing through X. On the other hand, in the tube of A (l) (= B (l) i for some i ∈ {1, . . . , m}), which is transformed into by the sequence of admissible operations, there is a sectional path from the indecomposable injective A (l)module I u 2 to the simple A (l) -module S σ (u 2 ) , but there is no ray passing through both. In general, we can create such a ray by the application of an admissible operation of type (ad 2), (ad 4) or (ad 5). Since does not contain an exceptional configurations of modules, we get that only way to create that ray is by the application of an admissible operation of type (ad 2) with pivot the indecomposable injective A (l) -module I τ σ (u 2 ) .
(ii) It follows from the discussion before the lemma, the assumption that there are no exceptional configurations of modules and from [14, Lemma 2.5 (b)].

Proof of Theorem 1.1
Let A be an algebra with a separating family of almost cyclic coherent components in A without exceptional configurations of modules. The necessity is obvious. To prove the sufficiency, observe first that by [27,Theorem C], there is a unique full convex subcategory m of A which is a tubular coextension of a concealed canonical algebra C = C 1 × . . . × C m such that A is obtained from A (l) by a sequence of admissible operations of types (ad 1)-(ad 5). We shall prove our claim by induction on the number of admissible operations leading from A (l) to the algebra A. Note that we can apply an admissible operation (ad 2), (ad 3), (ad 4) or (ad 5) if the number of all successors of the module Y i (which occurs in the definitions of the above admissible operations) is finite for each 1 ≤ i ≤ t. Indeed, if it is not the case, then the family of generalized multicoils obtained after applying such admissible operation is not sincere, and then it is not separating. Let C = A 0 , . . . , A p = A (l) , A p+1 , . . . , A n = A be an admissible sequence for A and assume that A p+1 = A. In this case, if the admissible operation performed is (ad 1), (ad 2) or (ad 3), then m = 1. Moreover, since A does not contain exceptional configuration of modules, we get that if the admissible operation performed is (ad 4) or (ad 5), then m = 2. Let v 0 be the corresponding extension point. We note that v 0 is the root of a branch of A (r) if the operation performed is (ad 1).
Assume that A is not strongly simply connected. Then, by Theorem 2.6, the bound quiver (Q A , I ) of A contains an irreducible cycle which is not a contour, or an irreducible contour which is not naturally contractible in (Q A , I ). As in [14, (3.1)] or [2, (4.4)], we consider two cases. w = (p, q) be an irreducible contour from v to h which is not naturally contractible. Since w lies neither inside A (l) nor inside A (r) , and since A (l) is closed under successors and A (r) is closed under predecessors, we get that v = v 0 and h ∈ A (l) \ A (r) .

Case 1 Let
Let α 1 : v → a 1 and α 2 : v → a 2 be the starting arrows of p and q, respectively. Since v = v 0 is separating, by [6, Section 2], α 1 and α 2 belong to the same equivalence class. We consider two subcases.

Subcase 1a
Assume that there exists a minimal relation λ 1 α 1 x 1 + λ 2 α 2 x 2 + j ≥3 λ j y j (see [6, (2.4)]) from v to a vertex u ∈ A (l) involving α 1 and α 2 , where 0 = λ i ∈ k. If the admissible operation performed is (ad 1), then u ∈ C. Since h does not belong to A (r) , it follows that u = h and there is no path from h to u. Moreover, there is no path from u to h because w is irreducible. If the admissible operation performed is (ad 2) or (ad 3), we can take u as the vertex in A (l) \ A (r) described in Lemma 4.3. Since w is not naturally contractible, it follows that u = h and there is no path from h to u. Also, there is no path from u to h because w is irreducible. Assume that the admissible operation performed is (ad 4) or (ad 5). Since, by the assumption, the sectional paths occurring in the definitions of the operations (ad 4), (fad 4) come from two components of two connected algebras, we infer that u ∈ C or u ∈ A (l) ∩ A (r) \ C. Then in both cases, since h ∈ A (r) , we get that u = h and there is no path from h to u. Also, there is no path from u to h because w is irreducible.
Let b 1 (or b 2 ) be the last common vertex between x 1 and p (or x 2 and q, respectively) and u be the first common vertex between x 1 and x 2 . Denote by x 1 (or x 2 ) the subpath of x 1 (or x 2 ) from b 1 (or b 2 , respectively) to u , and by p (or q ) the subpath of p (or q) from b 1 (or b 2 ) to h.
• Consider now the walk w = x 2 −1 q p −1 x 1 . Note that this walk is a cycle. Indeed, there is no intersection between p and q , nor is there one between x 1 and x 2 , and the existence of an intersection between x 2 x 1 −1 and p q −1 would contradict the irreducibility of w. Moreover, w is irreducible because w is. Finally, w is not a contour because it has two different sinks u and h. Since w lies inside A (l) , we obtain a contradiction.
Subcase 1b Assume now that does not exist a minimal relation involving α 1 and α 2 . Since the equivalent class of an arrow starting at v is the smallest equivalent relation, we have that α 1 and α 2 belong to the same equivalent class if there exists a sequence of arrows β 1 , . . . , β s , s ≥ 1 such that there exists a sequence of minimal relations ρ 1 , . . . , ρ s+1 satisfying the following sequence of conditions: α 1 and β 1 are arrows in ρ 1 , β 2 and β 3 are arrows in ρ 2 , . . ., β s and α 2 are arrows in ρ s+1 . Using this sequence of minimal relations and the same arguments as in the previous subcase, we will get an analogous cycle and a corresponding contradiction.

Case 2
Let w be an irreducible cycle which is not a contour. Denote by v 1 , . . . , v t its sources, by h 1 , . . . , h t its sinks and for 1 ≤ i ≤ t, by w i the path from v i to h i , and by w i the path from v i to h i+1 (where h t+1 = h 1 ). Since w lies neither inside A (l) nor inside A (r) , we may assume that v 1 = v 0 and h s ∈ A (l) \ A (r) , where 1 ≤ s ≤ t. Let α 1 : v 1 → a 1 and α 2 : v 1 → a 2 be the starting arrows of w 1 and w 1 , respectively. Since v 1 is separating, applying [6, Section 2], we conclude that α 1 and α 2 belong to the same equivalence class. As before, we consider two subcases.

Subcase 2a
Assume that there exists a minimal relation λ 1 α 1 x 1 + λ 2 α 2 x 2 + j ≥3 λ j y j (see [6, (2.4)]) from v 1 to a vertex u ∈ A (l) involving α 1 and α 2 , where 0 = λ i ∈ k. If the admissible operation performed is (ad 1), then u ∈ C and u = h s and there is no path from h s to u. Also, by the irreducibility of w, there is no path from u to h s . If the admissible operation performed is (ad 2) or (ad 3), then u can be taken as the vertex in A (l) \ A (r) described in Lemma 4.3. Since w is irreducible, it follows that u = h s , there is no path from from u to h s and none from h s to u (otherwise, h l is the root of the branch in which u lies and there is a path from v 1 to h s that reduces the cycle). If the admissible operation performed is (ad 4) or (ad 5) then, by the assumption, the sectional paths occurring in the definitions of the operations (ad 4), (fad 4) come from two components of two connected algebras, we get that u ∈ C or u ∈ A (l) ∩ A (r) \ C. Then in both cases, since h s ∈ A (r) , we obtain that u = h s and there is no path from h s to u. Again, by the irreducibility of w, we have that there is no path from u to h s . Let b 1 (or b 2 ) be the last common vertex of x 1 (or x 2 ) and w 1 (or w 1 , respectively), and u be the first common vertex of x 1 and x 2 . Denote by x 1 (or x 2 ) the subpath of x 1 (or x 2 ) from b 1 (or b 2 , respectively) to u , and by w 1 (or w 2 ) the subpath of w 1 (or w 1 ) from b 1 (or b 2 ) to h 1 (or h 2 , respectively).
Note that either t ≥ 3 or, if t = 2, then b 1 = h 1 or b 2 = h 2 . Indeed, if t = 2, then one of h 1 or h 2 is equal to h s and we have just shown that there is no path from h s to u. Consider now the walk w = w 1 w t −1 w t . . . w 2 w 2 −1 x 2 x 1 −1 . This walk is a cycle. Indeed, the walk w 1 w t −1 w t . . . w 2 w 2 −1 has no self-intersection because it is a subwalk of w, the walk x 2 x 1 −1 has no self-intersection by definition, and these two do not intersect because w is irreducible. Moreover, w is irreducible because w is. Finally, it is not a contour because it has at least two sinks (u and h s ). Since w lies inside A (l) , we get a contradiction.

Subcase 2b
Assume now that does not exist a minimal relation involving α 1 and α 2 . Since the equivalent class of an arrow starting at v 1 is the smallest equivalent relation, we have that α 1 and α 2 belong to the same equivalent class if there exists a sequence of arrows β 1 , . . . , β s , s ≥ 1 such that there exists a sequence of minimal relations ρ 1 , . . . , ρ s+1 satisfying the following sequence of conditions: α 1 and β 1 are arrows in ρ 1 , β 2 and β 3 are arrows in ρ 2 , . . ., β s and α 2 are arrows in ρ s+1 . Using this sequence of minimal relations and the same arguments as in the previous subcase, we will get an analogous cycle and a corresponding contradiction.
The contradictions obtained in both cases show that A p+1 is strongly simply connected. Let k > p, A = A k+1 and assume that A k is strongly simply connected, but A is not. Let v 0 be the extension point of A k and X ∈ ind A k be the pivot of the admissible operation. If X is the pivot of the admissible operation (ad 1), then X| C i = 0 for some i ∈ {1, . . . , m}. Indeed, otherwise, by Lemma 4.2(ii), X would be uniserial, and since A is obtained from A k by the one-point extension by X followed by several one-point coextensions by simple modules, by [4,Theorem 3.4], A would be strongly simply connected, contrary to our assumption.
As before, by Theorem 2.6, the bound quiver (Q A , I ) of A contains an irreducible cycle which is not a contour, or an irreducible contour which is not naturally contractible in (Q A , I ).
Case 1 Let w = (p, q) be an irreducible contour from v to h which is not naturally contractible. Since w lies neither inside A k nor inside A (r) , we deduce that v = v 0 and h ∈ A (l) \ A (r) . Now, we can then proceed as in the first step of induction to obtain an irreducible cycle which is not a contour lying inside A k , thus contradicting our assumption.

Case 2
Let w be an irreducible cycle which is not a contour. Denote by v 1 , . . . , v t its sources, by h 1 , . . . , h t its sinks and for 1 ≤ i ≤ t, by w i the path from v i to h i , and by w i the path from v i to h i+1 (where h t+1 = h 1 ). As above, we may assume that v 1 = v 0 and h s ∈ A (l) \ A (r) , where 1 ≤ s ≤ t. Moreover, let α 1 : v 1 → a 1 and α 2 : v 1 → a 2 be the starting arrows of w 1 and w 1 , respectively. Since v 1 is separating, we have that α 1 and α 2 belong to the same equivalence class. As before, we consider two subcases.
Assume first that there exists a minimal relation λ 1 α 1 x 1 + λ 2 α 2 x 2 + j ≥3 λ j y j from v 1 to a vertex u ∈ A k involving α 1 and α 2 , where 0 = λ i ∈ k. If the admissible operation applied is (ad 1), then u ∈ C (more precisely, u ∈ C i fore some i ∈ {1, . . . , m}) and we can proceed as in the first step of induction. If the admissible operation applied is (ad 2) or (ad 3), then u is the vertex in A (l) \ A (r) described in Lemma 4.3. Again the irreducibility of w gives that u = h s , there is no path from from u to h s and none from h s to u in case h s is a coextension point of C. If h s is not a coextension point of C, then by Corollary 4.9 there is no path from h s to u. Indeed, otherwise v 0 is not constructible since S(X) does not have the proper shape. If the admissible operation applied is (ad 4) or (ad 5) then, by the assumption, the sectional paths occurring in the definitions of the operations (ad 4), (fad 4) come from two components of two connected algebras, we get that u ∈ C i or u ∈ A (l) . . , m}. Then in both cases, since h s ∈ A (r) i , we obtain that u = h s and there is no path from h s to u. Again, by the irreducibility of w, we have that there is no path from u to h s . Now, we can proceed as in the subcase 2a in the first step of induction to obtain an irreducible cycle which is not a contour lying inside A k , contrary to our assumption.
Assume now that does not exist a minimal relation involving α 1 and α 2 . We can proceed as in the subcase 2b in the first step of induction to obtain a corresponding contradiction.
Finally, the contradictions obtained in both cases show that A k+1 is strongly simply connected. This finishes the proof of the theorem.

Proof of Corollary 1.3
Let A be an algebra. We know from [40,Theorem 4.1] that A is cycle-finite and A admits a sincere stable tube if and only if A is either tame concealed or tubular algebra. Applying now the description of the structure of the Auslander-Reiten quiver of a tame generalized multicoil algebra [25,Theorem 4.8] (that is, a tame generalized multicoil enlargement of a finite family of tame concealed algebras), we conclude that any such algebra is cycle-finite. Using additionally [27,Theorem A] in the tame case, we receive that A is tame and A admits a separating family of almost cyclic coherent components if and only if A is cyclefinite and A admits a separating family of almost cyclic coherent components. Now, the Corollary 1.3 is a direct consequence of Theorem 1.2. We apply (ad 1 * ) with pivot the simple regular C-module with vector a 1 , and with parameter t = 0. The modified algebra B 1 is given by the quiver with vertices 1, 2, . . . , 8 bound by αβ = 0. Now, we apply (ad 1 * ) with pivot the indecomposable B 1 -module with vector a 2 , and with parameter t = 2. The modified algebra B 2 is given by the quiver with vertices 1, 2, . . . , 11 bound by αβ = 0. Next, we apply (ad 1 * ) with pivot the indecomposable B 2 -module with vector a 3 , and with parameter t = 3. The modified algebra B 3 is given by the quiver with vertices 1, 2, . . . , 15 bound by αβ = 0, γ δ = 0. In the next step we apply (ad 1 * ) with pivot the indecomposable B 3 -module with vector a 4 , and with parameter t = 0. The modified algebra B 4 is given by the quiver with vertices 1, 2, . . . , 16 bound by αβ = 0, γ δ = 0, ηε = 0. Next, we apply the admissible operation (ad 5) in three steps. and with parameter r = 4. The modified algebra is equal to A. Then the left quasitilted algebra A (l) of A is the convex subcategory of A being the bound quiver algebra kQ (l) /I (l) , where Q (l) is a full subquiver of Q given by the vertices 1, 2, . . . , 16 and I (l) = kQ (l) ∩ I is the ideal in kQ (l) . The right quasitilted algebra A (r) of A is the convex subcategory of A being the bound quiver algebra kQ (r) /I (r) , where Q (r) is a full subquiver of Q given by the vertices 1, 2, . . . , 7, 14, 15, . . . , 25 and I (r) = kQ (r) ∩ I is the ideal in kQ (r) . It follows from [27, Theorem A] and the above construction that the Auslander-Reiten quiver A of the tame algebra A = kQ/I admits a separating family of almost cyclic coherent components with exceptional configurations of modules. Moreover, by [30, Theorem 1.1], the first Hochschild cohomology space H 1 (A) ∼ = k 2 , and so A is not strongly simply connected. We also note that, since A (l) and A (r) are tame tilted algebras of Euclidean type D such that H 1 (A (l) ) = 0 and H 1 (A (r) ) = 0, it follows from [5,Theorem] that A (l) and A (r) are strongly simply connected. We refer to [27,Example 4.1] (see also [29,Example 9.13]) for the example of the tame algebra with a separating family of almost cyclic coherent components which is not strongly simply connected, because contains a full convex subcategory which is hereditary of type A 5 . and I the ideal in the path algebra kQ of Q over k generated by the elements αβ, γ δ, ηε, κλ , ϕ 3 ψ, ϕ 4 ψ, ξ 3 ω 1 , ζ 1 ϕ 1 , ζ 1 ϕ 2 , ζ 2 ξ 3 ξ 2 ξ 1 − ζ 1 ψ, π 1 ξ 2 , π 1 ω 1 − π 2 ω 2 , μκλ, νξ 1

. Then
A is a generalized multicoil enlargement of a concealed canonical algebra C = C 1 × C 2 , where C 1 is the hereditary algebra of Euclidean type D 6 given by the vertices 1, 2, . . . , 7, and C 2 is the hereditary algebra of Euclidean type D 5 given by the vertices 20, 21, . . . , 24.
Then the left quasitilted algebra A (l) of A is the convex subcategory of A being the product A (l) = A 2 ). Moreover, it follows from [27, Theorem A] and the above construction that A is tame and A admits a separating family of almost cyclic coherent components without exceptional configurations of modules. Then, by Theorem 1.2, the algebra A is strongly simply connected.

Further applications
As an application of Theorem 1.1 and results of [1,3,4,24,38] we shall describe the strongly simply connected tame iterated generalized multicoil enlargements of a tame concealed algebra.

Fundamental properties
Recall that if A is a domestic tubular extension of the tame concealed algebra, then its module category may be described as follows: mod A = P A ∪ T A ∪ Q A , where P A is a postprojective component, Q A is a preinjective component and T A is a tubular P 1 (k)-family separating P A from Q (see [34, (4.9)]).
If A is a tubular algebra, then we know from [34, (5.2)] that A is non-domestic of polynomial growth and where P A 0 is a semi-regular postprojective component, Q A ∞ is a semi-regular preinjective component, T A 0 is a P 1 (k)-family of ray tubes separating P A (because A is also a cotubular algebra), and each T A γ , γ ∈ Q + , where Q + is the set of all positive rationals, is a P 1 (k)-family of stable tubes separating Remark 9.2 In [24, Section 6] we introduced tame iterated generalized coil enlargements of a tame concealed algebra using admissible operations of types (ad 1), (ad 2), (ad 3) and (ad 4) for r = 0. In the case of (ad 4), r ≥ 1, we add r new projective modules, but in a strictly defined and controlled way (see definition of (ad 4)). Since in the definition of admissible operation (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), this construction can be easily generalized, as we do below.

Iterated generalized multicoil algebras
Domestic tubular extensions and coextensions and tubular algebras are obtained from a tame concealed algebra by performing a sequence of admissible operations (ad 1) or (ad 1 * ) in the stable tubes of its separating tubular family. We call these algebras 0-tame iterated generalized multicoil enlargements.
Let B 0 be a branch coextension of a (not necessarily connected) tame concealed algebra C 0 , and assume that B 0 is domestic or tubular (each connected algebra of B 0 is domestic or tubular). Then ind B 0 = P B 0 ∪ T B 0 ∪ Q B 0 , where Q B 0 is the preinjective component of B 0 , and T B 0 is a separating tubular family containing injectives. Applying admissible operations (ad 1)-(ad 5) we insert projectives in the coinserted and stable tubes of T B 0 . We obtain a generalized multicoil enlargement A 1 of C 0 with A (l) 1 = B 0 . If the branch extension B 1 = A (r) 1 of C 0 is tame, we call 1 = A 1 a 1-tame iterated generalized multicoil enlargement.
If B 1 is domestic, the iteration process stops. If B 1 is a tubular algebra, then it is a branch coextension of a tame concealed algebra C 1 , and we can write ind 1 = P 1 ∪ T 1 ∪ Q 1 , where Q 1 is the preinjective component of 1 , and T 1 is a separating tubular family containing injectives. More precisely, T 1 and Q 1 are, respectively, the last separating tubular family and the preinjective component of B 1 (see [24,Section 6]). Applying admissible operations (ad 1)-(ad 5) we insert projectives in the coinserted and stable tubes of T 1 . We obtain a generalized multicoil enlargement A 2 of C 1 with A (l) 2 = A (r) 1 = B 1 . If the branch extension B 2 = A (r) 2 of C 1 is tame, we call 2 obtained from 1 = A 1 by inserting projectives in the tubes of T 1 a 2-tame iterated generalized multicoil enlargement.
Again, if B 2 is domestic, the iteration process stops, and if B 2 is a tubular algebra we are able to iterate the process one more. Inductively, assume that n−1 is an (n − 1)-tame iterated generalized multicoil enlargement such that we are able to iterate the process one more. Therefore, there exists a generalized multicoil enlargement A n−1 of a tame concealed algebra C n−2 such that the branch extension B n−1 = A (r) n−1 of C n−2 is a tubular algebra. Hence B n−1 is a branch coextension of a tame concealed algebra C n−1 , and we can write ind n−1 = P n−1 ∪ T n−1 ∪ Q n−1 , where Q n−1 is the preinjective component of n−1 , and T n−1 is a separating tubular family containing injectives. Applying admissible operations (ad 1)-(ad 5) we insert projectives in the coinserted and stable tubes of T n−1 . We obtain a generalized multicoil enlargement A n of C n−1 with A (l) n = A (r) n−1 = B n−1 . If the branch extension B n = A (r) n of C n−1 is tame, we call n obtained from n−1 by inserting projectives in the tubes of T n−1 an n-tame iterated generalized multicoil enlargement.

Theorem 9.5 Let
A be an n-iterated generalized multicoil algebra such that A does not contain exceptional configurations of modules. Then the following statements are equivalent: (i) A is strongly simply connected. Proof Clearly, (i) implies (ii), and (ii) implies (iii). Since the algebras B 0 and B n are, respectively, either cotilted and tilted of Euclidean type or tubular algebras, and the remaining B i are tubular algebras, the equivalence of (iii), (iv), (v) and (vi) follows from [ . It remains to show that (vi) implies (i). We proceed by induction on n. If n = 1, the statement follows from Theorem 1.1. Assume that n > 1, and that the statement holds for any k ≤ n − 1. If A = n is not strongly simply connected, then by Theorem 2.6, it contains an irreducible cycle w which is not a contour, or an irreducible contour w which is not naturally contractible. By induction, w must contain a source lying in B n but not in C n−1 , and a sink lying in B 0 but not in C 0 . As in the proof of Theorem 1.1, we may replace w by an irreducible cycle w which is not a contour, but lies in n−1 , a contradiction to the induction hypothesis.

Concluding remarks
As a direct consequence of Theorem 1.2, Corollary 1.3 and results of [38] we obtain the following corollaries. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.