The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

Piotr Malicki

doi:10.1007/s10468-021-10053-x

The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

Malicki, Piotr 2022-08-01 00:00:00 We study the simple connectedness of the class of finite-dimensional algebras over an alge- braically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes. Keywords Simply connected algebra · Hochschild cohomology · Auslander–Reiten quiver · Tame algebra · Generalized multicoil algebra Mathematics Subject Classiﬁcation (2010) Primary 16G70 · Secondary 16G20 1 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 /I,where Q = (Q ,Q ) is the ordinary quiver of A with the set of vertices Q and the set of A 0 1 0 arrows Q and I is an admissible ideal in the path algebra kQ of Q . If the quiver Q has 1 A A A 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 the Jacobson radical of mod A, and by ∞ i rad the intersection of all powers rad , i ≥ 1, of rad . Moreover, we denote by A A A A the Auslander–Reiten quiver of A, and by τ and τ the Auslander–Reiten translations Dedicated to Claus Michael Ringel on the occasion of his 75th birthday Presented by: Christof Geiss Piotr Malicki pmalicki@mat.umk.pl Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, ´ Poland 924 P. Malicki D Tr and Tr D, respectively. We will not distinguish between a module in ind A and the vertex of corresponding to it. Following [45], a family C of components is said to be generalized standard if rad (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 -orbits in C are τ -periodic (see [45, (2.3)]). We refer to [37] for the structure and A A homological properties of arbitrary generalized standard Auslander–Reiten components of algebras. Following Assem and Skowronski ´ [7], a triangular algebra A is called simply connected if, for any presentation A kQ /I of A as a bound quiver algebra, the fundamental group π (Q ,I) of (Q ,I) is trivial (see Section 2). The importance of these algebras follows 1 A A from the fact that often we may reduce (using techniques of Galois coverings) the study of the module category of an algebra to that for the corresponding simply connected alge- bras. Let us note that to prove that an algebra is simply connected seems to be a difficult problem, because one has to check that various fundamental groups are trivial. Therefore, it is worth looking for a simpler characterization of simple connectedness. In [44, Problem 1] Skowronski ´ has asked, whether it is true that a tame triangular algebra A is simply con- nected if and only if the first Hochschild cohomology space H (A) of A vanishes. This equivalence is true for representation-finite algebras [3, Proposition 3.7] (see also [12]for the general case), for tilted algebras (see [5] for the tame case and [25] for the general case), for quasitilted algebras (see [3] for the tame case and [26] for the general case), for piecewise hereditary algebras of type any quiver [25], and for weakly shod algebras [4]. 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 fam- ily of tubes has been introduced by Ringel in [40, 41] 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, Skowronski ´ and Tomei ´ n[10](seealso[33]). A family C = (C ) of components i i∈I of the Auslander–Reiten quiver of an algebra A is called separating in mod A if the A A A components of split into three disjoint families P , C = C and Q such that: (S1) C is a sincere generalized standard family of components; A A A A A A (S2) Hom (Q , P ) = 0, Hom (Q , C ) = 0, Hom (C , P ) = 0; A A A A A (S3) any homomorphism from P to Q in mod A factors through the additive category A A add(C ) of C . A A A A A A Then we say that C separates P from Q and write = P ∪ C ∪ Q .We A A A note that then P and Q are uniquely determined by C (see [10, (2.1)] or [41, (3.1)]). Moreover, C is called sincere if any simple A-module occurs as a composition factor of a A A module in C . We note that if A is an algebra of finite representation type that C = is A A trivially a unique separating component of , with P and Q being empty. Frequently, we may recover A completely from the shape and categorical behavior of the separating family C of components of . For example, the tilted algebras [24, 41], or more generally double tilted algebras [39](the strict shod algebras in the sense of [15]), are determined by their (separating) connecting components. Further, it was proved in [28] that the class of algebras with a separating family of stable tubes coincides with the class of concealed canonical algebras. This was extended in [29] 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 [33] 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 is called almost cyclic if all but A Simply Connected Algebras 925 finitely many modules in lie on oriented cycles contained entirely in . Moreover, a component of 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 → X →· · · → X → X → X → ··· (that is, X = τ X for 1 2 i i+1 i+2 i A i+2 any i ≥ 1) in ; (C2) For each injective module I in there is an infinite sectional path ··· → Y → Y → Y →···→ Y → Y = I (that is, Y = τ Y for j +2 j +1 j 2 1 j +2 A j any j ≥ 1) in . We are now in position to formulate the first main result of the paper, which answers positively the above mentioned question of Skowronski ´ [44, Problem 1] for tame algebras with separating almost cyclic coherent Auslander–Reiten components. Theorem 1.1 Let A be a tame algebra with a separating family of almost cyclic coherent components in .Then A is simply connected if and only if H (A) = 0. It has been proved in [33, Theorem A] that the Auslander–Reiten quiver of an alge- bra A admits a separating family C of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a finite product of concealed canonical algebras C ,...,C by an iterated application of admissible algebra operations of types 1 m (ad 1)–(ad 5) and their duals. These algebras are called generalized multicoil algebras (see Section 3 for details). Note that for such an algebra A,wehavethat A is triangular, gl. dim A ≤ 3, and pd M ≤ 2orid M ≤ 2 for any module M in ind A (see [33, Corollary A A A B and Theorem E]). Moreover, let = P ∪ C ∪ Q be the induced decomposi- tion of . Then, by [33, Theorem C], there are uniquely determined quotient algebras (l) (l) (r) (r) (l) (r) A = A ×···× A and A = A ×···× A of A which are the quasitilted algebras m m 1 1 (l) (r) A A A A of canonical type such that P = P and Q = Q . Let A be a generalized multicoil algebra obtained from a concealed canonical algebra C = C ×· · ·× C and C = A ,A ,...,A = A be an admissible sequence for A (see 1 m 0 1 n Section 3). In order to formulate the next result 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 A , i ∈{0,...,n − 1}, then we will say that contains an exceptional configuration of i+1 modules. The following theorem is the second main result of the paper. Theorem 1.2 Let A be a generalized multicoil algebra obtained from a family C ,...,C 1 m of simply connected concealed canonical algebras. Assume moreover that does not (l) contain exceptional configurations of modules. Then there are quotient algebras A = (l) (l) (r) (r) (r) A × ··· × A and A = A ×· · ·× A of A such that the following statements are m m 1 1 equivalent: (i) A is simply connected. (l) (r) (ii) A and A are simply connected, for any i ∈{1,...,m}. i i (iii) H (A) = 0. (l) (r) 1 1 (iv) H (A ) = 0 and H (A ) = 0, for any i ∈{1,...,m}. i i (v) A is strongly simply connected. 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 926 P. Malicki 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 and 6 are devoted to the proofs of Theorem 1.1 and Theorem 1.2, respectively. The aim of the final Section 7 is to present examples illustrating the main results of the paper. For basic background on the representation theory of algebras we refer to the books [6, 41–43], for more information on simply connected algebras we refer to the survey article [2], and for more details on algebras with separating families of Auslander–Reiten components and their representation theory to the survey article [35]. 2 Preliminaries 2.1 Let A be an algebra and A = kQ /I beapresentationof A as a bound quiver algebra. Then the algebra A = kQ /I can equivalently be considered as a k-linear category, of which the object class A is the set of points of Q , and the set of morphisms A(x, y) from 0 A x to y is the quotient of the k-vector space kQ (x, y) of all formal linear combinations of paths in Q from x to y by the subspace I (x, y) = kQ (x, y) ∩ I (see [11]). A full A A 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 we denote by S the corresponding simple A-module, A v and by P (respectively, I ) the projective cover (respectively, the injective envelope) of S . v v v 2.2 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 B 0 B[M]= Mk with the usual addition and multiplication of matrices. The quiver of B[M] contains Q as a convex subquiver and there is an additional (extension) point which is a source. B[M]- modules are usually identified with triples (V , X, ϕ),where V is a k-vector space, X a B-module and ϕ : V → Hom (M, X) a k-linear map. A B[M]-linear map (V ,X,ϕ) → (V ,X ,ϕ ) is then identified with a pair (f, g),where f : V → V is k-linear, g : X → X is B-linear and ϕ f = Hom (M, g)ϕ. One defines dually the one-point coextension [M]B of B by M (see [41]). 2.3 Tameness and Wildness Let A be an algebra and K[x] the polynomial algebra in one variable x. Following [17], the algebra A is said to be tame if, for any positive integer d, there exists a finite number of K[x]− A-bimodules M ,1 ≤ i ≤ n , which are finitely i d generated and free as left K[x]-modules, and all but a finite number of isoclasses of inde- composable A-modules of dimension d are of the form K[x]/(x − λ) ⊗ M for some K[x] i λ ∈ K and some i ∈{1,...,n }. Recall that, following [17], the algebra A is wild if there is a kx, y-A-bimodule M, free of finite rank as left kx, y-module, and the functor −⊗ M : mod kx, y→ mod A preserves the indecomposability of modules and sends kx,y nonisomorphic modules to nonisomorphic modules. From Drozd’s Tame and Wild Theo- rem [17] the class of algebras may be divided into two disjoint classes. One class consists of the tame algebras and the second class is formed by the wild algebras whose representa- tion theory comprises the representation theories of all finite dimensional algebras over k. Simply Connected Algebras 927 Hence, a classification of the finite dimensional modules is only feasible for tame algebras. It has been shown by Crawley-Boevey [16] that, if A is a tame algebra, then, for any positive integer d ≥ 1, all but finitely many isomorphism classes of indecomposable A-modules of dimension d are invariant on the action of τ , and hence, by a result due to Hoshino [23], lie in stable tubes of rank one in . 2.4 Hochschild Cohomology of Algebras Let A be an algebra. Denote by C A the • i i i i Hochschild complex C = (C ,d ) defined as follows: C = 0, d = 0for i< 0, i∈Z 0 i ⊗i ⊗i C = A , C = Hom (A ,A) for i> 0, where A denotes the i-fold tensor product A A k 0 0 over k of A with itself, d : A → Hom (A, A) with (d x)(a) = ax − xa for x, a ∈ A, i i i+1 d : C → C with i j (d f )(a ⊗···⊗a ) = a f(a ⊗· · ·⊗a )+ (−1) f(a ⊗· · ·⊗a a ⊗· · ·⊗a ) 1 i+1 1 2 i+1 1 j j +1 i+1 j =1 i+1 +(−1) f(a ⊗· · ·⊗ a )a 1 i i+1 i i i • for f ∈ C and a ,a ,...,a ∈ A.Then H (A) = H (C A) is called 1 2 i+1 the i-th Hochschild cohomology space of A (see [14, Chapter IX]). Recall that the first Hochschild cohomology space H (A) of an algebra A is isomorphic to the space Der(A, A)/ Der (A, A) of outer derivations of A,where Der(A, A) ={δ ∈ Hom (A, A) | δ(ab) = aδ(b) + δ(a)b, for a, b ∈ A} is the space of k-linear derivations of A and Der (A, A) is the subspace {δ ∈ Hom (A, A) | δ (a) = ax − xa, for a ∈ A} of x k x inner derivations of A. 2.5 Concealed Canonical Algebras An important role in our considerations will be played by certain tilts of canonical algebras introduced by Ringel [41]. Let p ,p ,...,p be a 1 2 t sequence of positive integers with t ≥ 2, 1 ≤ p ≤ p ≤ ... ≤ p ,and p ≥ 2if t ≥ 3. 1 2 t 1 Denote by (p ,...,p ) the quiver of the form 1 t α 1p −1 12 1 ◦ ◦ ··· ◦ ◦ α 1p 11 1 α α α α 2p −1 2p 21 22 2 2 ◦◦ ◦ ··· ◦ ◦ ◦ α α t1 tp ◦ ◦ ··· ◦ ◦ α α t2 tp −1 For t ≥ 3, consider a (t + 1)-tuple of pairwise different elements of P (k) = k ∪{∞}, normalized such that λ =∞, λ = 0, λ = 1, and the admissible ideal I(λ ,λ ,...,λ ) 1 2 3 1 2 t in the path algebra k(p ,...,p ) of (p ,...,p ) generated by the elements 1 t 1 t α ...α α + α ...α α + λ α ...α α , 3 ≤ i ≤ t. ip i2 i1 2p 22 21 i 1p 12 11 i 2 1 Then the bound quiver algebra (p, λ) = k(p ,...,p )/I (λ ,λ ,...,λ ) is said to be 1 t 1 2 t the canonical algebra of type p = (p ,...,p ). Moreover, for t = 2, the path algebra 1 t (p) = k(p ,p ) is said to be the canonical algebra of type p = (p ,p ). It has been 1 2 1 2 provedin[41, Theorem 3.7] that if is a canonical algebra of type (p ,...,p ) then 1 t = P ∪ T ∪ Q for a P (k)-family T of stable tubes of tubular type (p ,...,p ), 1 1 t separating P from Q . Following [27], a connected algebra C is called a concealed canonical algebra of type (p ,...,p ) if C is the endomorphism algebra End (T ),for 1 t 928 P. Malicki some canonical algebra of type (p ,...,p ) and a tilting -module T whose indecom- 1 t posable direct summands belong to P . Then the images of modules from T via the functor Hom (T , −) form a separating family T of stable tubes of , and in particular C C C we have a decomposition = P ∪ T ∪ Q . It has been proved by Lenzing and de la Pena ˜ [28, Theorem 1.1] that the class of (connected) concealed canonical algebras coin- cides with the class of all connected algebras with a separating family of stable tubes. It is also known that the class of concealed canonical algebras of type (p ,p ) coincides with 1 2 the class of hereditary algebras of Euclidean types A , m ≥ 1(see[22]). Recall also that the canonical algebras of types (2, 2, 2, 2), (3, 3, 3), (2, 4, 4) and (2, 3, 6) are called the tubular canonical algebras, and an algebra which is tilting-cotilting equivalent to a tubular canonical algebra is called a tubular algebra (see [18, 21, 41]). 2.6 Simple Connectedness Let (Q, I ) be a connected bound quiver. A relation = λ w ∈ I (x, y) is minimal if m ≥ 2 and, for any nonempty proper subset J ⊂ i i i=1 −1 {1,...,m},wehave λ w ∈ / I (x, y). We denote by α the formal inverse of an j j j ∈J ε ε ε 1 2 t arrow α ∈ Q .A walk in Q from x to y is a formal composition α α ...α (where 1 2 α ∈ Q and ε ∈{−1, 1} for all i) with source x and target y.Wedenoteby e the triv- i 1 i x ial path at x.Let ∼ be the homotopy relation on (Q, I ), that is, the smallest equivalence relation on the set of all walks in Q such that: −1 −1 (a) If α : x → y is an arrow, then α α ∼ e and αα ∼ e . y x (b) If = λ w is a minimal relation, then w ∼ w for all i, j. i i i j i=1 (c) If u ∼ v,then wuw ∼ wvw whenever these compositions make sense. Let x ∈ Q be arbitrary. The set π (Q, I, x) of equivalence classes u of closed walks u 0 1 starting and ending at u has a group structure defined by the operation u · v = uv .Since Q is connected, π (Q,I,x) does not depend on the choice of x.Wedenoteitby π (Q, I ) and 1 1 call it the fundamental group of (Q, I ). Let A = kQ /I be a presentation of a triangular algebra A as a bound quiver algebra. The fundamental group π (Q ,I) depends essentially on I, so is not an invariant of A. 1 A A triangular algebra A is called simply connected if, for any presentation A kQ /I of A as a bound quiver algebra, the fundamental group π (Q ,I) of (Q ,I) is trivial 1 A A [7]. Example 2.7 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form λ δ 51 2 4 and I the ideal in the path algebra kQ of Q over k generated by the elements γβ, δα − aδβ, αλ,where a ∈ k \{0}.Then π (Q, I ) is trivial. Moreover, the triangular algebra A is simply connected. Indeed, any choice of a basis of rad /rad will lead to at least one minimal relation with target 1 and source i ∈{3, 4} or with target 5 and source 2. Remark 2.8 It is known, for example, that the following important classes of algebras are simply connected: the iterated tilted algebras of Dynkin type (see [1, Proposition 3.5]), the iterated tilted algebras of Euclidean types D , E , n ≥ 4, p = 6, 7, 8, the tubular algebras n p (see [7, Corollary 1.4]), and the pg-critical algebras (see [38, Corollary 3.3]). Simply Connected Algebras 929 3 Almost Cyclic Coherent Auslander–Reiten components 3.1 Generalized Multicoil Algebras It has been proved in [32, Theorem A] that a con- nected component of an Auslander–Reiten quiver 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 [33, Section 3]. Given a generalized standard component of , and an indecomposable module X in ,the support S(X) of the functor Hom (X, −)| is the k-linear category defined as follows [9]. Let H denote the full subcategory of consisting of the indecomposable modules M in such that Hom (X, M) = 0, and I denote the ideal of H consisting of A X X the morphisms f : M → N (with M, N in H ) such that Hom (X, f ) = 0. We define X A S(X) to be the quotient category H /I . Following the above convention, we usually X X identify the k-linear category S(X) with its quiver. Recall that a module X in mod A is called a brick if End (X) k. Let A be an algebra and be a family of generalized standard infinite components of . 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 (X, −)| . These admissible operations yield in each case a modified algebra A such that the mod- ified translation quiver is a family of generalized standard infinite components in the Auslander–Reiten quiver of A (see [32, Section 2] or [35, Section 4] for the figures illustrating the modified translation quiver ). (ad 1) Assume S(X) consists of an infinite sectional path starting at X: X = X → X → X →· · · 0 1 2 Let t ≥ 1 be a positive integer, D be the full t ×t lower triangular matrix algebra, and Y , ..., Y denote the indecomposable injective D-modules with Y = Y the unique indecomposable t 1 projective-injective D-module. We set A = (A × D)[X ⊕ Y ]. In this case, is obtained by inserting in the rectangle consisting of the modules Z = k, X ⊕ Y , for i ≥ 0, ij i j 1 ≤ j ≤ t,and X = (k, X , 1) for i ≥ 0. If t = 0weset A = A[X] and the rectangle reduces to the sectional path consisting of the modules X , 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: Y ← ··· ← Y ← Y ← X = X → X → X →· · · t 2 1 0 1 2 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 Z = k, X ⊕ Y , ij i j for i ≥ 1, 1 ≤ j ≤ t,and X = (k, X , 1) for i ≥ 0. (ad 3) Assume S(X) is the mesh-category of two parallel sectional paths: Y → Y →···→ Y 1 2 t ↑↑ ↑ X = X → X →···→ X → X →· · · 0 1 t −1 t with the upper sectional path finite and t ≥ 2. In particular, X is necessarily injective. t −1 Moreover, we consider the translation quiver of obtained by deleting the arrows Y → −1 τ Y . We assume that the union of connected components of containing the modules i−1 −1 τ Y ,2 ≤ i ≤ t, is a finite translation quiver. Then is a disjoint union of and i−1 a cofinite full translation subquiver , containing the pivot X.Weset A = A[X].Inthis 930 P. Malicki case, is obtained from by inserting the rectangle consisting of the modules Z = ij k, X ⊕ Y , for i ≥ 1, 1 ≤ j ≤ i,and X = (k, X , 1) for i ≥ 0. i j i 1 i (ad 4) Suppose that S(X) consists of an infinite sectional path, starting at X X = X → X → X →· · · and Y = Y → Y →· · · → Y 0 1 2 1 2 t with t ≥ 1, is a finite sectional path in .Let r be a positive integer. Moreover, we −1 consider the translation quiver of obtained by deleting the arrows Y → τ Y .We i i−1 −1 assume that the union of connected components of containing the vertices τ Y , i−1 2 ≤ i ≤ t, is a finite translation quiver. Then is a disjoint union of and a cofinite full translation subquiver , containing the pivot X.For r = 0weset A = A[X ⊕ Y ]. In this case, is obtained from by inserting the rectangle consisting of the modules Z = k, X ⊕ Y , for i ≥ 0, 1 ≤ j ≤ t,and X = (k, X , 1) for i ≥ 0. ij i j i 1 i For r ≥ 1, let G be the full r × r lower triangular matrix algebra, U , U , 1,t +1 2,t +1 ..., U denote the indecomposable projective G-modules, U , U , ..., U r,t +1 r,t +1 r,t +2 r,t +r denote the indecomposable injective G-modules, with U the unique indecomposable r,t +1 projective-injective G-module. We define the matrix algebra ⎡ ⎤ A 00 ... 00 ⎢ ⎥ Yk 0 ... 00 ⎢ ⎥ ⎢ ⎥ Y k k ... 00 ⎢ ⎥ A = ⎢ . . . . . ⎥ . . . . . . ⎢ ⎥ . . . . . ⎢ ⎥ ⎣ ⎦ Y k k ... k 0 X ⊕ Y kk... kk with r + 2 columns and rows. In this case, is obtained from by inserting the following modules (k, Y , 1) for s = 1, 1 ≤ l ≤ t, ⎨ l 1 for i ≥ 0 and U = (k, U , 1) for 2 ≤ s ≤ r,1 ≤ l< t + s, Z = k, X ⊕ U , sl s,l−1 ij i rj 1 1 ≤ j ≤ t + r, (k, 0, 0) for 2 ≤ s ≤ r, l = t + s, and X = (k, X , 1) for i ≥ 0. In the above formulas U is treated as a module over the i sl algebra A = A [U ],where A = A and U = Y (in other words A is an algebra s s−1 s−1,1 0 01 s consisting of matrices obtained from the matrices belonging to A by choosing the first s +1 rows and columns). We note that the quiver Q of A is obtained from the quiver of the double one-point extension A[X][Y ] by adding a path of length r + 1 with source at the extension vertex of A[X] and sink at the extension vertex of A[Y ]. 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 → X → X →··· (in the definitions of (ad 1), (ad 2), 0 1 2 (ad 3), (ad 4)) by the finite sectional paths of the form X → X → X → ··· → X . 0 1 2 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 is injective (see the figures for (fad 1)–(fad 4) in [32, Section 2] or [35, 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 Simply Connected Algebras 931 (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). Moreover, together with each of the admissible operations (ad 1)–(ad 5), we consider ∗ ∗ its dual, denoted by (ad 1 )–(ad 5 ). These dual operations are also called admissible. Fol- lowing [32], a connected translation quiver is said to be a generalized multicoil if can be obtained from a finite family T , T ,..., T of stable tubes by an iterated application of 1 2 s ∗ ∗ ∗ ∗ admissible operations (ad 1), (ad 1 ), (ad 2), (ad 2 ), (ad 3), (ad 3 ), (ad 4), (ad 4 ), (ad 5) or (ad 5 ). If s = 1, such a translation quiver is said to be a generalized coil.The admis- ∗ ∗ sible operations of types (ad 1)–(ad 3), (ad 1 )–(ad 3 ) have been introduced in [8–10], and the admissible operations (ad 4) and (ad 4 )for r = 0in[30]. Finally, let C be a (not necessarily connected) concealed canonical algebra and T a separating family of stable tubes of . Following [33] we say that an algebra A is a gen- eralized multicoil enlargement of C using modules from T if there exists a sequence of algebras C = A ,A ,...,A = A such that A is obtained from A by an admissi- 0 1 n i+1 i ∗ ∗ ble operation of one of the types (ad 1)–(ad 5), (ad 1 )–(ad 5 ) performed either on stable A A i i tubes of T , or on generalized multicoils obtained from stable tubes of T by means of operations done so far. The sequence C = A ,A ,...,A = A is then called an admissi- 0 1 n ble sequence for A. Observe that this definition extends the concept of a coil enlargement of a concealed canonical algebra introduced in [10]. 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 [41]. 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. Proposition 3.2 [33, Proposition 3.7] Let C be a concealed canonical algebra, T asep- arating family of stable tubes of , and A a generalized multicoil enlargement of C using C A modules from T .Then admits a generalized standard family C of generalized mul- ticoils obtained from the family T of stable tubes by a sequence of admissible operations corresponding to the admissible operations leading from C to A. The following theorem, proved in [33, Theorem A], will be crucial for our further considerations. Theorem 3.3 Let A be an algebra. The following statements are equivalent: (i) admits a separating family of almost cyclic coherent components. (ii) A is a generalized multicoil enlargement of a concealed canonical algebra C. Remark 3.4 The concealed canonical algebra C is called the core of A and the number m of connected summands of C is a numerical invariant of A. We note that m can be arbitrary large, even if A is connected. Let us also note that the class of algebras with generalized standard almost cyclic coherent Auslander–Reiten components is large (see [34, Proposition 2.9] and the following comments). We note that the class of tubular extensions (respectively, tubular coextensions) of con- cealed canonical algebras coincides with the class of algebras having a separating family of ray tubes (respectively, coray tubes) in their Auslander–Reiten quiver (see [27, 29]). Moreover, these algebras are quasitilted algebras of canonical type. 932 P. Malicki We recall also the following theorem on the structure of the module category of an algebra with a separating family of almost cyclic coherent Auslander–Reiten components provedin[33, Theorems C and F]. Theorem 3.5 Let A be an algebra with a separating family C of almost cyclic coherent A A A components in , and = P ∪ C ∪ Q the associated decomposition of .Then A A A the following statements hold. (l) (l) (l) (i) There is a unique full convex subcategory A = A ×· · · × A of A which is a tubular coextension of a concealed canonical algebra C = C × ... × C such 1 m (l) (l) (l) (l) A A A A that (l) = P ∪ T ∪ Q for a separating family T of coray tubes (l) A A (l) in , P = P , and A is obtained from A by a sequence of admissible (l) (l) operations of types (ad 1)–(ad 5) using modules from T . (r) (r) (r) (ii) There is a unique full convex subcategory A = A ×· · · × A of A which is a tubular extension of a concealed canonical algebra C = C × ... × C such 1 m (r) (r) (r) (r) A A A A that (r) = P ∪ T ∪ Q for a separating family T of ray tubes (r) A A (r) in , Q = Q , and A is obtained from A by a sequence of admissible (r) (r) ∗ ∗ A operations of types (ad 1 )–(ad 5 ) using modules from T . (l) (r) (iii) A is tame if and only if A and A are tame. (l) (r) In the above notation, the algebras A and A are called the left and right quasitilted (l) (r) (l) (r) algebras of A. Moreover, the algebras A and A are tame if and only if A and A are products of tilted algebras of Euclidean type or tubular algebras. Recall that an algebra A is strongly simply connected if every convex subcategory of A is simply connected (see [44]). Clearly, if A is strongly simply connected then A is simply connected. We need the following result proved in [31, Theorem 1.1]. Theorem 3.6 Let A be an algebra with a separating family of almost cyclic coherent components in without exceptional configurations of modules. Then there are quotient (l) (l) (r) (r) (l) (r) algebras A = A ×···× A and A = A ×···× A of A such that the following m m 1 1 statements are equivalent: (i) A is strongly simply connected. (l) (r) (ii) A and A are strongly simply connected, for any i ∈{1,...,m}. i i 4 Preliminary Results 4.1 Branch Extensions and Coextensions Let A be an algebra and A kQ /I be a pre- = A sentation of A as a bound quiver algebra. For a given vertex v in Q , we denote by v (respectively, by v) the set of all arrows of the quiver Q starting at v (respectively, ter- minating at v). Let now K be a branch at a vertex v ∈ Q and E ∈ mod A. Recall that the branch extension A[E, K] by the branch K [41, (4.4)] is constructed in the follow- ing way: to the one-point extension A[E] with extension vertex w (that is, rad P = E) we add the branch K by identifying the vertices v and w.If E ,...,E ∈ mod A and 1 n K ,...,K is a set of branches, then the branch extension A[E ,K ] is defined induc- 1 n i i i=1 n−1 tively as: A[E ,K ] = (A[E ,K ] )[E ,K ]. The concept of branch coextension is i i i i n n i=1 i=1 defined dually. Simply Connected Algebras 933 Lemma 4.2 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C = C ×· · · × C . Moreover, let C = A ,...,A = A ,A ,...,A = A be an 1 m 0 p p+1 n admissible sequence for A, j ≥ p, X ∈ ind A be an (ad 2) or (ad 3)-pivot, and A be j j +1 the modified algebra of A .If v is the corresponding extension point then there is a unique (l) (r) vertex u ∈ A \ A that satisfies: (i) Each α ∈ v is the starting point of a nonzero path ω ∈ A(v, u). → → (ii) There are at least two different arrows in v . Moreover, if α, β ∈ v , and α = β, then ω − ω ∈ I . α β (l) Proof We know from [33, Section 4] that A is a unique maximal convex branch coex- (l) (l) (l) (l) tension of C = C ×· · ·× C inside A,thatis, A = B ×· · ·× B ,where B is a 1 m m unique maximal convex branch coextension of C inside A, i ∈{1,...,m}. More precisely, (l) t B = [K ,E ]C ,where K ,...,K are branches, i ∈{1,...,m}. Then there exists j j i 1 t i j =1 (l) s ∈{1,...,m} such that u ∈ B and A = A [X].If X is an (ad 2)-pivot (respectively, s j +1 j (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 ,...,Y in the sup- 1 t port of Hom (X, −) restricted to the generalized multicoil containing X - see definition of (ad 3)). The operations done after must not affect the support of Hom (X, −) restricted to the generalized multicoil containing X. Note that in general, in the sequence of earlier admissible operations, there can be an operation of type (ad 5) which contains an opera- tion (fad 4) which gives rise to the pivot X of (ad 2) (respectively, to the pivot X of (ad 3)) but from Lemma [33, 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 = A [X],and u, u ,...,u (where X = I , Y = j +1 j 1 t u i I for i ∈{1,...,t } - see definition of (ad 2)) be the points in the quiver Q of A u A j i j corresponding to the new indecomposable injective A -modules obtained after performing ∗ ∗ the above admissible operation (ad 1 ) or the operation (fad 1 ). Then u, u ,...,u ∈ 1 t (l) A .Since X = rad P , there must be a nonzero path from v to each vertex w which is a predecessor of u. Hence, each α ∈ v is the starting arrow of a nonzero path from v to u, and there are at least two arrows in v , namely: one from v to u and one from v to a point in Supp X ,where X is the immediate successor of X on the infinite sectional path 1 1 in S(X) (see definition of (ad 2)). Moreover, since P (u) = X(u) = k, all paths from v to u are congruent modulo I . The bound quiver Q of A is of the form j +1 A j +1 j +1 u Supp X u ··· u 1 t where A (v, u) is one-dimensional. From the proofs of [33, Theorems A and C], we j +1 (l) (r) (r) (l) (l) (r) have u ∈ A \ A , v ∈ A \ A ,and u ,...,u ∈ A ∩ A . 1 t Let now X be an (ad 3)-pivot, A = A [X], and assume that we had r consecutive j +1 j ∗ ∗ admissible operations of types (ad 1 )or(fad1 ), the first of which had X as a pivot, and t 934 P. Malicki these admissible operations built up a branch K in A with points u, u ,...,u in Q ,so j 1 t A that X and Y are the indecomposable injective A -modules corresponding respectively t −1 t j −1 to u and u , and both Y and τ Y are coray modules in the generalized multicoil con- 1 1 1 taining the (ad 3)-pivot X (where X, X ,X ,Y and Y are as in the definition of (ad 3)). t −1 t 1 t (l) Then u, u ∈ A and X is the indecomposable A -module given by: X(w) = 0if w< u , 1 j 1 X(w) = k if u <w,and X(w) = X (w) in any other case. Since X = rad P ,there 1 t −1 v must be a nonzero path from v to each vertex w whichisapredecessorof u, but those which are predecessors of u . Hence, each α ∈ v is the starting arrow of a nonzero path from v to u, and there are at least two arrows in v , namely: one from v to u and one from v to a point in Supp X ,where X is the immediate successor of X on the infinite sectional t t t −1 path in S(X) (see definition of (ad 3)). Moreover, since P (u) = X (u) = k, all paths v t −1 from v to u are congruent modulo I . The bound quiver Q of A is of the form j +1 A j +1 j +1 Supp X uv rest of K where A (v, u) is one-dimensional. Again, from the proofs of [33, Theorems A and C], j +1 (l) (r) (r) (l) we have u ∈ A \ A , v ∈ A \ A , u and the vertices of the branch K belong to (l) (r) A ∩ A . Lemma 4.3 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C = C ×· · · × C . Moreover, let C = A ,...,A = A ,A ,...,A = A be an 1 m 0 p p+1 n admissible sequence for A, j ≥ p, X ∈ ind A be an (ad 1)-pivot, A be the modified j j +1 algebra of A , and v be the corresponding extension point. Then the following statements hold. (l) (r) → (i) If there is a vertex u ∈ A \ A such that each α ∈ v is the starting point of a nonzero path ω ∈ A(v, u), then: (a) The vertex u is unique. (b) There are at least two different arrows in v . (c) If α, β ∈ v , and α = β,then ω − ω ∈ I . α β (ii) If X| = 0 for any i ∈{1,...,m},then X is uniserial. Proof Since X is an (ad 1)-pivot, the support S(X) consists of an infinite sectional path X = X → X → X →··· starting at X.Let t ≥ 1 be a positive integer, D be the full 0 1 2 t × t lower triangular matrix algebra, and Y , ..., Y be the indecomposable injective D- 1 t modules with Y the unique indecomposable projective-injective D-module (see definition of (ad 1)). (l) (i) Again, we know from [33, Section 4] that A is a unique maximal convex branch (l) (l) (l) coextension of C = C × ··· × C inside A,thatis, A = B ×· · ·× B ,where 1 m (l) B is a unique maximal convex branch coextension of C inside A, i ∈{1,...,m}.More (l) t precisely, B = [K ,E ]C ,where K ,...,K are branches, i ∈{1,...,m}. Assume j j i 1 t i j =1 (l) (r) → that there is a vertex u ∈ A \ A such that each α ∈ v is the starting point of a Simply Connected Algebras 935 (l) nonzero path ω ∈ A(v, u). Then there exists s ∈{1,...,m} such that u ∈ B . Moreover, α s A = (A × D)[X ⊕ Y ] and the bound quiver Q | is of the form j +1 j 1 A Supp X j +1 Q (l) ◦ uv v ··· v 1 t >? <= :; ./ ,- *+ () &' $% "# ! where v ,...,v are the points in the quiver Q of A corresponding to the new 1 t A j +1 j +1 (l) indecomposable projective A -modules. Then A is the extension of B at X by j +1 j +1 s the extension branch K consisting of the points v, v ,...,v , that is, we have A = 1 t j +1 (r) → A [X, K].Since u does not belong to A and for any α ∈ v it is the starting point of a nonzero path ω ∈ A(v, u), we get that u is the coextension point of the admissible oper- ∗ ∗ ∗ ∗ ation (ad 2 )or(ad3 ). By [10, Lemma 3.1] the admissible operations (ad 2 )and (ad3 ) ∗ ∗ commute with (ad 1), so we can apply (ad 2 ) after (ad 1) (respectively, (ad 3 ) after (ad 1)). Using now [10, Lemma 3.3] (respectively, [10, Lemma 3.4]), we are able to replace (ad 1) ∗ ∗ ∗ followedby(ad2 ) (respectively, (ad 1) followed by (ad 3 )) by an operation of type (ad 1 ) followed by an operation of type (ad 2) (respectively, (ad 1 ) followed by an operation of type (ad 3)). Therefore, the statements (a), (b) and (c) follow from Lemma 4.2. (ii) A case by case inspection (which admissible operation gives rise to the (ad 1)-pivot X)shows that X is either simple module or the support of X is a linearly ordered quiver of type A . Lemma 4.4 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C = C ×· · · × C . Moreover, let C = A ,...,A = A ,A ,...,A = A be an 1 m 0 p p+1 n admissible sequence for A, j ≥ p, X ∈ ind A be an (ad 4) or (ad 5)-pivot, A be the j j +1 modified algebra of A , and v be the corresponding extension point. If there is a vertex (l) (r) → u ∈ A \ A such that for pairwise different arrows α ,...,α ∈ v , q ≥ 2 there are 1 q paths ω ,...,ω ∈ A(v, u), then for arbitrary f, g ∈{1,...,q},f = g, one of the α α 1 q following cases holds: (i) At least one of ω ,ω is zero path. α α f g (ii) The paths ω ,ω are nonzero and ω − ω ∈ I . α α α α g g f f (l) Proof It follows from [33, Section 4] that A is a unique maximal convex branch coex- (l) (l) (l) (l) tension of C = C ×· · ·× C inside A,thatis, A = B ×· · ·× B ,where B is a 1 m m 1 i unique maximal convex branch coextension of C inside A, i ∈{1,...,m}. More precisely, (l) t B = [K ,E ]C ,where K ,...,K are branches, i ∈{1,...,m}. Assume that there j j i 1 t i i j =1 (l) (r) → is a vertex u ∈ A \ A such that for pairwise different arrows α ,...,α ∈ v , q ≥ 2, 1 q (l) there are paths ω ,...,ω ∈ A(v, u). Then there exists s ∈{1,...,m} such that u ∈ B . α α q s Let X be an (ad 4)-pivot and Y → Y → ··· → Y with t ≥ 1, be a finite sectional path 1 2 t in (as in the definition of (ad 4)). Note that this finite sectional path is the linearly ori- ented quiver of type A and its support algebra (given by the vertices corresponding to the simple composition factors of the modules Y ,Y ,...,Y ) is a tilted algebra of the path 1 2 t 936 P. Malicki algebra D of the linearly oriented quiver of type A .From[41, (4.4)(2)] we know that is a bound quiver algebra given by a branch in x,where x corresponds to the unique projective- injective D-module. Let be a generalized multicoil of obtained by applying the j +1 admissible operation (ad 4), where X is the pivot contained in the generalized multicoil , and Y is the starting vertex of a finite sectional path contained in the generalized multicoil or .So, is obtained from or from the disjoint union of two generalized multi- 1 2 1 coils , by the corresponding translation quiver admissible operations. In general, 1 2 1 and are components of the same connected algebra or two connected algebras. Hence, we get two cases. In the first case X, Y ∈ or X ∈ , Y ∈ and , are two 1 1 1 1 2 1 2 components of the same connected algebra. In the second case X ∈ , Y ∈ and , 1 1 2 1 are two components of two connected algebras. Therefore, the bound quiver Q of 2 A j +1 A in the first case is of the form j +1 u Supp X v d+1 w ··· w w d 1 β β β d 2 1 for r = 0and u Supp X v v d+1 w ··· w w v ··· v d 1 r 2 β β β d 2 1 for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of A [X], w is the extension point of A [Y ], w ,...,w belong to the branch in w generated j j 1 1 d by the support of Y ⊕ ··· ⊕ Y ,and αβ ...β = 0for some h ∈{1,...,d + 1}.Inthe 1 t 1 h second case the bound quiver Q of A is of the form A j +1 j +1 u Supp X v yw ··· w w d 1 β β β β d+1 d 2 1 for r = 0and Supp X u v v yw ··· w w v ··· v d 1 r 2 β β β β d+1 d 2 1 Simply Connected Algebras 937 for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of A [X], w is the extension point of A [Y ], w ,...,w belong to the branch in w generated j j 1 1 d by the support of Y ⊕· · ·⊕ Y , αβ ...β = 0forsome h ∈{1,...,d + 1},and y is 1 t 1 h (l) (l) (r) the coextension point of A such that y ∈ A \ A . More precisely, y ∈ B ,where s ∈{1,...,m} and s = s. Moreover in both cases, we have P (u) = X(u) = k or P (u) = X(u) = 0, and hence all nonzero paths from v to u are congruent modulo I .So, v j +1 A (v, u) is at most one-dimensional. We note that in the first case, the definition of (ad 4) j +1 (see the shape of the bound quiver Q of A ) implies that if the paths ω ,ω ∈ A j +1 α α j +1 f g A (v, u) are nonzero and ω − ω ∈ I, then there is also a zero path ω ∈ A (v, u) j +1 α α α j +1 f g h for some h ∈{1,...,q},h = f = g. Let X be an (ad 5)-pivot and be a generalized multicoil of obtained by apply- j +1 ing this admissible operation with pivot X.Then is obtained from the disjoint union of the finite family of generalized multicoils , ,..., by the corresponding translation 1 2 e quiver admissible operations, 1 ≤ e ≤ l,where l is the number of stable tubes of 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. Remark 4.5 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C. We know from Theorems 3.3 and 3.5 that A can be obtained from A by a sequence of (r) admissible operations of types (ad 1)–(ad 5) or A can be obtained from A by a sequence ∗ ∗ of admissible operations of types (ad 1 )–(ad 5 ). We note that all presented above lemmas ∗ ∗ can be formulated and proved for dual operations (ad 1 )–(ad 5 ) in a similar way. 4.6 The Separating Vertex Let A be a triangular algebra. Recall that a vertex v of Q is called separating if the radical of P is a direct sum of pairwise nonisomorphic indecom- posable modules whose supports are contained in different connected components of the subquiver Q(v) of Q obtained by deleting all those vertices u of Q being the source of a A A path with target v (including the trivial path from v to v). We have the following lemma which follows from the proof of [44, Proposition 2.3] (see also [2, Lemma 2.3]). Lemma 4.7 Let A be a triangular algebra and assume that A = B[X],where v is the extension vertex and X = rad P .If B is simply connected and v is separating, then A is A v simply connected. Let D be the same as in the definition of (ad 1), that is, the full t × t lower triangular matrix algebra. Denote by Y , ..., Y the indecomposable injective D-modules with Y = Y 1 t 1 the unique indecomposable projective-injective D-module. Lemma 4.8 Let A be a triangular algebra and assume that A = (B × D)[X ⊕ Y ],where v is the extension vertex and X ⊕ Y = rad P .If B is simply connected and v is separating, A v then A is simply connected. Proof Since the module P is a sink in the full subcategory of ind A consisting of projec- tives, the vertex v is a source in Q . Moreover, A = (B × D)[X ⊕ Y ],where X is the indecomposable direct summand of rad P that belongs to mod B and Y is a directing A v 938 P. Malicki module (that is, an indecomposable module which does not lie on a cycle in ind A) such that rad P = X ⊕ Y . Therefore, the proof follows from the proof of [44, Proposition 2.3] (see A v also the proof of Lemma 2.3 in [2]). 4.9 The Pointed Bound Quiver In order to carry out the construction of the free product of two fundamental groups of bound quivers, and in analogy with algebraic topology where pointed spaces are considered, one can define a pointed bound quiver (Q,I,x),thatis,a bound quiver (Q, I ) together with a distinguished vertex x (see [13, Section 3]). Given two pointed bound quivers Q = (Q ,I ,x ) and Q = (Q ,I ,x ), we can assume, without loss of generality, that Q ∩ Q = Q ∩ Q =∅. We define the quiver Q = Q Q 0 0 1 1 as follows: Q is Q ∪ Q in which we identify x and x toasinglenew vertex x,and 0 0 Q = Q ∪Q . Then, Q and Q are identified to two full convex subquivers of Q,sowalks 1 1 on Q or Q can be considered as walks on Q. Thus, I and I generate two-sided ideals of kQ which we denote again by I and I .Wedefine I to be the ideal I +I of kQ. It follows from this definition that the minimal relations of I are precisely the minimal relations of I together with the minimal relations of I give the minimal relations needed to determine the homotopy relation on (Q, I ). Moreover, we can consider an element w ∈ π (Q ,I ,x ) as an element w ∈ π (Q,I,x) (we denote by w the homotopy class of a walk w). Conversely, any (reduced) walk w in Q has a decomposition w = w w w w ...w w ,where w and 1 1 2 2 n n i w are walks in Q and Q for i ∈{1,...,n}, respectively. Moreover, this decomposition is unique, up to reduced walk, and compatible with the homotopy relations involved. This leads us to the following proposition. Proposition 4.10 [13, Proposition 3.1] With the notations above we have: (i) (Q,I,x) is the coproduct of (Q ,I ,x ) and (Q ,I ,x ) in the category of pointed bound quivers. (i) π (Q, I, x) π (Q ,I ,x ) π (Q ,I ,x ). 1 1 1 5 Proof of Theorem 1.1 The aim of this section is to prove Theorem 1.1 and recall the relevant facts. We know from Theorem 3.3 that the Auslander–Reiten quiver of A admits a separat- ing family of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a concealed canonical algebra C.Let C = C × C ×· · ·× C × C ×···× 1 2 l l+1 C be a decomposition of C into product of connected algebras such that C ,C ,...,C m 1 2 l are of type (p ,p ) and C ,C ,...,C are of type (p ,...,p ) with t ≥ 3. Following 1 2 l+1 l+2 m 1 t [36], by h we denote the number of all stable tubes of rank one from with 1 ≤ i ≤ l, i C used in the whole process of creating A from C,and h = 0, if l + 1 ≤ i ≤ m. Moreover, let 0if C is of type (p ,...,p ) with t ≥ 3 i 1 t 1if C is of type (p ,p ) with p ,p ≥ 2 i 1 2 1 2 e = 2if C is of type (p ,p ) with p = 1, p ≥ 2 i 1 2 1 2 3if C is of type (p ,p ) with p = p = 1, i 1 2 1 2 for i ∈{1,...,m}.Wedefinealso f = max{e − h , 0},for i ∈{1,...,m} and set C i i m l f = f = f . Note that we can apply the operations (ad 4), (fad 4), (ad 4 ), A C C i=1 i i=1 i (fad 4 ) 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 Simply Connected Algebras 939 connected algebras. By d we denote the number of all operations (ad 4), (fad 4), (ad 4 )or (fad 4 ) which are of the first type, used in the whole process of creating A from C. The Hochschild cohomology of a connected generalized multicoil algebra A has been described in [36, Theorem 1.1] using the numerical invariants of A (f , d and the others), A A depending on the types of admissible operations (ad 1)–(ad 5) and their duals, leading from a product C of concealed canonical algebras to A. Here, we will only need information about the first Hochschild cohomology of A, namely from [36, Theorem 1.1(iii)] we have: Theorem 5.1 Let A be a connected generalized multicoil algebra. Then dim H (A) = d + f . A A We are now able to complete the proof of Theorem 1.1. Since A is tame, we may restrict to the generalized multicoil enlargements of tame concealed algebras. Namely, we have the following consequence of Theorem 3.3 and [33, Theorem F]: A is tame and admits a separating family of almost cyclic coherent components if and only if A is a tame generalized multicoil enlargement of a finite family C ,...,C of tame concealed algebras (concealed canonical algebras of Euclidean type). 1 m We first show the necessity. Suppose that A is simply connected. We must show that the 1 1 first Hochschild cohomology H (A) of A vanishes. Assume to the contrary that H (A) = 0. Then by Theorem 5.1, d + f = 0. If d = 0, then it follows from the proof of A A A Lemma 4.4 (and its dual version) that A is not simply connected, a contradiction. Therefore, we may assume that d = 0and f = 0. Since f = max{e − h , 0} = 0, we get A A A i i i=1 that max{e − h , 0} = 0forsome j ∈{1,...,l}.Notethat, fromLemmas4.2,4.3,4.4 and j j their proofs (and also from their dual versions - see Remark 4.5), we know how the bound quiver algebra changes after applying a given admissible operation. We have three cases to consider: (1) Assume that the algebra C is of type (p ,p ) with p ,p ≥ 2. Then e = 1and j 1 2 1 2 j h = 0. The bound quiver algebra A = kQ/I is given by the quiver Q which can be visualized as follows: ◦ ◦ part B ◦ ε ω 2 1 p −1 γ α 1 2 1 ◦◦ ◦ ··· ◦ ◦ part D α p ◦◦ ◦ ··· ◦ ◦ ◦ ◦ β β β β 1 2 p −1 p 2 2 ξ σ part A ◦◦ ◦ ◦ ◦ ϕ σ σ 3 2 ◦ ◦ where I the ideal in the path algebra kQ of Q over k generated by the elements ε α , α γ , 1 1 2 1 ε γ − ε γ , β ξ, α ω, δα , σ β , σ σ ϕ, elements from parts A, B, D of Q,and 1 1 2 2 2 p −1 p 1 p −1 2 3 1 1 2 elements from C . Therefore, π (Q, I ) is not trivial and so A is not simply connected. More i 1 940 P. Malicki precisely, it follows from Proposition 4.10 that π (Q, I ) = Z π (A) π (B) π (D) 1 1 1 1 π (C ). 1 i (2) Assume that the algebra C is of type (p ,p ) with p = 1, p ≥ 2. Then e = 2, j 1 2 1 2 j h = 0or h = 1 and we have two subcases to consider. If e = 2and h = 0, then j j j j the bound quiver algebra A = kQ/I is given by the quiver Q which can be visualized as follows: ◦◦ β β 1 p ◦ ◦ ··· ◦ ◦ ◦ β β 2 p −1 ω ε ◦ part A ◦ part B ◦ ◦ ◦ C σ ϕ where I the ideal in the path algebra kQ of Q over k generated by the elements γβ , β ω, σ β , σ σ ϕ, elements from parts A, B of Q, and elements from C .There- p −1 1 1 2 3 i fore, π (Q, I ) is not trivial and so A is not simply connected. More precisely, it follows from Proposition 4.10 that π (Q, I ) = Z π (A) π (B) π (C ).If e = 2and 1 1 1 1 i j h = 1, then the bound quiver algebra A = kQ/J is given by the quiver Q which can be visualized as in the previous subcase with the ideal J of kQ generated by the elements γα − aγβ ...β β , β ω, σ β , σ σ ϕ, elements from parts A, B of Q, and elements 1 p 2 1 p −1 1 1 2 3 2 2 from C ,where a ∈ k\{0}. Note that in general, we can apply to a stable tube T of one of the following admissible operations: (ad 1), (ad 4), (ad 5) or their dual versions (with an infinite sectional path belonging to T ). Since h = 1, we applied (in the above visualization) an admissible operation from the set S ={(ad 1), (ad 4), (ad 5)} to the algebra C with pivot the regular C -module corresponding to the indecomposable representation of the form kk k k ··· k k 1 1 lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]), where a ∈ k \{0}.More precisely, if we apply (ad 1) with parameter t = 0, then we have to remove the arrow ε and the part B. Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/J , where the ideal J of kQ is generated by the elements of J \{γα − aγβ ...β β }∪{γα } and π (Q, J ) is not trivial. Again, it follows from 1 p 2 1 1 1 Proposition 4.10 that π (Q, J ) = Z π (A) π (B) π (C ). If we apply an admissible 1 1 1 1 i ∗ ∗ ∗ ∗ operation from the set S ={(ad 1 ), (ad 4 ), (ad 5 )} to the algebra C , the proof follows by dual arguments. (3) Assume that the algebra C is of type (p ,p ) with p = p = 1. Then e = 3, j 1 2 1 2 j h = 0, h = 1or h = 2 and we have three subcases to consider. Note that in this case all j j j stable tubes in have ranks equal to 1. Now, if e = 3and h = 0, then j = l = 1and C j j the path algebra A = kQ is given by the Kronecker quiver Q:◦◦ . Therefore, β Simply Connected Algebras 941 π (Q) = Z and so A is not simply connected. If e = 3and h = 1, then the bound quiver 1 j j algebra A = kQ/J is given by the quiver Q which can be visualized as follows: α γ ◦◦ ◦ part A with the ideal J in the path algebra kQ of Q over k generated by the element γα − aγβ and elements from part A (the rest of Q), where a ∈ k \{0}.Since h = 1, we applied (in the above visualization) an admissible operation from the set S to the algebra C with pivot the regular C -module corresponding to the indecomposable representation of the formkk lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]), where a ∈ k \{0}. More precisely, if we apply (ad 1) with parameter t = 0, then we have to remove the arrow ε and the part A. Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/J , where the ideal J of kQ is generated by the elements of J \{γα − aγβ}∪{γα} and π (Q, J ) is not trivial. Again, it follows from Proposition 4.10 that π (Q, J ) = Z π (A). Moreover, if we apply an admissible 1 1 operation from the set S to the algebra C , the proof follows by dual arguments. If e = 3 j j and h = 2, then the bound quiver algebra A = kQ/L is given by the quiver Q which can be visualized as follows: λ δ ε part A ◦◦ ◦ ◦ part B with the ideal L of kQ generated by the elements γα − aγβ, αδ − bβδ, γαδ and elements from parts A, B of Q,where a, b ∈ k \{0} and a = b.Since h = 2, we applied (in the above visualization) one admissible operation from the set S and one from the set S to the algebra C with pivots the regular C -modules corresponding to the indecomposable j j representations of the form 1 1 kk andkk lying in different stable tubes of rank 1 in (see [42, XIII.2.4(c)]), where a, b ∈ k \{0} and a = b. More precisely, if we apply (ad 1) (respectively, (ad 1 )) with parameter t = 0, then we have to remove the arrow ε and the part B (respectively, the arrow λ and the part A). Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/L , where the ideal L of kQ is generated by the elements of L \{γα − aγβ, αδ − bβδ}∪{γα, αδ} and π (Q, L ) is not trivial. Again, it follows from Proposition 4.10 that π (Q, L ) = Z π (A) π (B). In a similar way, one can show all the cases of applying 1 1 1 two admissible operations from the set S ∪ S to any two stable tubes of rank one from the Auslander–Reiten quiver of the Kronecker algebra. We now show the sufficiency. We know from Theorem 3.5 that there is a unique full (l) (l) (l) convex subcategory A = A × ··· × A of A which is a tubular coextension of the product C × ... × C = C of a family C ,...,C of tame concealed algebras (see 1 m 1 m (l) remarks immediately after Theorem 5.1) such that A is obtained from A by a sequence of admissible operations of types (ad 1)–(ad 5). We shall prove our claim by induction on (l) the number of admissible operations leading from A 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 (which occurs in the definitions of the above admissible operations) is finite for each 1 ≤ i ≤ t. Indeed, if this is not the case, then the family of generalized multicoils obtained after applying such admissible operation is not sincere, and then it is not 942 P. Malicki (l) separating. Let C = A ,...,A = A ,A ,...,A = A be an admissible sequence for 0 p p+1 n A and assume that A = A. In this case A is tame quasitilted algebra and our claim follows from [3, Theorem A]. Let k ≥ p, A = A and assume that A is simply connected. k+1 k Moreover, let v be the extension point of A and X ∈ ind A be the pivot of the admissible k k operation. Since H (A) = 0, the vertex v is separating, by [44, Lemma 3.2]. Note that if the admissible operation leading from A to A is of type (ad 1), (ad 2) or (ad 3), then A is k k a connected algebra. If X is an (ad 1)-pivot, then A = A [X] or A = (A × D)[X ⊕ Y ], where rad P = X k k A v or rad P = X ⊕ Y respectively, D is the full t × t lower triangular matrix algebra over A v k for some t ≥ 1, and Y is the unique indecomposable projective-injective D-module (see definition of (ad 1)). Applying Lemma 4.7 or Lemma 4.8 respectively, we conclude that A is simply connected. If X is an (ad 2)-pivot or (ad 3)-pivot, then A = A [X], where rad P = X. Applying k A v Lemma 4.7, we conclude that A is simply connected. Let X be an (ad 4)-pivot and Y = Y → Y → ··· → Y with t ≥ 1 be a finite sectional 1 2 t path in . Then, for r = 0, A = A [X ⊕ Y ],andfor r ≥ 1, A k ⎡ ⎤ A 00 ... 00 Yk 0 ... 00 ⎢ ⎥ ⎢ ⎥ Y k k ... 00 ⎢ ⎥ A = ⎢ ⎥ . . . . . . . . . . . ⎢ ⎥ . . . . . ⎢ ⎥ ⎣ ⎦ Y k k ... k 0 X ⊕ Y kk... kk with r +2 columns and rows (see definition of (ad 4)). We note that Y is directing A-module for each 1 ≤ i ≤ t. Indeed, since H (A) = 0, we get d = 0, and so A is not connected. A k Now, if r = 0, then A = A [X ⊕ Y ] and rad P = X ⊕ Y . Then it follows from k A v Lemma 4.7 that A is simply connected. If r ≥ 1, then observe that the modified algebra A of A can be obtained by applying (0) (1) (0) r + 1 one-point extensions in the following way: A = A [U ], A = A [U ], k 01 11 k k k (2) (1) (r−1) (r−2) (r) (r−1) A = A [U ], ..., A = A [U ] and finally A = A = A [X ⊕ U ], 21 r−1,1 r1 k k k k k k (j −1) where U = Y , U is a projective A -module such that rad (j −1) U = U ,for 01 j1 j1 j −1,1 (j −1) r ≥ 1, 1 ≤ j ≤ r. We denote by v the extension vertex of A ,for 1 ≤ j ≤ r. Since the vertex v of Q (0) is separating and rad (0) P = U , applying Lemma 4.7, we conclude 1 v 01 A A k k (0) that the algebra A is simply connected. Further, since the vertex v of Q is separating, (1) (0) (1) rad (1) P = U ,and A is simply connected, it follows from Lemma 4.7 that A is v 11 k k (r−1) simply connected. Iterating a finite number of times the same arguments, we get that A is simply connected. Finally, since the vertex v of Q is separating and rad P = X ⊕ U , A A v r1 applying again Lemma 4.7, we get that A is simply connected. Let X be an (ad 5)-pivot. 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 admis- Simply Connected Algebras 943 sible operation (ad 4), we conclude that the required statement follows from the above considerations. This finishes the proof of Theorem 1.1. 6 Proof of Theorem 1.2 Let A be a generalized multicoil algebra. Then A is a connected generalized multicoil enlargement of a concealed canonical algebra C.Let C = C × C ×· · ·× C × C ×···× 1 2 l l+1 C be a decomposition of C into product of connected algebras such that C ,C ,...,C m 1 2 l are of type (p ,p ) and C ,C ,...,C are of type (p ,...,p ) with t ≥ 3. Since C , 1 2 l+1 l+2 m 1 t i i ∈{1,...,m}, are simply connected, we get l = 0. Moreover, by the assumption, the sec- ∗ ∗ tional paths occuring in the definitions of the operations (ad 4), (fad 4), (ad 4 ), (fad 4 ) come from two components of two connected algebras. Applying Theorems 3.3 and 3.5 we (l) (l) (l) infer that there exists a unique factor algebra A = A ×· · ·× A of A which is a tubu- lar coextension of a concealed canonical algebra C = C × ... × C , and a unique factor 1 m (r) (r) (r) algebra A = A ×· · ·× A of A which is a tubular extension of a concealed canoni- (l) (r) cal algebra C = C × ... × C .Since A and A are quasitilted algebras (of canonical 1 m types), the equivalence (ii) and (iv) follows from [26, Theorem 1]. Clearly, (v) implies (i). We now show that (i) implies (iii). Since all algebras C ,...,C are of type (p ,...,p ) 1 m 1 t with t ≥ 3(l = 0), we get f = 0. Assume to the contrary that H (A) = 0. Then, by Theorem 5.1, d + f = 0. Therefore, d = 0 and it follows from the proof of Lemma 4.4 A A A (and its dual version) that A is not simply connected, a contradiction with (i). We show that (iii) implies (iv). Assume to the contrary that there exists i ∈{1,...,m} (l) (r) 1 1 such that H (A ) = 0or H (A ) = 0. Without loss of generality, we may assume i i (l) (l) that H (A ) = 0for some i ∈{1,...,m}.Since A is a tubular coextension of a con- i i (l) cealed canonical algebra C ,wehavethat A is a generalized multicoil enlargement of C , i i (l) and so, by Theorem 5.1, dim H (A ) = d (l) + f (l). Moreover, by our assumption on A A i i C ,wehave f (l) = 0. Hence d (l) = 0. Since d ≥ d (l), we get a contradiction with i A A A A i i i (iii). (l) In order to finish the proof we will show that (iv) implies (v). Assume that H (A ) = (r) 0and H (A ) = 0, for any i ∈{1,...,m}. We know that for each i ∈{1,...,m}, (l) (r) A (respectively, A ) is a tubular coextension (respectively, extension) of a concealed i i canonical algebra C of type (p ,...,p ), t ≥ 3and H (C ) = 0, by [20, Theorem 2.4]. i 1 t i (l) (r) Then H (B) = 0 for every full convex subcategory B of A (respectively, A ). Therefore, i i (l) (r) it follows from [44, Theorem 4.1] that A and A are strongly simply connected, for any i i i ∈{1,...,m}. Moreover, by our assumption on A, the Auslander–Reiten quiver does not contain exceptional configurations of modules. Applying now Theorems 3.3 and 3.6 we infer that A is strongly simply connected. 7 Examples We start this section with the following remark. 944 P. Malicki Remark 7.1 We can apply Theorem 1.1 to important classes of algebras. For example, to the cycle-finite algebras with separating families of almost cyclic coherent Auslander–Reiten components. Indeed, it is known (see [8]) that every cycle-finite algebra is tame. Example 7.2 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 1 5 ν α 18 2 4 6 3 8 7 19 9 10 δ 11 15 12 13 14 17 20 21 22 and I the ideal in the path algebra kQ of Q over k generated by the elements αβ, γδ, ηε, κλ, ξκλ, να.Then A is a generalized multicoil enlargement of a concealed canonical algebra C,where C is the hereditary algebra of Euclidean type D given by the vertices 1, 2,..., 7. Indeed, consider the dimension-vectors 00 00 00 000 100 00 00 010 000 000 a = 010 , a = 010 , a = 010 , a = , a = 00 , a = 00 . 1 2 3 4 5 6 00 010 10 00 00 0 011 0000 1 0 We apply (ad 1 ) with pivot the simple regular C-module with vector a , and with parameter t = 0. The modified algebra B is given by the quiver with vertices 1, 2,..., 8 bound by αβ = 0. Now, we apply (ad 1 ) with pivot the indecomposable B -module with vector a , 1 2 and with parameter t = 2. The modified algebra B is given by the quiver with vertices 1, 2,..., 11 bound by αβ = 0. Next, we apply (ad 1 ) with pivot the indecomposable B - module with vector a , and with parameter t = 3. The modified algebra B is given by 3 3 the quiver with vertices 1, 2,..., 15 bound by αβ = 0, γδ = 0. In the next step we apply (ad 1 ) with pivot the indecomposable B -module with vector a , and with parameter t = 0. 3 4 The modified algebra B is given by the quiver with vertices 1, 2,..., 16 bound by αβ = 0, γδ = 0, ηε = 0. Next, we apply the admissible operation (ad 5) in two steps. The first step: we apply the operation (fad 3) with pivot the indecomposable B -module with vector a , and with parameters t = 3, s = 2. The modified algebra B is given by the quiver with 5 5 vertices 1, 2,..., 17 bound by αβ = 0, γδ = 0, ηε = 0, κλ = 0. The second step: we apply the operation (ad 4) with pivot the indecomposable B -module with vector a ,and 5 6 Simply Connected Algebras 945 with a finite sectional path consisting of the indecomposable B -modules with dimension- vectors 00 00 00 00 000 000 000 000 000 000 000 000 00 → 00 → 00 → 00 00 00 01 00 0011 0011 0011 0001 1 0 0 0 and with parameter r = 4. The modified algebra is equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) bound quiver algebra kQ /I ,where Q is a full subquiver of Q givenbythe vertices (l) (l) (l) (r) 1, 2,..., 16 and I = kQ ∩ I is the ideal in kQ . The right quasitilted algebra A of (r) (r) (r) A is the convex subcategory of A being the bound quiver algebra kQ /I ,where Q is (r) (r) a full subquiver of Q givenbythevertices 1, 2,..., 7, 14, 15,..., 18 and I = kQ ∩ I (r) (l) (r) is the ideal in kQ . Note that A and A are tame. It follows from Theorems 3.3, 3.5(iii) and the above construction that the Auslander– Reiten quiver ofthetamealgebra A = kQ/I admits a separating family of almost cyclic coherent components. Further, π (Q, I ) = Z and hence A is not simply connected. Moreover, by Theorem 5.1, the first Hochschild cohomology space H (A) k (d = (l) (r) 1,f = 0). We also note that, since A and A are tame tilted algebras of Euclidean 1 (l) 1 (r) (l) type D such that H (A ) = 0and H (A ) = 0, it follows from [5, Theorem] that A (r) and A are simply connected (and even strongly simply connected from [5, Corollary]). We refer to [33, Example 4.1] (see also [35, Example 9.13]) for a more extensive example of the tame algebra with a separating family of almost cyclic coherent components which is not simply connected. Finally, we also mention that A is a generalized multicoil algebra such that contains the exceptional configurations of modules. Example 7.3 We borrow the following example from [31]. Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 1 520 23 ϕ ϕ 1 3 2 4 6 25 22 ϕ ϕ 2 4 β ξ ζ 1 1 3 8 719 26 21 32 24 ξ ζ 2 2 γ ξ 9 10 18 27 28 μ ω δ 11 15 1 33 η π λ κ 12 13 14 17 29 30 31 and I the ideal in the path algebra kQ of Q over k generated by the elements αβ, γδ, ηε, κλ, ϕ ψ, ϕ ψ, ξ ω , ζ ϕ , ζ ϕ , ζ ξ ξ ξ − ζ ψ, π ξ , π ω − π ω , μκλ, νξ .Then A is 3 4 3 1 1 1 1 2 2 3 2 1 1 1 2 1 1 2 2 1 a generalized multicoil enlargement of a concealed canonical algebra C = C × C ,where 1 2 C is the hereditary algebra of Euclidean type D given by the vertices 1, 2,..., 7, and C 1 6 2 is the hereditary algebra of Euclidean type D given by the vertices 20, 21,..., 24. Indeed, 5 946 P. Malicki we apply (ad 1 )to C with pivot the simple regular C -module S , and with parameter 1 1 4 t = 0. The modified algebra B is given by the quiver with the vertices 1, 2,..., 8 bound by αβ = 0. Next, we apply (ad 1 )to B with pivot the indecomposable injective B - 1 1 module I , and with parameter t = 2. The modified algebra B is given by the quiver with 8 2 the vertices 1, 2,..., 11 bound by αβ = 0. Now, we apply (ad 1 )to B with pivot the indecomposable B -module τ S , and with parameter t = 3. The modified algebra B 2 B 10 3 is given by the quiver with the vertices 1, 2,..., 15 bound by αβ = 0, γδ = 0. Next, we apply (ad 1 )to B with pivot the simple B -module S , and with parameter t = 0. The 3 3 14 modified algebra B is given by the quiver with the vertices 1, 2,..., 16 bound by αβ = 0, γδ = 0, ηε = 0. In the next step we apply (ad 1 )to C with pivot the simple regular C -module S , and with parameter t = 3. The modified algebra B is given by the quiver 2 22 5 with the vertices 20, 21,..., 28 bound by ϕ ψ = 0, ϕ ψ = 0. Now, we apply (ad 1 )to 3 4 B with pivot the simple B -module S , and with parameter t = 2. The modified algebra 5 5 B is given by the quiver with the vertices 20, 21,..., 31 bound by ϕ ψ = 0, ϕ ψ = 0, 6 3 4 ξ ω = 0. Next, we apply (ad 2) to B with pivot the indecomposable injective B -module 3 1 6 6 I , and with parameter t = 3. The modified algebra B is given by the quiver with the 25 7 vertices 20, 21,..., 32 bound by ϕ ψ = 0, ϕ ψ = 0, ξ ω = 0, ζ ϕ = 0, ζ ϕ = 0, 3 4 3 1 1 1 1 2 ζ ξ ξ ξ = ζ ψ. Now, we apply (ad 3) to B with pivot the indecomposable B -module 2 3 2 1 1 7 7 τ S , and with parameter t = 2. The modified algebra B is given by the quiver with the B 30 8 vertices 20, 21,..., 33 bound by ϕ ψ = 0, ϕ ψ = 0, ξ ω = 0, ζ ϕ = 0, ζ ϕ = 0, 3 4 3 1 1 1 1 2 ζ ξ ξ ξ = ζ ψ, π ξ = 0, π ω = π ω . Finally, we apply (ad 5) to B × B in two 2 3 2 1 1 1 2 1 1 2 2 4 8 steps. The first step: we apply (fad 3) with pivot the indecomposable B -module τ S , 4 B 14 and with parameters t = 3, s = 2. The modified algebra B is given by the quiver with the vertices 1, 2,..., 17 bound by αβ = 0, γδ = 0, ηε = 0, κλ = 0. The second step: we apply (ad 4) with pivot the simple B -module S , and with the finite sectional path 8 26 I → τ S → I → S consisting of the indecomposable B -modules, and with 16 B 15 14 17 9 parameters t = 4, r = 1. The modified algebra is equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) (l) (l) (l) product A = A × A ,where A = kQ /I is the branch coextension of the tame 1 2 1 1 1 (l) concealed algebra C , Q is a full subquiver of Q givenbythe vertices1, 2,..., 16 and (l) (l) (l) (l) (l) (l) I = kQ ∩I is the ideal in kQ , A = kQ /I is the branch coextension of the tame 1 1 1 2 2 2 (l) concealed algebra C , Q is a full subquiver of Q given by the vertices 20, 21,..., 31 and (l) (l) (l) (r) I = kQ ∩ I is the ideal in kQ . The right quasitilted algebra A of A is the convex 2 2 2 (r) (r) (r) (r) (r) (r) (r) subcategory of A being the product A = A ×A ,where A = C , A = kQ /I 1 2 1 2 2 2 (r) is the branch extension of the tame concealed algebra C , Q is a full subquiver of Q given (r) (r) by the vertices 14, 15,..., 24, 26, 27, 28, 30, 31, 32, 33 and I = kQ ∩ I is the ideal in 2 2 (r) (l) (l) (r) (r) kQ . Note that A , A , A and A are tame. 2 1 2 1 2 It follows from Theorems 3.3, 3.5(iii) and the above construction that A is tame and admits a separating family of almost cyclic coherent components. Moreover, by Theo- rem 5.1, the first Hochschild cohomology space H (A) = 0(d = 0,f = 0). Then, a A A direct application of Theorem 1.1 shows that the algebra A is simply connected. In fact, it follows from [31, Theorem 1.2] that A is strongly simply connected. We also note that, since (l) (l) (r) (r) (l) A , A , A and A are tame tilted algebras of Euclidean type D such that H (A ) = 0, 1 2 1 2 1 (l) (r) (r) (l) (l) 1 1 1 H (A ) = 0, H (A ) = 0and H (A ) = 0 it follows from [5, Theorem] that A , A , 2 1 2 1 2 (r) (r) A and A are simply connected (and even strongly simply connected from [5, Corol- 1 2 lary]). Finally, we mention that C , C are simply connected, A is a generalized multicoil 1 2 Simply Connected Algebras 947 algebra, does not contain exceptional configurations of modules, and so this example illustrates also Theorem 1.2. Example 7.4 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 13 14 15 ψ ζ β μ λ α ε κ ν 16 1 2 3 11 12 67 8 ξ ω 5 4 10 9 17 η π and I the ideal in the path algebra kQ of Q over k generated by the elements aγβαλ − δλ, γε, bπ ωνμ − πηξ, ζμ, ϕψκ,where a, b ∈ k \{0}.Then A is a generalized multicoil enlargement of a concealed canonical algebra C = C × C ,where C is the hereditary 1 2 1 algebra of Euclidean type A given by the vertices 1, 2,..., 5, and C is the hereditary 4 2 algebra of Euclidean type A givenbythe vertices6, 7,..., 10. Indeed, we apply (ad 1 ) to C with pivot the simple regular C -module S , and with parameter t = 2. The modified 1 1 3 algebra B is given by the quiver with the vertices 1, 2,..., 5, 11, 12, 13 bound by γε = 0. Next, we apply (ad 4) to B × C with pivot the simple regular C -module S and with 1 2 2 7 the finite sectional path I → S consisting of the indecomposable B -modules, and 12 13 1 with parameters t = 2, r = 1. The modified algebra B is given by the quiver with the vertices 1, 2,..., 15 bound by γε = 0, ζμ = 0, ϕψκ = 0. Now, we apply (ad 1 ) with parameter t = 0 to the algebra B with pivot the regular C -module corresponding to the 2 1 indecomposable representation of the form a 1 kk k 1 1 k k lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]). The modified algebra B is C 3 given by the quiver with the vertices 1, 2,..., 16 bound by γε = 0, ζμ = 0, ϕψκ = 0, aγβαλ = δλ,where a ∈ k \{0}. Finally, we apply (ad 1) with parameter t = 0 to the algebra B with pivot the regular C -module corresponding to the indecomposable 3 2 representation of the form kk k 1 1 k k lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]). The modified algebra is then equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) (l) (l) (l) product A = A × A ,where A = kQ /I is the branch coextension of C , 1 2 1 1 1 (l) (l) Q is a full subquiver of Q given by the vertices 1, 2,..., 5, 11, 12, 13, 16 and I = 1 1 (l) (l) (l) (r) kQ ∩ I is the ideal in kQ , A = C . The right quasitilted algebra A of A is the 1 1 2 (r) (r) (r) (r) (r) convex subcategory of A being the product A = A × A ,where A = C , A = 1 2 1 2 (r) (r) (r) kQ /I is the branch extension of C , Q is a full subquiver of Q givenbythe vertices 2 2 2 948 P. Malicki (r) (r) (r) (l) 6, 7,..., 10, 12, 13, 14, 15, 17 and I = kQ ∩ I is the ideal in kQ . Note that A , 2 2 2 1 (l) (r) (r) A , A and A are tame. 2 1 2 It follows from Theorems 3.3, 3.5(iii) and the above construction that A is tame and admits a separating family of almost cyclic coherent components. Moreover, we have h = 1, e = 1, h = 1, e = 1, f = 0, f = 0, f = f + f = 0, and 1 1 2 2 C C A C C 1 2 1 2 d = 0. Therefore, by Theorem 5.1, the first Hochschild cohomology space H (A) = 0. Then, a direct application of Theorem 1.1 shows that the algebra A is simply connected. We (l) (r) (l) (r) 1 1 ∼ ∼ note that, by [19, Proposition 1.6], H (A ) k, H (A ) k.Since A and A are = = 2 1 1 2 (l) (r) 1 1 generalized multicoil algebras, we get by Theorem 5.1 that H (A ) = 0, H (A ) = 0. 1 2 (r) (l) (l) (r) We also mention that A = C , A = C are not simply connected, A , A are simply 1 2 1 2 1 2 connected, by [3, Theorem A], and so A is not strongly simply connected. Moreover, by the above construction we know that A is a generalized multicoil algebra, such that does not contain exceptional configurations of modules. Therefore, this example shows that simple connectedness assumption imposed on the considered concealed canonical algebras is essential for the validity of Theorem 1.2. We end this section with an example of a wild generalized multicoil algebra, illustrating Theorem 1.2. Example 7.5 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 20 21 23 26 ϕ ϕ 3 4 1 θ θ η η 2 2 1 1 22 32 31 25 γ γ ϕ ϕ 2 1 2 1 β β β β 4 3 2 1 02 3 4 8 34 24 1 27 α α 4 1 ρ α α ε 1 3 2 1 19 5 6 7 18 35 33 28 ρ ξ σ δ ε ψ 2 1 2 1 1 2 17 16 12910 11 36 29 ξ δ 2 2 13 14 15 38 37 30 σ σ ω 3 4 1 and I the ideal in the path algebra kQ of Q over k generated by the elements α α α α + 1 2 3 4 β β β β + γ γ , α δ , α σ , ξ α , ε α , ε δ − ε δ , α ρ , ξ ρ − ξ ρ , ν γ , γ θ , 1 2 3 4 1 2 1 1 2 1 1 4 1 2 1 1 2 2 3 1 1 1 2 2 1 2 1 2 ν θ − ν θ , ϕ ψ , ϕ ψ , η ϕ , η ϕ , ψ κ , η κ − η ψ κ , ω κ , ω σ σ σ .Then A is a 1 2 2 1 1 1 4 1 1 2 1 3 2 1 2 2 1 1 1 2 2 1 4 3 2 generalized multicoil algebra. Indeed, A is a generalized multicoil enlargement of a canon- ical algebra C = C × C ,where C is the tubular canonical algebra of type (2, 4, 4) given 1 2 1 by the vertices 0, 1,..., 8 bound by α α α α + β β β β + γ γ = 0, and C is the 1 2 3 4 1 2 3 4 1 2 2 canonical algebra of Euclidean type D given by the vertices 23, 24,..., 27. It is known that admits an infinite family T , λ ∈ P (k), of pairwise orthogonal stable tubes, hav- C 1 ing a stable tube, say T , of rank 4 with the mouth formed by the modules S = τ S , 5 C 6 1 1 S = τ S , S = τ E, E = τ S ,where E is the unique indecomposable C -module 6 C 7 7 C C 5 1 1 1 1 with the dimension vector dimE = 11111 , and a unique stable tube, say T ,ofrank2with 000 Simply Connected Algebras 949 the mouth formed by the modules S = τ F , F = τ S ,where F is the unique indecom- 1 C C 1 1 1 posable C -module with the dimension vector dimF = (see [41, (3.7)]). Moreover, admits an infinite family T , μ ∈ P (k), of pairwise orthogonal stable tubes, hav- C μ 1 ing a stable tube, say T , of rank 2 with the mouth formed by the modules S = τ G, 25 C G = τ S ,where G is the unique indecomposable C -module with the dimension vec- C 25 2 tor dimG = 1 . We have the following sequence of the modified algebras. First, we apply (ad 1 )to C with pivot the simple regular C -module S , and with parameter t = 2. The 1 1 7 modified algebra B is given by the quiver with the vertices 0, 1,..., 11 bound by α δ = 0. 1 1 1 Next, we apply (ad 1 )to B with pivot the simple B -module S , and with parameter 1 1 6 t = 3. The modified algebra B is given by the quiver with the vertices 0, 1,..., 15 bound by α δ = 0, α σ = 0. Now, we apply (ad 1) to B with pivot the simple B -module S , 1 1 2 1 2 2 and with parameter t = 1. The modified algebra B is given by the quiver with the ver- tices 0, 1,..., 17 bound by α δ = 0, α σ = 0, ξ α = 0. Next, we apply (ad 3) to B 1 1 2 1 1 4 3 with pivot the indecomposable B -module τ I , and with parameter t = 2. The modi- 3 B 10 fied algebra B is given by the quiver with the vertices 0, 1,..., 18 bound by α δ = 0, 4 1 1 α σ = 0, ξ α = 0, ε α = 0, ε δ = ε δ . Further, we apply (ad 2 )to B with pivot the 2 1 1 4 1 2 1 1 2 2 4 indecomposable projective B -module P , and with parameter t = 1. The modified alge- 4 16 bra B is given by the quiver with the vertices 0, 1,..., 19 bound by α δ = 0, α σ = 0, 5 1 1 2 1 ξ α = 0, ε α = 0, ε δ = ε δ , α ρ = 0, ξ ρ = ξ ρ . Now, we apply (ad 1) to B 1 4 1 2 1 1 2 2 3 1 1 1 2 2 5 with pivot the simple regular B -module S , and with parameter t = 1. The modified alge- 5 1 bra B is given by the quiver with the vertices 0, 1,..., 21 bound by α δ = 0, α σ = 0, 6 1 1 2 1 ξ α = 0, ε α = 0, ε δ = ε δ , α ρ = 0, ξ ρ = ξ ρ , ν γ = 0. Next, we apply 1 4 1 2 1 1 2 2 3 1 1 1 2 2 1 2 (ad 2 )to B with pivot the indecomposable projective B -module P , and with parameter 6 6 21 t = 1. The modified algebra B is given by the quiver with the vertices 0, 1,..., 22 bound by α δ = 0, α σ = 0, ξ α = 0, ε α = 0, ε δ = ε δ , α ρ = 0, ξ ρ = ξ ρ , 1 1 2 1 1 4 1 2 1 1 2 2 3 1 1 1 2 2 ν γ = 0, γ θ = 0, ν θ = ν θ .Now,weapply(ad1 )to C with pivot the simple regular 1 2 1 2 1 2 2 1 2 C -module S , and with parameter t = 2. The modified algebra B is given by the quiver 2 25 8 with the vertices 23, 24,..., 30 bound by ϕ ψ = 0, ϕ ψ = 0. Next, we apply (ad 1) to 1 1 4 1 B with pivot the indecomposable B -module τ S , and with parameter t = 1. The mod- 8 8 B 29 ified algebra B is given by the quiver with the vertices 23, 24,..., 32 bound by ϕ ψ = 0, 9 1 1 ϕ ψ = 0, η ϕ = 0, η ϕ = 0. Now, we apply (ad 2 )to B with pivot the indecomposable 4 1 1 2 1 3 9 projective B -module P , and with parameter t = 1. The modified algebra B is given 9 31 10 by the quiver with the vertices 23, 24,..., 33 bound by ϕ ψ = 0, ϕ ψ = 0, η ϕ = 0, 1 1 4 1 1 2 η ϕ = 0, ψ κ = 0, η κ = η ψ κ . Next, we apply (ad 4) to B × B with pivot the 1 3 2 1 2 2 1 1 1 7 10 simple B -module S , and with the finite sectional path I → I → S consisting of 10 32 13 14 15 the indecomposable B -modules, and with parameters t = 3, r = 4. The modified algebra is then equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) (l) (l) (l) (l) product A = A × A ,where A = kQ /I is the branch coextension of C , Q is 1 2 1 1 1 1 (l) (l) a full subquiver of Q givenbythe vertices0, 1,..., 15, 17, 19, 21, 22 and I = kQ ∩ I 1 1 (l) (l) (l) (l) (l) is the ideal in kQ , A = kQ /I is the branch coextension of C , Q is a full 1 2 2 2 2 (l) (l) subquiver of Q given by the vertices 23, 24,..., 30, 33 and I = kQ ∩ I is the ideal 2 2 (l) (r) in kQ . The right quasitilted algebra A of A is the convex subcategory of A being the (r) (r) (r) (r) (r) (r) (r) product A = A × A ,where A = kQ /I is the branch extension of C , Q is 1 2 1 1 1 1 (r) a full subquiver of Q givenbythevertices0, 1,..., 8, 10, 11, 16, 17, 18, 20, 21 and I = (r) (r) (r) (r) (r) (r) kQ ∩ I is the ideal in kQ , A = kQ /I is the branch extension of C , Q is a 1 1 2 2 2 2 950 P. Malicki full subquiver of Q given by the vertices 13, 14, 15, 23, 24,..., 27, 31, 32, 34, 35,..., 38 (r) (r) (r) (l) (r) and I = kQ ∩ I is the ideal in kQ . Then, A and A are the quasitilted algebras 2 2 2 1 1 (l) (r) of wild types (4, 4, 13), (4, 4, 9), respectively. Moreover, A and A are tame. 2 2 It follows from [7, Corollary 1.4] that C is simply connected. Moreover, C is also sim- 1 2 ply connected. By the above construction we know that A is a generalized multicoil algebra obtained from C , C and does not contain exceptional configurations of modules. Fur- 1 2 A ther, by Theorem 5.1, the first Hochschild cohomology space H (A) = 0(d = 0, f = 0) A A (l) (l) (r) (r) 1 1 1 1 and H (A ) = 0, H (A ) = 0, H (A ) = 0, H (A ) = 0. Then, a direct application 1 2 1 2 (l) (l) (r) (r) of Theorem 1.2 shows that the algebras A , A , A , A and A are simply connected. 1 2 1 2 Acknowledgements I thank an anonymous referee for useful comments. Data Availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study. 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. 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Abstract

We study the simple connectedness of the class of finite-dimensional algebras over an alge- braically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes. Keywords Simply connected algebra · Hochschild cohomology · Auslander–Reiten quiver · Tame algebra · Generalized multicoil algebra Mathematics Subject Classiﬁcation (2010) Primary 16G70 · Secondary 16G20 1 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 /I,where Q = (Q ,Q ) is the ordinary quiver of A with the set of vertices Q and the set of A 0 1 0 arrows Q and I is an admissible ideal in the path algebra kQ of Q . If the quiver Q has 1 A A A 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 the Jacobson radical of mod A, and by ∞ i rad the intersection of all powers rad , i ≥ 1, of rad . Moreover, we denote by A A A A the Auslander–Reiten quiver of A, and by τ and τ the Auslander–Reiten translations Dedicated to Claus Michael Ringel on the occasion of his 75th birthday Presented by: Christof Geiss Piotr Malicki pmalicki@mat.umk.pl Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, ´ Poland 924 P. Malicki D Tr and Tr D, respectively. We will not distinguish between a module in ind A and the vertex of corresponding to it. Following [45], a family C of components is said to be generalized standard if rad (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 -orbits in C are τ -periodic (see [45, (2.3)]). We refer to [37] for the structure and A A homological properties of arbitrary generalized standard Auslander–Reiten components of algebras. Following Assem and Skowronski ´ [7], a triangular algebra A is called simply connected if, for any presentation A kQ /I of A as a bound quiver algebra, the fundamental group π (Q ,I) of (Q ,I) is trivial (see Section 2). The importance of these algebras follows 1 A A from the fact that often we may reduce (using techniques of Galois coverings) the study of the module category of an algebra to that for the corresponding simply connected alge- bras. Let us note that to prove that an algebra is simply connected seems to be a difficult problem, because one has to check that various fundamental groups are trivial. Therefore, it is worth looking for a simpler characterization of simple connectedness. In [44, Problem 1] Skowronski ´ has asked, whether it is true that a tame triangular algebra A is simply con- nected if and only if the first Hochschild cohomology space H (A) of A vanishes. This equivalence is true for representation-finite algebras [3, Proposition 3.7] (see also [12]for the general case), for tilted algebras (see [5] for the tame case and [25] for the general case), for quasitilted algebras (see [3] for the tame case and [26] for the general case), for piecewise hereditary algebras of type any quiver [25], and for weakly shod algebras [4]. 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 fam- ily of tubes has been introduced by Ringel in [40, 41] 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, Skowronski ´ and Tomei ´ n[10](seealso[33]). A family C = (C ) of components i i∈I of the Auslander–Reiten quiver of an algebra A is called separating in mod A if the A A A components of split into three disjoint families P , C = C and Q such that: (S1) C is a sincere generalized standard family of components; A A A A A A (S2) Hom (Q , P ) = 0, Hom (Q , C ) = 0, Hom (C , P ) = 0; A A A A A (S3) any homomorphism from P to Q in mod A factors through the additive category A A add(C ) of C . A A A A A A Then we say that C separates P from Q and write = P ∪ C ∪ Q .We A A A note that then P and Q are uniquely determined by C (see [10, (2.1)] or [41, (3.1)]). Moreover, C is called sincere if any simple A-module occurs as a composition factor of a A A module in C . We note that if A is an algebra of finite representation type that C = is A A trivially a unique separating component of , with P and Q being empty. Frequently, we may recover A completely from the shape and categorical behavior of the separating family C of components of . For example, the tilted algebras [24, 41], or more generally double tilted algebras [39](the strict shod algebras in the sense of [15]), are determined by their (separating) connecting components. Further, it was proved in [28] that the class of algebras with a separating family of stable tubes coincides with the class of concealed canonical algebras. This was extended in [29] 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 [33] 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 is called almost cyclic if all but A Simply Connected Algebras 925 finitely many modules in lie on oriented cycles contained entirely in . Moreover, a component of 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 → X →· · · → X → X → X → ··· (that is, X = τ X for 1 2 i i+1 i+2 i A i+2 any i ≥ 1) in ; (C2) For each injective module I in there is an infinite sectional path ··· → Y → Y → Y →···→ Y → Y = I (that is, Y = τ Y for j +2 j +1 j 2 1 j +2 A j any j ≥ 1) in . We are now in position to formulate the first main result of the paper, which answers positively the above mentioned question of Skowronski ´ [44, Problem 1] for tame algebras with separating almost cyclic coherent Auslander–Reiten components. Theorem 1.1 Let A be a tame algebra with a separating family of almost cyclic coherent components in .Then A is simply connected if and only if H (A) = 0. It has been proved in [33, Theorem A] that the Auslander–Reiten quiver of an alge- bra A admits a separating family C of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a finite product of concealed canonical algebras C ,...,C by an iterated application of admissible algebra operations of types 1 m (ad 1)–(ad 5) and their duals. These algebras are called generalized multicoil algebras (see Section 3 for details). Note that for such an algebra A,wehavethat A is triangular, gl. dim A ≤ 3, and pd M ≤ 2orid M ≤ 2 for any module M in ind A (see [33, Corollary A A A B and Theorem E]). Moreover, let = P ∪ C ∪ Q be the induced decomposi- tion of . Then, by [33, Theorem C], there are uniquely determined quotient algebras (l) (l) (r) (r) (l) (r) A = A ×···× A and A = A ×···× A of A which are the quasitilted algebras m m 1 1 (l) (r) A A A A of canonical type such that P = P and Q = Q . Let A be a generalized multicoil algebra obtained from a concealed canonical algebra C = C ×· · ·× C and C = A ,A ,...,A = A be an admissible sequence for A (see 1 m 0 1 n Section 3). In order to formulate the next result 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 A , i ∈{0,...,n − 1}, then we will say that contains an exceptional configuration of i+1 modules. The following theorem is the second main result of the paper. Theorem 1.2 Let A be a generalized multicoil algebra obtained from a family C ,...,C 1 m of simply connected concealed canonical algebras. Assume moreover that does not (l) contain exceptional configurations of modules. Then there are quotient algebras A = (l) (l) (r) (r) (r) A × ··· × A and A = A ×· · ·× A of A such that the following statements are m m 1 1 equivalent: (i) A is simply connected. (l) (r) (ii) A and A are simply connected, for any i ∈{1,...,m}. i i (iii) H (A) = 0. (l) (r) 1 1 (iv) H (A ) = 0 and H (A ) = 0, for any i ∈{1,...,m}. i i (v) A is strongly simply connected. 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 926 P. Malicki 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 and 6 are devoted to the proofs of Theorem 1.1 and Theorem 1.2, respectively. The aim of the final Section 7 is to present examples illustrating the main results of the paper. For basic background on the representation theory of algebras we refer to the books [6, 41–43], for more information on simply connected algebras we refer to the survey article [2], and for more details on algebras with separating families of Auslander–Reiten components and their representation theory to the survey article [35]. 2 Preliminaries 2.1 Let A be an algebra and A = kQ /I beapresentationof A as a bound quiver algebra. Then the algebra A = kQ /I can equivalently be considered as a k-linear category, of which the object class A is the set of points of Q , and the set of morphisms A(x, y) from 0 A x to y is the quotient of the k-vector space kQ (x, y) of all formal linear combinations of paths in Q from x to y by the subspace I (x, y) = kQ (x, y) ∩ I (see [11]). A full A A 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 we denote by S the corresponding simple A-module, A v and by P (respectively, I ) the projective cover (respectively, the injective envelope) of S . v v v 2.2 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 B 0 B[M]= Mk with the usual addition and multiplication of matrices. The quiver of B[M] contains Q as a convex subquiver and there is an additional (extension) point which is a source. B[M]- modules are usually identified with triples (V , X, ϕ),where V is a k-vector space, X a B-module and ϕ : V → Hom (M, X) a k-linear map. A B[M]-linear map (V ,X,ϕ) → (V ,X ,ϕ ) is then identified with a pair (f, g),where f : V → V is k-linear, g : X → X is B-linear and ϕ f = Hom (M, g)ϕ. One defines dually the one-point coextension [M]B of B by M (see [41]). 2.3 Tameness and Wildness Let A be an algebra and K[x] the polynomial algebra in one variable x. Following [17], the algebra A is said to be tame if, for any positive integer d, there exists a finite number of K[x]− A-bimodules M ,1 ≤ i ≤ n , which are finitely i d generated and free as left K[x]-modules, and all but a finite number of isoclasses of inde- composable A-modules of dimension d are of the form K[x]/(x − λ) ⊗ M for some K[x] i λ ∈ K and some i ∈{1,...,n }. Recall that, following [17], the algebra A is wild if there is a kx, y-A-bimodule M, free of finite rank as left kx, y-module, and the functor −⊗ M : mod kx, y→ mod A preserves the indecomposability of modules and sends kx,y nonisomorphic modules to nonisomorphic modules. From Drozd’s Tame and Wild Theo- rem [17] the class of algebras may be divided into two disjoint classes. One class consists of the tame algebras and the second class is formed by the wild algebras whose representa- tion theory comprises the representation theories of all finite dimensional algebras over k. Simply Connected Algebras 927 Hence, a classification of the finite dimensional modules is only feasible for tame algebras. It has been shown by Crawley-Boevey [16] that, if A is a tame algebra, then, for any positive integer d ≥ 1, all but finitely many isomorphism classes of indecomposable A-modules of dimension d are invariant on the action of τ , and hence, by a result due to Hoshino [23], lie in stable tubes of rank one in . 2.4 Hochschild Cohomology of Algebras Let A be an algebra. Denote by C A the • i i i i Hochschild complex C = (C ,d ) defined as follows: C = 0, d = 0for i< 0, i∈Z 0 i ⊗i ⊗i C = A , C = Hom (A ,A) for i> 0, where A denotes the i-fold tensor product A A k 0 0 over k of A with itself, d : A → Hom (A, A) with (d x)(a) = ax − xa for x, a ∈ A, i i i+1 d : C → C with i j (d f )(a ⊗···⊗a ) = a f(a ⊗· · ·⊗a )+ (−1) f(a ⊗· · ·⊗a a ⊗· · ·⊗a ) 1 i+1 1 2 i+1 1 j j +1 i+1 j =1 i+1 +(−1) f(a ⊗· · ·⊗ a )a 1 i i+1 i i i • for f ∈ C and a ,a ,...,a ∈ A.Then H (A) = H (C A) is called 1 2 i+1 the i-th Hochschild cohomology space of A (see [14, Chapter IX]). Recall that the first Hochschild cohomology space H (A) of an algebra A is isomorphic to the space Der(A, A)/ Der (A, A) of outer derivations of A,where Der(A, A) ={δ ∈ Hom (A, A) | δ(ab) = aδ(b) + δ(a)b, for a, b ∈ A} is the space of k-linear derivations of A and Der (A, A) is the subspace {δ ∈ Hom (A, A) | δ (a) = ax − xa, for a ∈ A} of x k x inner derivations of A. 2.5 Concealed Canonical Algebras An important role in our considerations will be played by certain tilts of canonical algebras introduced by Ringel [41]. Let p ,p ,...,p be a 1 2 t sequence of positive integers with t ≥ 2, 1 ≤ p ≤ p ≤ ... ≤ p ,and p ≥ 2if t ≥ 3. 1 2 t 1 Denote by (p ,...,p ) the quiver of the form 1 t α 1p −1 12 1 ◦ ◦ ··· ◦ ◦ α 1p 11 1 α α α α 2p −1 2p 21 22 2 2 ◦◦ ◦ ··· ◦ ◦ ◦ α α t1 tp ◦ ◦ ··· ◦ ◦ α α t2 tp −1 For t ≥ 3, consider a (t + 1)-tuple of pairwise different elements of P (k) = k ∪{∞}, normalized such that λ =∞, λ = 0, λ = 1, and the admissible ideal I(λ ,λ ,...,λ ) 1 2 3 1 2 t in the path algebra k(p ,...,p ) of (p ,...,p ) generated by the elements 1 t 1 t α ...α α + α ...α α + λ α ...α α , 3 ≤ i ≤ t. ip i2 i1 2p 22 21 i 1p 12 11 i 2 1 Then the bound quiver algebra (p, λ) = k(p ,...,p )/I (λ ,λ ,...,λ ) is said to be 1 t 1 2 t the canonical algebra of type p = (p ,...,p ). Moreover, for t = 2, the path algebra 1 t (p) = k(p ,p ) is said to be the canonical algebra of type p = (p ,p ). It has been 1 2 1 2 provedin[41, Theorem 3.7] that if is a canonical algebra of type (p ,...,p ) then 1 t = P ∪ T ∪ Q for a P (k)-family T of stable tubes of tubular type (p ,...,p ), 1 1 t separating P from Q . Following [27], a connected algebra C is called a concealed canonical algebra of type (p ,...,p ) if C is the endomorphism algebra End (T ),for 1 t 928 P. Malicki some canonical algebra of type (p ,...,p ) and a tilting -module T whose indecom- 1 t posable direct summands belong to P . Then the images of modules from T via the functor Hom (T , −) form a separating family T of stable tubes of , and in particular C C C we have a decomposition = P ∪ T ∪ Q . It has been proved by Lenzing and de la Pena ˜ [28, Theorem 1.1] that the class of (connected) concealed canonical algebras coin- cides with the class of all connected algebras with a separating family of stable tubes. It is also known that the class of concealed canonical algebras of type (p ,p ) coincides with 1 2 the class of hereditary algebras of Euclidean types A , m ≥ 1(see[22]). Recall also that the canonical algebras of types (2, 2, 2, 2), (3, 3, 3), (2, 4, 4) and (2, 3, 6) are called the tubular canonical algebras, and an algebra which is tilting-cotilting equivalent to a tubular canonical algebra is called a tubular algebra (see [18, 21, 41]). 2.6 Simple Connectedness Let (Q, I ) be a connected bound quiver. A relation = λ w ∈ I (x, y) is minimal if m ≥ 2 and, for any nonempty proper subset J ⊂ i i i=1 −1 {1,...,m},wehave λ w ∈ / I (x, y). We denote by α the formal inverse of an j j j ∈J ε ε ε 1 2 t arrow α ∈ Q .A walk in Q from x to y is a formal composition α α ...α (where 1 2 α ∈ Q and ε ∈{−1, 1} for all i) with source x and target y.Wedenoteby e the triv- i 1 i x ial path at x.Let ∼ be the homotopy relation on (Q, I ), that is, the smallest equivalence relation on the set of all walks in Q such that: −1 −1 (a) If α : x → y is an arrow, then α α ∼ e and αα ∼ e . y x (b) If = λ w is a minimal relation, then w ∼ w for all i, j. i i i j i=1 (c) If u ∼ v,then wuw ∼ wvw whenever these compositions make sense. Let x ∈ Q be arbitrary. The set π (Q, I, x) of equivalence classes u of closed walks u 0 1 starting and ending at u has a group structure defined by the operation u · v = uv .Since Q is connected, π (Q,I,x) does not depend on the choice of x.Wedenoteitby π (Q, I ) and 1 1 call it the fundamental group of (Q, I ). Let A = kQ /I be a presentation of a triangular algebra A as a bound quiver algebra. The fundamental group π (Q ,I) depends essentially on I, so is not an invariant of A. 1 A A triangular algebra A is called simply connected if, for any presentation A kQ /I of A as a bound quiver algebra, the fundamental group π (Q ,I) of (Q ,I) is trivial 1 A A [7]. Example 2.7 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form λ δ 51 2 4 and I the ideal in the path algebra kQ of Q over k generated by the elements γβ, δα − aδβ, αλ,where a ∈ k \{0}.Then π (Q, I ) is trivial. Moreover, the triangular algebra A is simply connected. Indeed, any choice of a basis of rad /rad will lead to at least one minimal relation with target 1 and source i ∈{3, 4} or with target 5 and source 2. Remark 2.8 It is known, for example, that the following important classes of algebras are simply connected: the iterated tilted algebras of Dynkin type (see [1, Proposition 3.5]), the iterated tilted algebras of Euclidean types D , E , n ≥ 4, p = 6, 7, 8, the tubular algebras n p (see [7, Corollary 1.4]), and the pg-critical algebras (see [38, Corollary 3.3]). Simply Connected Algebras 929 3 Almost Cyclic Coherent Auslander–Reiten components 3.1 Generalized Multicoil Algebras It has been proved in [32, Theorem A] that a con- nected component of an Auslander–Reiten quiver 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 [33, Section 3]. Given a generalized standard component of , and an indecomposable module X in ,the support S(X) of the functor Hom (X, −)| is the k-linear category defined as follows [9]. Let H denote the full subcategory of consisting of the indecomposable modules M in such that Hom (X, M) = 0, and I denote the ideal of H consisting of A X X the morphisms f : M → N (with M, N in H ) such that Hom (X, f ) = 0. We define X A S(X) to be the quotient category H /I . Following the above convention, we usually X X identify the k-linear category S(X) with its quiver. Recall that a module X in mod A is called a brick if End (X) k. Let A be an algebra and be a family of generalized standard infinite components of . 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 (X, −)| . These admissible operations yield in each case a modified algebra A such that the mod- ified translation quiver is a family of generalized standard infinite components in the Auslander–Reiten quiver of A (see [32, Section 2] or [35, Section 4] for the figures illustrating the modified translation quiver ). (ad 1) Assume S(X) consists of an infinite sectional path starting at X: X = X → X → X →· · · 0 1 2 Let t ≥ 1 be a positive integer, D be the full t ×t lower triangular matrix algebra, and Y , ..., Y denote the indecomposable injective D-modules with Y = Y the unique indecomposable t 1 projective-injective D-module. We set A = (A × D)[X ⊕ Y ]. In this case, is obtained by inserting in the rectangle consisting of the modules Z = k, X ⊕ Y , for i ≥ 0, ij i j 1 ≤ j ≤ t,and X = (k, X , 1) for i ≥ 0. If t = 0weset A = A[X] and the rectangle reduces to the sectional path consisting of the modules X , 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: Y ← ··· ← Y ← Y ← X = X → X → X →· · · t 2 1 0 1 2 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 Z = k, X ⊕ Y , ij i j for i ≥ 1, 1 ≤ j ≤ t,and X = (k, X , 1) for i ≥ 0. (ad 3) Assume S(X) is the mesh-category of two parallel sectional paths: Y → Y →···→ Y 1 2 t ↑↑ ↑ X = X → X →···→ X → X →· · · 0 1 t −1 t with the upper sectional path finite and t ≥ 2. In particular, X is necessarily injective. t −1 Moreover, we consider the translation quiver of obtained by deleting the arrows Y → −1 τ Y . We assume that the union of connected components of containing the modules i−1 −1 τ Y ,2 ≤ i ≤ t, is a finite translation quiver. Then is a disjoint union of and i−1 a cofinite full translation subquiver , containing the pivot X.Weset A = A[X].Inthis 930 P. Malicki case, is obtained from by inserting the rectangle consisting of the modules Z = ij k, X ⊕ Y , for i ≥ 1, 1 ≤ j ≤ i,and X = (k, X , 1) for i ≥ 0. i j i 1 i (ad 4) Suppose that S(X) consists of an infinite sectional path, starting at X X = X → X → X →· · · and Y = Y → Y →· · · → Y 0 1 2 1 2 t with t ≥ 1, is a finite sectional path in .Let r be a positive integer. Moreover, we −1 consider the translation quiver of obtained by deleting the arrows Y → τ Y .We i i−1 −1 assume that the union of connected components of containing the vertices τ Y , i−1 2 ≤ i ≤ t, is a finite translation quiver. Then is a disjoint union of and a cofinite full translation subquiver , containing the pivot X.For r = 0weset A = A[X ⊕ Y ]. In this case, is obtained from by inserting the rectangle consisting of the modules Z = k, X ⊕ Y , for i ≥ 0, 1 ≤ j ≤ t,and X = (k, X , 1) for i ≥ 0. ij i j i 1 i For r ≥ 1, let G be the full r × r lower triangular matrix algebra, U , U , 1,t +1 2,t +1 ..., U denote the indecomposable projective G-modules, U , U , ..., U r,t +1 r,t +1 r,t +2 r,t +r denote the indecomposable injective G-modules, with U the unique indecomposable r,t +1 projective-injective G-module. We define the matrix algebra ⎡ ⎤ A 00 ... 00 ⎢ ⎥ Yk 0 ... 00 ⎢ ⎥ ⎢ ⎥ Y k k ... 00 ⎢ ⎥ A = ⎢ . . . . . ⎥ . . . . . . ⎢ ⎥ . . . . . ⎢ ⎥ ⎣ ⎦ Y k k ... k 0 X ⊕ Y kk... kk with r + 2 columns and rows. In this case, is obtained from by inserting the following modules (k, Y , 1) for s = 1, 1 ≤ l ≤ t, ⎨ l 1 for i ≥ 0 and U = (k, U , 1) for 2 ≤ s ≤ r,1 ≤ l< t + s, Z = k, X ⊕ U , sl s,l−1 ij i rj 1 1 ≤ j ≤ t + r, (k, 0, 0) for 2 ≤ s ≤ r, l = t + s, and X = (k, X , 1) for i ≥ 0. In the above formulas U is treated as a module over the i sl algebra A = A [U ],where A = A and U = Y (in other words A is an algebra s s−1 s−1,1 0 01 s consisting of matrices obtained from the matrices belonging to A by choosing the first s +1 rows and columns). We note that the quiver Q of A is obtained from the quiver of the double one-point extension A[X][Y ] by adding a path of length r + 1 with source at the extension vertex of A[X] and sink at the extension vertex of A[Y ]. 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 → X → X →··· (in the definitions of (ad 1), (ad 2), 0 1 2 (ad 3), (ad 4)) by the finite sectional paths of the form X → X → X → ··· → X . 0 1 2 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 is injective (see the figures for (fad 1)–(fad 4) in [32, Section 2] or [35, 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 Simply Connected Algebras 931 (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). Moreover, together with each of the admissible operations (ad 1)–(ad 5), we consider ∗ ∗ its dual, denoted by (ad 1 )–(ad 5 ). These dual operations are also called admissible. Fol- lowing [32], a connected translation quiver is said to be a generalized multicoil if can be obtained from a finite family T , T ,..., T of stable tubes by an iterated application of 1 2 s ∗ ∗ ∗ ∗ admissible operations (ad 1), (ad 1 ), (ad 2), (ad 2 ), (ad 3), (ad 3 ), (ad 4), (ad 4 ), (ad 5) or (ad 5 ). If s = 1, such a translation quiver is said to be a generalized coil.The admis- ∗ ∗ sible operations of types (ad 1)–(ad 3), (ad 1 )–(ad 3 ) have been introduced in [8–10], and the admissible operations (ad 4) and (ad 4 )for r = 0in[30]. Finally, let C be a (not necessarily connected) concealed canonical algebra and T a separating family of stable tubes of . Following [33] we say that an algebra A is a gen- eralized multicoil enlargement of C using modules from T if there exists a sequence of algebras C = A ,A ,...,A = A such that A is obtained from A by an admissi- 0 1 n i+1 i ∗ ∗ ble operation of one of the types (ad 1)–(ad 5), (ad 1 )–(ad 5 ) performed either on stable A A i i tubes of T , or on generalized multicoils obtained from stable tubes of T by means of operations done so far. The sequence C = A ,A ,...,A = A is then called an admissi- 0 1 n ble sequence for A. Observe that this definition extends the concept of a coil enlargement of a concealed canonical algebra introduced in [10]. 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 [41]. 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. Proposition 3.2 [33, Proposition 3.7] Let C be a concealed canonical algebra, T asep- arating family of stable tubes of , and A a generalized multicoil enlargement of C using C A modules from T .Then admits a generalized standard family C of generalized mul- ticoils obtained from the family T of stable tubes by a sequence of admissible operations corresponding to the admissible operations leading from C to A. The following theorem, proved in [33, Theorem A], will be crucial for our further considerations. Theorem 3.3 Let A be an algebra. The following statements are equivalent: (i) admits a separating family of almost cyclic coherent components. (ii) A is a generalized multicoil enlargement of a concealed canonical algebra C. Remark 3.4 The concealed canonical algebra C is called the core of A and the number m of connected summands of C is a numerical invariant of A. We note that m can be arbitrary large, even if A is connected. Let us also note that the class of algebras with generalized standard almost cyclic coherent Auslander–Reiten components is large (see [34, Proposition 2.9] and the following comments). We note that the class of tubular extensions (respectively, tubular coextensions) of con- cealed canonical algebras coincides with the class of algebras having a separating family of ray tubes (respectively, coray tubes) in their Auslander–Reiten quiver (see [27, 29]). Moreover, these algebras are quasitilted algebras of canonical type. 932 P. Malicki We recall also the following theorem on the structure of the module category of an algebra with a separating family of almost cyclic coherent Auslander–Reiten components provedin[33, Theorems C and F]. Theorem 3.5 Let A be an algebra with a separating family C of almost cyclic coherent A A A components in , and = P ∪ C ∪ Q the associated decomposition of .Then A A A the following statements hold. (l) (l) (l) (i) There is a unique full convex subcategory A = A ×· · · × A of A which is a tubular coextension of a concealed canonical algebra C = C × ... × C such 1 m (l) (l) (l) (l) A A A A that (l) = P ∪ T ∪ Q for a separating family T of coray tubes (l) A A (l) in , P = P , and A is obtained from A by a sequence of admissible (l) (l) operations of types (ad 1)–(ad 5) using modules from T . (r) (r) (r) (ii) There is a unique full convex subcategory A = A ×· · · × A of A which is a tubular extension of a concealed canonical algebra C = C × ... × C such 1 m (r) (r) (r) (r) A A A A that (r) = P ∪ T ∪ Q for a separating family T of ray tubes (r) A A (r) in , Q = Q , and A is obtained from A by a sequence of admissible (r) (r) ∗ ∗ A operations of types (ad 1 )–(ad 5 ) using modules from T . (l) (r) (iii) A is tame if and only if A and A are tame. (l) (r) In the above notation, the algebras A and A are called the left and right quasitilted (l) (r) (l) (r) algebras of A. Moreover, the algebras A and A are tame if and only if A and A are products of tilted algebras of Euclidean type or tubular algebras. Recall that an algebra A is strongly simply connected if every convex subcategory of A is simply connected (see [44]). Clearly, if A is strongly simply connected then A is simply connected. We need the following result proved in [31, Theorem 1.1]. Theorem 3.6 Let A be an algebra with a separating family of almost cyclic coherent components in without exceptional configurations of modules. Then there are quotient (l) (l) (r) (r) (l) (r) algebras A = A ×···× A and A = A ×···× A of A such that the following m m 1 1 statements are equivalent: (i) A is strongly simply connected. (l) (r) (ii) A and A are strongly simply connected, for any i ∈{1,...,m}. i i 4 Preliminary Results 4.1 Branch Extensions and Coextensions Let A be an algebra and A kQ /I be a pre- = A sentation of A as a bound quiver algebra. For a given vertex v in Q , we denote by v (respectively, by v) the set of all arrows of the quiver Q starting at v (respectively, ter- minating at v). Let now K be a branch at a vertex v ∈ Q and E ∈ mod A. Recall that the branch extension A[E, K] by the branch K [41, (4.4)] is constructed in the follow- ing way: to the one-point extension A[E] with extension vertex w (that is, rad P = E) we add the branch K by identifying the vertices v and w.If E ,...,E ∈ mod A and 1 n K ,...,K is a set of branches, then the branch extension A[E ,K ] is defined induc- 1 n i i i=1 n−1 tively as: A[E ,K ] = (A[E ,K ] )[E ,K ]. The concept of branch coextension is i i i i n n i=1 i=1 defined dually. Simply Connected Algebras 933 Lemma 4.2 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C = C ×· · · × C . Moreover, let C = A ,...,A = A ,A ,...,A = A be an 1 m 0 p p+1 n admissible sequence for A, j ≥ p, X ∈ ind A be an (ad 2) or (ad 3)-pivot, and A be j j +1 the modified algebra of A .If v is the corresponding extension point then there is a unique (l) (r) vertex u ∈ A \ A that satisfies: (i) Each α ∈ v is the starting point of a nonzero path ω ∈ A(v, u). → → (ii) There are at least two different arrows in v . Moreover, if α, β ∈ v , and α = β, then ω − ω ∈ I . α β (l) Proof We know from [33, Section 4] that A is a unique maximal convex branch coex- (l) (l) (l) (l) tension of C = C ×· · ·× C inside A,thatis, A = B ×· · ·× B ,where B is a 1 m m unique maximal convex branch coextension of C inside A, i ∈{1,...,m}. More precisely, (l) t B = [K ,E ]C ,where K ,...,K are branches, i ∈{1,...,m}. Then there exists j j i 1 t i j =1 (l) s ∈{1,...,m} such that u ∈ B and A = A [X].If X is an (ad 2)-pivot (respectively, s j +1 j (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 ,...,Y in the sup- 1 t port of Hom (X, −) restricted to the generalized multicoil containing X - see definition of (ad 3)). The operations done after must not affect the support of Hom (X, −) restricted to the generalized multicoil containing X. Note that in general, in the sequence of earlier admissible operations, there can be an operation of type (ad 5) which contains an opera- tion (fad 4) which gives rise to the pivot X of (ad 2) (respectively, to the pivot X of (ad 3)) but from Lemma [33, 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 = A [X],and u, u ,...,u (where X = I , Y = j +1 j 1 t u i I for i ∈{1,...,t } - see definition of (ad 2)) be the points in the quiver Q of A u A j i j corresponding to the new indecomposable injective A -modules obtained after performing ∗ ∗ the above admissible operation (ad 1 ) or the operation (fad 1 ). Then u, u ,...,u ∈ 1 t (l) A .Since X = rad P , there must be a nonzero path from v to each vertex w which is a predecessor of u. Hence, each α ∈ v is the starting arrow of a nonzero path from v to u, and there are at least two arrows in v , namely: one from v to u and one from v to a point in Supp X ,where X is the immediate successor of X on the infinite sectional path 1 1 in S(X) (see definition of (ad 2)). Moreover, since P (u) = X(u) = k, all paths from v to u are congruent modulo I . The bound quiver Q of A is of the form j +1 A j +1 j +1 u Supp X u ··· u 1 t where A (v, u) is one-dimensional. From the proofs of [33, Theorems A and C], we j +1 (l) (r) (r) (l) (l) (r) have u ∈ A \ A , v ∈ A \ A ,and u ,...,u ∈ A ∩ A . 1 t Let now X be an (ad 3)-pivot, A = A [X], and assume that we had r consecutive j +1 j ∗ ∗ admissible operations of types (ad 1 )or(fad1 ), the first of which had X as a pivot, and t 934 P. Malicki these admissible operations built up a branch K in A with points u, u ,...,u in Q ,so j 1 t A that X and Y are the indecomposable injective A -modules corresponding respectively t −1 t j −1 to u and u , and both Y and τ Y are coray modules in the generalized multicoil con- 1 1 1 taining the (ad 3)-pivot X (where X, X ,X ,Y and Y are as in the definition of (ad 3)). t −1 t 1 t (l) Then u, u ∈ A and X is the indecomposable A -module given by: X(w) = 0if w< u , 1 j 1 X(w) = k if u <w,and X(w) = X (w) in any other case. Since X = rad P ,there 1 t −1 v must be a nonzero path from v to each vertex w whichisapredecessorof u, but those which are predecessors of u . Hence, each α ∈ v is the starting arrow of a nonzero path from v to u, and there are at least two arrows in v , namely: one from v to u and one from v to a point in Supp X ,where X is the immediate successor of X on the infinite sectional t t t −1 path in S(X) (see definition of (ad 3)). Moreover, since P (u) = X (u) = k, all paths v t −1 from v to u are congruent modulo I . The bound quiver Q of A is of the form j +1 A j +1 j +1 Supp X uv rest of K where A (v, u) is one-dimensional. Again, from the proofs of [33, Theorems A and C], j +1 (l) (r) (r) (l) we have u ∈ A \ A , v ∈ A \ A , u and the vertices of the branch K belong to (l) (r) A ∩ A . Lemma 4.3 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C = C ×· · · × C . Moreover, let C = A ,...,A = A ,A ,...,A = A be an 1 m 0 p p+1 n admissible sequence for A, j ≥ p, X ∈ ind A be an (ad 1)-pivot, A be the modified j j +1 algebra of A , and v be the corresponding extension point. Then the following statements hold. (l) (r) → (i) If there is a vertex u ∈ A \ A such that each α ∈ v is the starting point of a nonzero path ω ∈ A(v, u), then: (a) The vertex u is unique. (b) There are at least two different arrows in v . (c) If α, β ∈ v , and α = β,then ω − ω ∈ I . α β (ii) If X| = 0 for any i ∈{1,...,m},then X is uniserial. Proof Since X is an (ad 1)-pivot, the support S(X) consists of an infinite sectional path X = X → X → X →··· starting at X.Let t ≥ 1 be a positive integer, D be the full 0 1 2 t × t lower triangular matrix algebra, and Y , ..., Y be the indecomposable injective D- 1 t modules with Y the unique indecomposable projective-injective D-module (see definition of (ad 1)). (l) (i) Again, we know from [33, Section 4] that A is a unique maximal convex branch (l) (l) (l) coextension of C = C × ··· × C inside A,thatis, A = B ×· · ·× B ,where 1 m (l) B is a unique maximal convex branch coextension of C inside A, i ∈{1,...,m}.More (l) t precisely, B = [K ,E ]C ,where K ,...,K are branches, i ∈{1,...,m}. Assume j j i 1 t i j =1 (l) (r) → that there is a vertex u ∈ A \ A such that each α ∈ v is the starting point of a Simply Connected Algebras 935 (l) nonzero path ω ∈ A(v, u). Then there exists s ∈{1,...,m} such that u ∈ B . Moreover, α s A = (A × D)[X ⊕ Y ] and the bound quiver Q | is of the form j +1 j 1 A Supp X j +1 Q (l) ◦ uv v ··· v 1 t >? <= :; ./ ,- *+ () &' $% "# ! where v ,...,v are the points in the quiver Q of A corresponding to the new 1 t A j +1 j +1 (l) indecomposable projective A -modules. Then A is the extension of B at X by j +1 j +1 s the extension branch K consisting of the points v, v ,...,v , that is, we have A = 1 t j +1 (r) → A [X, K].Since u does not belong to A and for any α ∈ v it is the starting point of a nonzero path ω ∈ A(v, u), we get that u is the coextension point of the admissible oper- ∗ ∗ ∗ ∗ ation (ad 2 )or(ad3 ). By [10, Lemma 3.1] the admissible operations (ad 2 )and (ad3 ) ∗ ∗ commute with (ad 1), so we can apply (ad 2 ) after (ad 1) (respectively, (ad 3 ) after (ad 1)). Using now [10, Lemma 3.3] (respectively, [10, Lemma 3.4]), we are able to replace (ad 1) ∗ ∗ ∗ followedby(ad2 ) (respectively, (ad 1) followed by (ad 3 )) by an operation of type (ad 1 ) followed by an operation of type (ad 2) (respectively, (ad 1 ) followed by an operation of type (ad 3)). Therefore, the statements (a), (b) and (c) follow from Lemma 4.2. (ii) A case by case inspection (which admissible operation gives rise to the (ad 1)-pivot X)shows that X is either simple module or the support of X is a linearly ordered quiver of type A . Lemma 4.4 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C = C ×· · · × C . Moreover, let C = A ,...,A = A ,A ,...,A = A be an 1 m 0 p p+1 n admissible sequence for A, j ≥ p, X ∈ ind A be an (ad 4) or (ad 5)-pivot, A be the j j +1 modified algebra of A , and v be the corresponding extension point. If there is a vertex (l) (r) → u ∈ A \ A such that for pairwise different arrows α ,...,α ∈ v , q ≥ 2 there are 1 q paths ω ,...,ω ∈ A(v, u), then for arbitrary f, g ∈{1,...,q},f = g, one of the α α 1 q following cases holds: (i) At least one of ω ,ω is zero path. α α f g (ii) The paths ω ,ω are nonzero and ω − ω ∈ I . α α α α g g f f (l) Proof It follows from [33, Section 4] that A is a unique maximal convex branch coex- (l) (l) (l) (l) tension of C = C ×· · ·× C inside A,thatis, A = B ×· · ·× B ,where B is a 1 m m 1 i unique maximal convex branch coextension of C inside A, i ∈{1,...,m}. More precisely, (l) t B = [K ,E ]C ,where K ,...,K are branches, i ∈{1,...,m}. Assume that there j j i 1 t i i j =1 (l) (r) → is a vertex u ∈ A \ A such that for pairwise different arrows α ,...,α ∈ v , q ≥ 2, 1 q (l) there are paths ω ,...,ω ∈ A(v, u). Then there exists s ∈{1,...,m} such that u ∈ B . α α q s Let X be an (ad 4)-pivot and Y → Y → ··· → Y with t ≥ 1, be a finite sectional path 1 2 t in (as in the definition of (ad 4)). Note that this finite sectional path is the linearly ori- ented quiver of type A and its support algebra (given by the vertices corresponding to the simple composition factors of the modules Y ,Y ,...,Y ) is a tilted algebra of the path 1 2 t 936 P. Malicki algebra D of the linearly oriented quiver of type A .From[41, (4.4)(2)] we know that is a bound quiver algebra given by a branch in x,where x corresponds to the unique projective- injective D-module. Let be a generalized multicoil of obtained by applying the j +1 admissible operation (ad 4), where X is the pivot contained in the generalized multicoil , and Y is the starting vertex of a finite sectional path contained in the generalized multicoil or .So, is obtained from or from the disjoint union of two generalized multi- 1 2 1 coils , by the corresponding translation quiver admissible operations. In general, 1 2 1 and are components of the same connected algebra or two connected algebras. Hence, we get two cases. In the first case X, Y ∈ or X ∈ , Y ∈ and , are two 1 1 1 1 2 1 2 components of the same connected algebra. In the second case X ∈ , Y ∈ and , 1 1 2 1 are two components of two connected algebras. Therefore, the bound quiver Q of 2 A j +1 A in the first case is of the form j +1 u Supp X v d+1 w ··· w w d 1 β β β d 2 1 for r = 0and u Supp X v v d+1 w ··· w w v ··· v d 1 r 2 β β β d 2 1 for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of A [X], w is the extension point of A [Y ], w ,...,w belong to the branch in w generated j j 1 1 d by the support of Y ⊕ ··· ⊕ Y ,and αβ ...β = 0for some h ∈{1,...,d + 1}.Inthe 1 t 1 h second case the bound quiver Q of A is of the form A j +1 j +1 u Supp X v yw ··· w w d 1 β β β β d+1 d 2 1 for r = 0and Supp X u v v yw ··· w w v ··· v d 1 r 2 β β β β d+1 d 2 1 Simply Connected Algebras 937 for r ≥ 1, where the index r is as in the definition of (ad 4), v is the extension point of A [X], w is the extension point of A [Y ], w ,...,w belong to the branch in w generated j j 1 1 d by the support of Y ⊕· · ·⊕ Y , αβ ...β = 0forsome h ∈{1,...,d + 1},and y is 1 t 1 h (l) (l) (r) the coextension point of A such that y ∈ A \ A . More precisely, y ∈ B ,where s ∈{1,...,m} and s = s. Moreover in both cases, we have P (u) = X(u) = k or P (u) = X(u) = 0, and hence all nonzero paths from v to u are congruent modulo I .So, v j +1 A (v, u) is at most one-dimensional. We note that in the first case, the definition of (ad 4) j +1 (see the shape of the bound quiver Q of A ) implies that if the paths ω ,ω ∈ A j +1 α α j +1 f g A (v, u) are nonzero and ω − ω ∈ I, then there is also a zero path ω ∈ A (v, u) j +1 α α α j +1 f g h for some h ∈{1,...,q},h = f = g. Let X be an (ad 5)-pivot and be a generalized multicoil of obtained by apply- j +1 ing this admissible operation with pivot X.Then is obtained from the disjoint union of the finite family of generalized multicoils , ,..., by the corresponding translation 1 2 e quiver admissible operations, 1 ≤ e ≤ l,where l is the number of stable tubes of 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. Remark 4.5 Let A be a generalized multicoil enlargement of a concealed canonical algebra (l) C. We know from Theorems 3.3 and 3.5 that A can be obtained from A by a sequence of (r) admissible operations of types (ad 1)–(ad 5) or A can be obtained from A by a sequence ∗ ∗ of admissible operations of types (ad 1 )–(ad 5 ). We note that all presented above lemmas ∗ ∗ can be formulated and proved for dual operations (ad 1 )–(ad 5 ) in a similar way. 4.6 The Separating Vertex Let A be a triangular algebra. Recall that a vertex v of Q is called separating if the radical of P is a direct sum of pairwise nonisomorphic indecom- posable modules whose supports are contained in different connected components of the subquiver Q(v) of Q obtained by deleting all those vertices u of Q being the source of a A A path with target v (including the trivial path from v to v). We have the following lemma which follows from the proof of [44, Proposition 2.3] (see also [2, Lemma 2.3]). Lemma 4.7 Let A be a triangular algebra and assume that A = B[X],where v is the extension vertex and X = rad P .If B is simply connected and v is separating, then A is A v simply connected. Let D be the same as in the definition of (ad 1), that is, the full t × t lower triangular matrix algebra. Denote by Y , ..., Y the indecomposable injective D-modules with Y = Y 1 t 1 the unique indecomposable projective-injective D-module. Lemma 4.8 Let A be a triangular algebra and assume that A = (B × D)[X ⊕ Y ],where v is the extension vertex and X ⊕ Y = rad P .If B is simply connected and v is separating, A v then A is simply connected. Proof Since the module P is a sink in the full subcategory of ind A consisting of projec- tives, the vertex v is a source in Q . Moreover, A = (B × D)[X ⊕ Y ],where X is the indecomposable direct summand of rad P that belongs to mod B and Y is a directing A v 938 P. Malicki module (that is, an indecomposable module which does not lie on a cycle in ind A) such that rad P = X ⊕ Y . Therefore, the proof follows from the proof of [44, Proposition 2.3] (see A v also the proof of Lemma 2.3 in [2]). 4.9 The Pointed Bound Quiver In order to carry out the construction of the free product of two fundamental groups of bound quivers, and in analogy with algebraic topology where pointed spaces are considered, one can define a pointed bound quiver (Q,I,x),thatis,a bound quiver (Q, I ) together with a distinguished vertex x (see [13, Section 3]). Given two pointed bound quivers Q = (Q ,I ,x ) and Q = (Q ,I ,x ), we can assume, without loss of generality, that Q ∩ Q = Q ∩ Q =∅. We define the quiver Q = Q Q 0 0 1 1 as follows: Q is Q ∪ Q in which we identify x and x toasinglenew vertex x,and 0 0 Q = Q ∪Q . Then, Q and Q are identified to two full convex subquivers of Q,sowalks 1 1 on Q or Q can be considered as walks on Q. Thus, I and I generate two-sided ideals of kQ which we denote again by I and I .Wedefine I to be the ideal I +I of kQ. It follows from this definition that the minimal relations of I are precisely the minimal relations of I together with the minimal relations of I give the minimal relations needed to determine the homotopy relation on (Q, I ). Moreover, we can consider an element w ∈ π (Q ,I ,x ) as an element w ∈ π (Q,I,x) (we denote by w the homotopy class of a walk w). Conversely, any (reduced) walk w in Q has a decomposition w = w w w w ...w w ,where w and 1 1 2 2 n n i w are walks in Q and Q for i ∈{1,...,n}, respectively. Moreover, this decomposition is unique, up to reduced walk, and compatible with the homotopy relations involved. This leads us to the following proposition. Proposition 4.10 [13, Proposition 3.1] With the notations above we have: (i) (Q,I,x) is the coproduct of (Q ,I ,x ) and (Q ,I ,x ) in the category of pointed bound quivers. (i) π (Q, I, x) π (Q ,I ,x ) π (Q ,I ,x ). 1 1 1 5 Proof of Theorem 1.1 The aim of this section is to prove Theorem 1.1 and recall the relevant facts. We know from Theorem 3.3 that the Auslander–Reiten quiver of A admits a separat- ing family of almost cyclic coherent components if and only if A is a generalized multicoil enlargement of a concealed canonical algebra C.Let C = C × C ×· · ·× C × C ×···× 1 2 l l+1 C be a decomposition of C into product of connected algebras such that C ,C ,...,C m 1 2 l are of type (p ,p ) and C ,C ,...,C are of type (p ,...,p ) with t ≥ 3. Following 1 2 l+1 l+2 m 1 t [36], by h we denote the number of all stable tubes of rank one from with 1 ≤ i ≤ l, i C used in the whole process of creating A from C,and h = 0, if l + 1 ≤ i ≤ m. Moreover, let 0if C is of type (p ,...,p ) with t ≥ 3 i 1 t 1if C is of type (p ,p ) with p ,p ≥ 2 i 1 2 1 2 e = 2if C is of type (p ,p ) with p = 1, p ≥ 2 i 1 2 1 2 3if C is of type (p ,p ) with p = p = 1, i 1 2 1 2 for i ∈{1,...,m}.Wedefinealso f = max{e − h , 0},for i ∈{1,...,m} and set C i i m l f = f = f . Note that we can apply the operations (ad 4), (fad 4), (ad 4 ), A C C i=1 i i=1 i (fad 4 ) 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 Simply Connected Algebras 939 connected algebras. By d we denote the number of all operations (ad 4), (fad 4), (ad 4 )or (fad 4 ) which are of the first type, used in the whole process of creating A from C. The Hochschild cohomology of a connected generalized multicoil algebra A has been described in [36, Theorem 1.1] using the numerical invariants of A (f , d and the others), A A depending on the types of admissible operations (ad 1)–(ad 5) and their duals, leading from a product C of concealed canonical algebras to A. Here, we will only need information about the first Hochschild cohomology of A, namely from [36, Theorem 1.1(iii)] we have: Theorem 5.1 Let A be a connected generalized multicoil algebra. Then dim H (A) = d + f . A A We are now able to complete the proof of Theorem 1.1. Since A is tame, we may restrict to the generalized multicoil enlargements of tame concealed algebras. Namely, we have the following consequence of Theorem 3.3 and [33, Theorem F]: A is tame and admits a separating family of almost cyclic coherent components if and only if A is a tame generalized multicoil enlargement of a finite family C ,...,C of tame concealed algebras (concealed canonical algebras of Euclidean type). 1 m We first show the necessity. Suppose that A is simply connected. We must show that the 1 1 first Hochschild cohomology H (A) of A vanishes. Assume to the contrary that H (A) = 0. Then by Theorem 5.1, d + f = 0. If d = 0, then it follows from the proof of A A A Lemma 4.4 (and its dual version) that A is not simply connected, a contradiction. Therefore, we may assume that d = 0and f = 0. Since f = max{e − h , 0} = 0, we get A A A i i i=1 that max{e − h , 0} = 0forsome j ∈{1,...,l}.Notethat, fromLemmas4.2,4.3,4.4 and j j their proofs (and also from their dual versions - see Remark 4.5), we know how the bound quiver algebra changes after applying a given admissible operation. We have three cases to consider: (1) Assume that the algebra C is of type (p ,p ) with p ,p ≥ 2. Then e = 1and j 1 2 1 2 j h = 0. The bound quiver algebra A = kQ/I is given by the quiver Q which can be visualized as follows: ◦ ◦ part B ◦ ε ω 2 1 p −1 γ α 1 2 1 ◦◦ ◦ ··· ◦ ◦ part D α p ◦◦ ◦ ··· ◦ ◦ ◦ ◦ β β β β 1 2 p −1 p 2 2 ξ σ part A ◦◦ ◦ ◦ ◦ ϕ σ σ 3 2 ◦ ◦ where I the ideal in the path algebra kQ of Q over k generated by the elements ε α , α γ , 1 1 2 1 ε γ − ε γ , β ξ, α ω, δα , σ β , σ σ ϕ, elements from parts A, B, D of Q,and 1 1 2 2 2 p −1 p 1 p −1 2 3 1 1 2 elements from C . Therefore, π (Q, I ) is not trivial and so A is not simply connected. More i 1 940 P. Malicki precisely, it follows from Proposition 4.10 that π (Q, I ) = Z π (A) π (B) π (D) 1 1 1 1 π (C ). 1 i (2) Assume that the algebra C is of type (p ,p ) with p = 1, p ≥ 2. Then e = 2, j 1 2 1 2 j h = 0or h = 1 and we have two subcases to consider. If e = 2and h = 0, then j j j j the bound quiver algebra A = kQ/I is given by the quiver Q which can be visualized as follows: ◦◦ β β 1 p ◦ ◦ ··· ◦ ◦ ◦ β β 2 p −1 ω ε ◦ part A ◦ part B ◦ ◦ ◦ C σ ϕ where I the ideal in the path algebra kQ of Q over k generated by the elements γβ , β ω, σ β , σ σ ϕ, elements from parts A, B of Q, and elements from C .There- p −1 1 1 2 3 i fore, π (Q, I ) is not trivial and so A is not simply connected. More precisely, it follows from Proposition 4.10 that π (Q, I ) = Z π (A) π (B) π (C ).If e = 2and 1 1 1 1 i j h = 1, then the bound quiver algebra A = kQ/J is given by the quiver Q which can be visualized as in the previous subcase with the ideal J of kQ generated by the elements γα − aγβ ...β β , β ω, σ β , σ σ ϕ, elements from parts A, B of Q, and elements 1 p 2 1 p −1 1 1 2 3 2 2 from C ,where a ∈ k\{0}. Note that in general, we can apply to a stable tube T of one of the following admissible operations: (ad 1), (ad 4), (ad 5) or their dual versions (with an infinite sectional path belonging to T ). Since h = 1, we applied (in the above visualization) an admissible operation from the set S ={(ad 1), (ad 4), (ad 5)} to the algebra C with pivot the regular C -module corresponding to the indecomposable representation of the form kk k k ··· k k 1 1 lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]), where a ∈ k \{0}.More precisely, if we apply (ad 1) with parameter t = 0, then we have to remove the arrow ε and the part B. Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/J , where the ideal J of kQ is generated by the elements of J \{γα − aγβ ...β β }∪{γα } and π (Q, J ) is not trivial. Again, it follows from 1 p 2 1 1 1 Proposition 4.10 that π (Q, J ) = Z π (A) π (B) π (C ). If we apply an admissible 1 1 1 1 i ∗ ∗ ∗ ∗ operation from the set S ={(ad 1 ), (ad 4 ), (ad 5 )} to the algebra C , the proof follows by dual arguments. (3) Assume that the algebra C is of type (p ,p ) with p = p = 1. Then e = 3, j 1 2 1 2 j h = 0, h = 1or h = 2 and we have three subcases to consider. Note that in this case all j j j stable tubes in have ranks equal to 1. Now, if e = 3and h = 0, then j = l = 1and C j j the path algebra A = kQ is given by the Kronecker quiver Q:◦◦ . Therefore, β Simply Connected Algebras 941 π (Q) = Z and so A is not simply connected. If e = 3and h = 1, then the bound quiver 1 j j algebra A = kQ/J is given by the quiver Q which can be visualized as follows: α γ ◦◦ ◦ part A with the ideal J in the path algebra kQ of Q over k generated by the element γα − aγβ and elements from part A (the rest of Q), where a ∈ k \{0}.Since h = 1, we applied (in the above visualization) an admissible operation from the set S to the algebra C with pivot the regular C -module corresponding to the indecomposable representation of the formkk lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]), where a ∈ k \{0}. More precisely, if we apply (ad 1) with parameter t = 0, then we have to remove the arrow ε and the part A. Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/J , where the ideal J of kQ is generated by the elements of J \{γα − aγβ}∪{γα} and π (Q, J ) is not trivial. Again, it follows from Proposition 4.10 that π (Q, J ) = Z π (A). Moreover, if we apply an admissible 1 1 operation from the set S to the algebra C , the proof follows by dual arguments. If e = 3 j j and h = 2, then the bound quiver algebra A = kQ/L is given by the quiver Q which can be visualized as follows: λ δ ε part A ◦◦ ◦ ◦ part B with the ideal L of kQ generated by the elements γα − aγβ, αδ − bβδ, γαδ and elements from parts A, B of Q,where a, b ∈ k \{0} and a = b.Since h = 2, we applied (in the above visualization) one admissible operation from the set S and one from the set S to the algebra C with pivots the regular C -modules corresponding to the indecomposable j j representations of the form 1 1 kk andkk lying in different stable tubes of rank 1 in (see [42, XIII.2.4(c)]), where a, b ∈ k \{0} and a = b. More precisely, if we apply (ad 1) (respectively, (ad 1 )) with parameter t = 0, then we have to remove the arrow ε and the part B (respectively, the arrow λ and the part A). Observe also that A is not simply connected, because A is isomorphic to the algebra A = kQ/L , where the ideal L of kQ is generated by the elements of L \{γα − aγβ, αδ − bβδ}∪{γα, αδ} and π (Q, L ) is not trivial. Again, it follows from Proposition 4.10 that π (Q, L ) = Z π (A) π (B). In a similar way, one can show all the cases of applying 1 1 1 two admissible operations from the set S ∪ S to any two stable tubes of rank one from the Auslander–Reiten quiver of the Kronecker algebra. We now show the sufficiency. We know from Theorem 3.5 that there is a unique full (l) (l) (l) convex subcategory A = A × ··· × A of A which is a tubular coextension of the product C × ... × C = C of a family C ,...,C of tame concealed algebras (see 1 m 1 m (l) remarks immediately after Theorem 5.1) such that A is obtained from A by a sequence of admissible operations of types (ad 1)–(ad 5). We shall prove our claim by induction on (l) the number of admissible operations leading from A 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 (which occurs in the definitions of the above admissible operations) is finite for each 1 ≤ i ≤ t. Indeed, if this is not the case, then the family of generalized multicoils obtained after applying such admissible operation is not sincere, and then it is not 942 P. Malicki (l) separating. Let C = A ,...,A = A ,A ,...,A = A be an admissible sequence for 0 p p+1 n A and assume that A = A. In this case A is tame quasitilted algebra and our claim follows from [3, Theorem A]. Let k ≥ p, A = A and assume that A is simply connected. k+1 k Moreover, let v be the extension point of A and X ∈ ind A be the pivot of the admissible k k operation. Since H (A) = 0, the vertex v is separating, by [44, Lemma 3.2]. Note that if the admissible operation leading from A to A is of type (ad 1), (ad 2) or (ad 3), then A is k k a connected algebra. If X is an (ad 1)-pivot, then A = A [X] or A = (A × D)[X ⊕ Y ], where rad P = X k k A v or rad P = X ⊕ Y respectively, D is the full t × t lower triangular matrix algebra over A v k for some t ≥ 1, and Y is the unique indecomposable projective-injective D-module (see definition of (ad 1)). Applying Lemma 4.7 or Lemma 4.8 respectively, we conclude that A is simply connected. If X is an (ad 2)-pivot or (ad 3)-pivot, then A = A [X], where rad P = X. Applying k A v Lemma 4.7, we conclude that A is simply connected. Let X be an (ad 4)-pivot and Y = Y → Y → ··· → Y with t ≥ 1 be a finite sectional 1 2 t path in . Then, for r = 0, A = A [X ⊕ Y ],andfor r ≥ 1, A k ⎡ ⎤ A 00 ... 00 Yk 0 ... 00 ⎢ ⎥ ⎢ ⎥ Y k k ... 00 ⎢ ⎥ A = ⎢ ⎥ . . . . . . . . . . . ⎢ ⎥ . . . . . ⎢ ⎥ ⎣ ⎦ Y k k ... k 0 X ⊕ Y kk... kk with r +2 columns and rows (see definition of (ad 4)). We note that Y is directing A-module for each 1 ≤ i ≤ t. Indeed, since H (A) = 0, we get d = 0, and so A is not connected. A k Now, if r = 0, then A = A [X ⊕ Y ] and rad P = X ⊕ Y . Then it follows from k A v Lemma 4.7 that A is simply connected. If r ≥ 1, then observe that the modified algebra A of A can be obtained by applying (0) (1) (0) r + 1 one-point extensions in the following way: A = A [U ], A = A [U ], k 01 11 k k k (2) (1) (r−1) (r−2) (r) (r−1) A = A [U ], ..., A = A [U ] and finally A = A = A [X ⊕ U ], 21 r−1,1 r1 k k k k k k (j −1) where U = Y , U is a projective A -module such that rad (j −1) U = U ,for 01 j1 j1 j −1,1 (j −1) r ≥ 1, 1 ≤ j ≤ r. We denote by v the extension vertex of A ,for 1 ≤ j ≤ r. Since the vertex v of Q (0) is separating and rad (0) P = U , applying Lemma 4.7, we conclude 1 v 01 A A k k (0) that the algebra A is simply connected. Further, since the vertex v of Q is separating, (1) (0) (1) rad (1) P = U ,and A is simply connected, it follows from Lemma 4.7 that A is v 11 k k (r−1) simply connected. Iterating a finite number of times the same arguments, we get that A is simply connected. Finally, since the vertex v of Q is separating and rad P = X ⊕ U , A A v r1 applying again Lemma 4.7, we get that A is simply connected. Let X be an (ad 5)-pivot. 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 admis- Simply Connected Algebras 943 sible operation (ad 4), we conclude that the required statement follows from the above considerations. This finishes the proof of Theorem 1.1. 6 Proof of Theorem 1.2 Let A be a generalized multicoil algebra. Then A is a connected generalized multicoil enlargement of a concealed canonical algebra C.Let C = C × C ×· · ·× C × C ×···× 1 2 l l+1 C be a decomposition of C into product of connected algebras such that C ,C ,...,C m 1 2 l are of type (p ,p ) and C ,C ,...,C are of type (p ,...,p ) with t ≥ 3. Since C , 1 2 l+1 l+2 m 1 t i i ∈{1,...,m}, are simply connected, we get l = 0. Moreover, by the assumption, the sec- ∗ ∗ tional paths occuring in the definitions of the operations (ad 4), (fad 4), (ad 4 ), (fad 4 ) come from two components of two connected algebras. Applying Theorems 3.3 and 3.5 we (l) (l) (l) infer that there exists a unique factor algebra A = A ×· · ·× A of A which is a tubu- lar coextension of a concealed canonical algebra C = C × ... × C , and a unique factor 1 m (r) (r) (r) algebra A = A ×· · ·× A of A which is a tubular extension of a concealed canoni- (l) (r) cal algebra C = C × ... × C .Since A and A are quasitilted algebras (of canonical 1 m types), the equivalence (ii) and (iv) follows from [26, Theorem 1]. Clearly, (v) implies (i). We now show that (i) implies (iii). Since all algebras C ,...,C are of type (p ,...,p ) 1 m 1 t with t ≥ 3(l = 0), we get f = 0. Assume to the contrary that H (A) = 0. Then, by Theorem 5.1, d + f = 0. Therefore, d = 0 and it follows from the proof of Lemma 4.4 A A A (and its dual version) that A is not simply connected, a contradiction with (i). We show that (iii) implies (iv). Assume to the contrary that there exists i ∈{1,...,m} (l) (r) 1 1 such that H (A ) = 0or H (A ) = 0. Without loss of generality, we may assume i i (l) (l) that H (A ) = 0for some i ∈{1,...,m}.Since A is a tubular coextension of a con- i i (l) cealed canonical algebra C ,wehavethat A is a generalized multicoil enlargement of C , i i (l) and so, by Theorem 5.1, dim H (A ) = d (l) + f (l). Moreover, by our assumption on A A i i C ,wehave f (l) = 0. Hence d (l) = 0. Since d ≥ d (l), we get a contradiction with i A A A A i i i (iii). (l) In order to finish the proof we will show that (iv) implies (v). Assume that H (A ) = (r) 0and H (A ) = 0, for any i ∈{1,...,m}. We know that for each i ∈{1,...,m}, (l) (r) A (respectively, A ) is a tubular coextension (respectively, extension) of a concealed i i canonical algebra C of type (p ,...,p ), t ≥ 3and H (C ) = 0, by [20, Theorem 2.4]. i 1 t i (l) (r) Then H (B) = 0 for every full convex subcategory B of A (respectively, A ). Therefore, i i (l) (r) it follows from [44, Theorem 4.1] that A and A are strongly simply connected, for any i i i ∈{1,...,m}. Moreover, by our assumption on A, the Auslander–Reiten quiver does not contain exceptional configurations of modules. Applying now Theorems 3.3 and 3.6 we infer that A is strongly simply connected. 7 Examples We start this section with the following remark. 944 P. Malicki Remark 7.1 We can apply Theorem 1.1 to important classes of algebras. For example, to the cycle-finite algebras with separating families of almost cyclic coherent Auslander–Reiten components. Indeed, it is known (see [8]) that every cycle-finite algebra is tame. Example 7.2 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 1 5 ν α 18 2 4 6 3 8 7 19 9 10 δ 11 15 12 13 14 17 20 21 22 and I the ideal in the path algebra kQ of Q over k generated by the elements αβ, γδ, ηε, κλ, ξκλ, να.Then A is a generalized multicoil enlargement of a concealed canonical algebra C,where C is the hereditary algebra of Euclidean type D given by the vertices 1, 2,..., 7. Indeed, consider the dimension-vectors 00 00 00 000 100 00 00 010 000 000 a = 010 , a = 010 , a = 010 , a = , a = 00 , a = 00 . 1 2 3 4 5 6 00 010 10 00 00 0 011 0000 1 0 We apply (ad 1 ) with pivot the simple regular C-module with vector a , and with parameter t = 0. The modified algebra B is given by the quiver with vertices 1, 2,..., 8 bound by αβ = 0. Now, we apply (ad 1 ) with pivot the indecomposable B -module with vector a , 1 2 and with parameter t = 2. The modified algebra B is given by the quiver with vertices 1, 2,..., 11 bound by αβ = 0. Next, we apply (ad 1 ) with pivot the indecomposable B - module with vector a , and with parameter t = 3. The modified algebra B is given by 3 3 the quiver with vertices 1, 2,..., 15 bound by αβ = 0, γδ = 0. In the next step we apply (ad 1 ) with pivot the indecomposable B -module with vector a , and with parameter t = 0. 3 4 The modified algebra B is given by the quiver with vertices 1, 2,..., 16 bound by αβ = 0, γδ = 0, ηε = 0. Next, we apply the admissible operation (ad 5) in two steps. The first step: we apply the operation (fad 3) with pivot the indecomposable B -module with vector a , and with parameters t = 3, s = 2. The modified algebra B is given by the quiver with 5 5 vertices 1, 2,..., 17 bound by αβ = 0, γδ = 0, ηε = 0, κλ = 0. The second step: we apply the operation (ad 4) with pivot the indecomposable B -module with vector a ,and 5 6 Simply Connected Algebras 945 with a finite sectional path consisting of the indecomposable B -modules with dimension- vectors 00 00 00 00 000 000 000 000 000 000 000 000 00 → 00 → 00 → 00 00 00 01 00 0011 0011 0011 0001 1 0 0 0 and with parameter r = 4. The modified algebra is equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) bound quiver algebra kQ /I ,where Q is a full subquiver of Q givenbythe vertices (l) (l) (l) (r) 1, 2,..., 16 and I = kQ ∩ I is the ideal in kQ . The right quasitilted algebra A of (r) (r) (r) A is the convex subcategory of A being the bound quiver algebra kQ /I ,where Q is (r) (r) a full subquiver of Q givenbythevertices 1, 2,..., 7, 14, 15,..., 18 and I = kQ ∩ I (r) (l) (r) is the ideal in kQ . Note that A and A are tame. It follows from Theorems 3.3, 3.5(iii) and the above construction that the Auslander– Reiten quiver ofthetamealgebra A = kQ/I admits a separating family of almost cyclic coherent components. Further, π (Q, I ) = Z and hence A is not simply connected. Moreover, by Theorem 5.1, the first Hochschild cohomology space H (A) k (d = (l) (r) 1,f = 0). We also note that, since A and A are tame tilted algebras of Euclidean 1 (l) 1 (r) (l) type D such that H (A ) = 0and H (A ) = 0, it follows from [5, Theorem] that A (r) and A are simply connected (and even strongly simply connected from [5, Corollary]). We refer to [33, Example 4.1] (see also [35, Example 9.13]) for a more extensive example of the tame algebra with a separating family of almost cyclic coherent components which is not simply connected. Finally, we also mention that A is a generalized multicoil algebra such that contains the exceptional configurations of modules. Example 7.3 We borrow the following example from [31]. Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 1 520 23 ϕ ϕ 1 3 2 4 6 25 22 ϕ ϕ 2 4 β ξ ζ 1 1 3 8 719 26 21 32 24 ξ ζ 2 2 γ ξ 9 10 18 27 28 μ ω δ 11 15 1 33 η π λ κ 12 13 14 17 29 30 31 and I the ideal in the path algebra kQ of Q over k generated by the elements αβ, γδ, ηε, κλ, ϕ ψ, ϕ ψ, ξ ω , ζ ϕ , ζ ϕ , ζ ξ ξ ξ − ζ ψ, π ξ , π ω − π ω , μκλ, νξ .Then A is 3 4 3 1 1 1 1 2 2 3 2 1 1 1 2 1 1 2 2 1 a generalized multicoil enlargement of a concealed canonical algebra C = C × C ,where 1 2 C is the hereditary algebra of Euclidean type D given by the vertices 1, 2,..., 7, and C 1 6 2 is the hereditary algebra of Euclidean type D given by the vertices 20, 21,..., 24. Indeed, 5 946 P. Malicki we apply (ad 1 )to C with pivot the simple regular C -module S , and with parameter 1 1 4 t = 0. The modified algebra B is given by the quiver with the vertices 1, 2,..., 8 bound by αβ = 0. Next, we apply (ad 1 )to B with pivot the indecomposable injective B - 1 1 module I , and with parameter t = 2. The modified algebra B is given by the quiver with 8 2 the vertices 1, 2,..., 11 bound by αβ = 0. Now, we apply (ad 1 )to B with pivot the indecomposable B -module τ S , and with parameter t = 3. The modified algebra B 2 B 10 3 is given by the quiver with the vertices 1, 2,..., 15 bound by αβ = 0, γδ = 0. Next, we apply (ad 1 )to B with pivot the simple B -module S , and with parameter t = 0. The 3 3 14 modified algebra B is given by the quiver with the vertices 1, 2,..., 16 bound by αβ = 0, γδ = 0, ηε = 0. In the next step we apply (ad 1 )to C with pivot the simple regular C -module S , and with parameter t = 3. The modified algebra B is given by the quiver 2 22 5 with the vertices 20, 21,..., 28 bound by ϕ ψ = 0, ϕ ψ = 0. Now, we apply (ad 1 )to 3 4 B with pivot the simple B -module S , and with parameter t = 2. The modified algebra 5 5 B is given by the quiver with the vertices 20, 21,..., 31 bound by ϕ ψ = 0, ϕ ψ = 0, 6 3 4 ξ ω = 0. Next, we apply (ad 2) to B with pivot the indecomposable injective B -module 3 1 6 6 I , and with parameter t = 3. The modified algebra B is given by the quiver with the 25 7 vertices 20, 21,..., 32 bound by ϕ ψ = 0, ϕ ψ = 0, ξ ω = 0, ζ ϕ = 0, ζ ϕ = 0, 3 4 3 1 1 1 1 2 ζ ξ ξ ξ = ζ ψ. Now, we apply (ad 3) to B with pivot the indecomposable B -module 2 3 2 1 1 7 7 τ S , and with parameter t = 2. The modified algebra B is given by the quiver with the B 30 8 vertices 20, 21,..., 33 bound by ϕ ψ = 0, ϕ ψ = 0, ξ ω = 0, ζ ϕ = 0, ζ ϕ = 0, 3 4 3 1 1 1 1 2 ζ ξ ξ ξ = ζ ψ, π ξ = 0, π ω = π ω . Finally, we apply (ad 5) to B × B in two 2 3 2 1 1 1 2 1 1 2 2 4 8 steps. The first step: we apply (fad 3) with pivot the indecomposable B -module τ S , 4 B 14 and with parameters t = 3, s = 2. The modified algebra B is given by the quiver with the vertices 1, 2,..., 17 bound by αβ = 0, γδ = 0, ηε = 0, κλ = 0. The second step: we apply (ad 4) with pivot the simple B -module S , and with the finite sectional path 8 26 I → τ S → I → S consisting of the indecomposable B -modules, and with 16 B 15 14 17 9 parameters t = 4, r = 1. The modified algebra is equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) (l) (l) (l) product A = A × A ,where A = kQ /I is the branch coextension of the tame 1 2 1 1 1 (l) concealed algebra C , Q is a full subquiver of Q givenbythe vertices1, 2,..., 16 and (l) (l) (l) (l) (l) (l) I = kQ ∩I is the ideal in kQ , A = kQ /I is the branch coextension of the tame 1 1 1 2 2 2 (l) concealed algebra C , Q is a full subquiver of Q given by the vertices 20, 21,..., 31 and (l) (l) (l) (r) I = kQ ∩ I is the ideal in kQ . The right quasitilted algebra A of A is the convex 2 2 2 (r) (r) (r) (r) (r) (r) (r) subcategory of A being the product A = A ×A ,where A = C , A = kQ /I 1 2 1 2 2 2 (r) is the branch extension of the tame concealed algebra C , Q is a full subquiver of Q given (r) (r) by the vertices 14, 15,..., 24, 26, 27, 28, 30, 31, 32, 33 and I = kQ ∩ I is the ideal in 2 2 (r) (l) (l) (r) (r) kQ . Note that A , A , A and A are tame. 2 1 2 1 2 It follows from Theorems 3.3, 3.5(iii) and the above construction that A is tame and admits a separating family of almost cyclic coherent components. Moreover, by Theo- rem 5.1, the first Hochschild cohomology space H (A) = 0(d = 0,f = 0). Then, a A A direct application of Theorem 1.1 shows that the algebra A is simply connected. In fact, it follows from [31, Theorem 1.2] that A is strongly simply connected. We also note that, since (l) (l) (r) (r) (l) A , A , A and A are tame tilted algebras of Euclidean type D such that H (A ) = 0, 1 2 1 2 1 (l) (r) (r) (l) (l) 1 1 1 H (A ) = 0, H (A ) = 0and H (A ) = 0 it follows from [5, Theorem] that A , A , 2 1 2 1 2 (r) (r) A and A are simply connected (and even strongly simply connected from [5, Corol- 1 2 lary]). Finally, we mention that C , C are simply connected, A is a generalized multicoil 1 2 Simply Connected Algebras 947 algebra, does not contain exceptional configurations of modules, and so this example illustrates also Theorem 1.2. Example 7.4 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 13 14 15 ψ ζ β μ λ α ε κ ν 16 1 2 3 11 12 67 8 ξ ω 5 4 10 9 17 η π and I the ideal in the path algebra kQ of Q over k generated by the elements aγβαλ − δλ, γε, bπ ωνμ − πηξ, ζμ, ϕψκ,where a, b ∈ k \{0}.Then A is a generalized multicoil enlargement of a concealed canonical algebra C = C × C ,where C is the hereditary 1 2 1 algebra of Euclidean type A given by the vertices 1, 2,..., 5, and C is the hereditary 4 2 algebra of Euclidean type A givenbythe vertices6, 7,..., 10. Indeed, we apply (ad 1 ) to C with pivot the simple regular C -module S , and with parameter t = 2. The modified 1 1 3 algebra B is given by the quiver with the vertices 1, 2,..., 5, 11, 12, 13 bound by γε = 0. Next, we apply (ad 4) to B × C with pivot the simple regular C -module S and with 1 2 2 7 the finite sectional path I → S consisting of the indecomposable B -modules, and 12 13 1 with parameters t = 2, r = 1. The modified algebra B is given by the quiver with the vertices 1, 2,..., 15 bound by γε = 0, ζμ = 0, ϕψκ = 0. Now, we apply (ad 1 ) with parameter t = 0 to the algebra B with pivot the regular C -module corresponding to the 2 1 indecomposable representation of the form a 1 kk k 1 1 k k lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]). The modified algebra B is C 3 given by the quiver with the vertices 1, 2,..., 16 bound by γε = 0, ζμ = 0, ϕψκ = 0, aγβαλ = δλ,where a ∈ k \{0}. Finally, we apply (ad 1) with parameter t = 0 to the algebra B with pivot the regular C -module corresponding to the indecomposable 3 2 representation of the form kk k 1 1 k k lying in a stable tube of rank 1 in (see [42, XIII.2.4(c)]). The modified algebra is then equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) (l) (l) (l) product A = A × A ,where A = kQ /I is the branch coextension of C , 1 2 1 1 1 (l) (l) Q is a full subquiver of Q given by the vertices 1, 2,..., 5, 11, 12, 13, 16 and I = 1 1 (l) (l) (l) (r) kQ ∩ I is the ideal in kQ , A = C . The right quasitilted algebra A of A is the 1 1 2 (r) (r) (r) (r) (r) convex subcategory of A being the product A = A × A ,where A = C , A = 1 2 1 2 (r) (r) (r) kQ /I is the branch extension of C , Q is a full subquiver of Q givenbythe vertices 2 2 2 948 P. Malicki (r) (r) (r) (l) 6, 7,..., 10, 12, 13, 14, 15, 17 and I = kQ ∩ I is the ideal in kQ . Note that A , 2 2 2 1 (l) (r) (r) A , A and A are tame. 2 1 2 It follows from Theorems 3.3, 3.5(iii) and the above construction that A is tame and admits a separating family of almost cyclic coherent components. Moreover, we have h = 1, e = 1, h = 1, e = 1, f = 0, f = 0, f = f + f = 0, and 1 1 2 2 C C A C C 1 2 1 2 d = 0. Therefore, by Theorem 5.1, the first Hochschild cohomology space H (A) = 0. Then, a direct application of Theorem 1.1 shows that the algebra A is simply connected. We (l) (r) (l) (r) 1 1 ∼ ∼ note that, by [19, Proposition 1.6], H (A ) k, H (A ) k.Since A and A are = = 2 1 1 2 (l) (r) 1 1 generalized multicoil algebras, we get by Theorem 5.1 that H (A ) = 0, H (A ) = 0. 1 2 (r) (l) (l) (r) We also mention that A = C , A = C are not simply connected, A , A are simply 1 2 1 2 1 2 connected, by [3, Theorem A], and so A is not strongly simply connected. Moreover, by the above construction we know that A is a generalized multicoil algebra, such that does not contain exceptional configurations of modules. Therefore, this example shows that simple connectedness assumption imposed on the considered concealed canonical algebras is essential for the validity of Theorem 1.2. We end this section with an example of a wild generalized multicoil algebra, illustrating Theorem 1.2. Example 7.5 Let A = kQ/I be the bound quiver algebra given by the quiver Q of the form 20 21 23 26 ϕ ϕ 3 4 1 θ θ η η 2 2 1 1 22 32 31 25 γ γ ϕ ϕ 2 1 2 1 β β β β 4 3 2 1 02 3 4 8 34 24 1 27 α α 4 1 ρ α α ε 1 3 2 1 19 5 6 7 18 35 33 28 ρ ξ σ δ ε ψ 2 1 2 1 1 2 17 16 12910 11 36 29 ξ δ 2 2 13 14 15 38 37 30 σ σ ω 3 4 1 and I the ideal in the path algebra kQ of Q over k generated by the elements α α α α + 1 2 3 4 β β β β + γ γ , α δ , α σ , ξ α , ε α , ε δ − ε δ , α ρ , ξ ρ − ξ ρ , ν γ , γ θ , 1 2 3 4 1 2 1 1 2 1 1 4 1 2 1 1 2 2 3 1 1 1 2 2 1 2 1 2 ν θ − ν θ , ϕ ψ , ϕ ψ , η ϕ , η ϕ , ψ κ , η κ − η ψ κ , ω κ , ω σ σ σ .Then A is a 1 2 2 1 1 1 4 1 1 2 1 3 2 1 2 2 1 1 1 2 2 1 4 3 2 generalized multicoil algebra. Indeed, A is a generalized multicoil enlargement of a canon- ical algebra C = C × C ,where C is the tubular canonical algebra of type (2, 4, 4) given 1 2 1 by the vertices 0, 1,..., 8 bound by α α α α + β β β β + γ γ = 0, and C is the 1 2 3 4 1 2 3 4 1 2 2 canonical algebra of Euclidean type D given by the vertices 23, 24,..., 27. It is known that admits an infinite family T , λ ∈ P (k), of pairwise orthogonal stable tubes, hav- C 1 ing a stable tube, say T , of rank 4 with the mouth formed by the modules S = τ S , 5 C 6 1 1 S = τ S , S = τ E, E = τ S ,where E is the unique indecomposable C -module 6 C 7 7 C C 5 1 1 1 1 with the dimension vector dimE = 11111 , and a unique stable tube, say T ,ofrank2with 000 Simply Connected Algebras 949 the mouth formed by the modules S = τ F , F = τ S ,where F is the unique indecom- 1 C C 1 1 1 posable C -module with the dimension vector dimF = (see [41, (3.7)]). Moreover, admits an infinite family T , μ ∈ P (k), of pairwise orthogonal stable tubes, hav- C μ 1 ing a stable tube, say T , of rank 2 with the mouth formed by the modules S = τ G, 25 C G = τ S ,where G is the unique indecomposable C -module with the dimension vec- C 25 2 tor dimG = 1 . We have the following sequence of the modified algebras. First, we apply (ad 1 )to C with pivot the simple regular C -module S , and with parameter t = 2. The 1 1 7 modified algebra B is given by the quiver with the vertices 0, 1,..., 11 bound by α δ = 0. 1 1 1 Next, we apply (ad 1 )to B with pivot the simple B -module S , and with parameter 1 1 6 t = 3. The modified algebra B is given by the quiver with the vertices 0, 1,..., 15 bound by α δ = 0, α σ = 0. Now, we apply (ad 1) to B with pivot the simple B -module S , 1 1 2 1 2 2 and with parameter t = 1. The modified algebra B is given by the quiver with the ver- tices 0, 1,..., 17 bound by α δ = 0, α σ = 0, ξ α = 0. Next, we apply (ad 3) to B 1 1 2 1 1 4 3 with pivot the indecomposable B -module τ I , and with parameter t = 2. The modi- 3 B 10 fied algebra B is given by the quiver with the vertices 0, 1,..., 18 bound by α δ = 0, 4 1 1 α σ = 0, ξ α = 0, ε α = 0, ε δ = ε δ . Further, we apply (ad 2 )to B with pivot the 2 1 1 4 1 2 1 1 2 2 4 indecomposable projective B -module P , and with parameter t = 1. The modified alge- 4 16 bra B is given by the quiver with the vertices 0, 1,..., 19 bound by α δ = 0, α σ = 0, 5 1 1 2 1 ξ α = 0, ε α = 0, ε δ = ε δ , α ρ = 0, ξ ρ = ξ ρ . Now, we apply (ad 1) to B 1 4 1 2 1 1 2 2 3 1 1 1 2 2 5 with pivot the simple regular B -module S , and with parameter t = 1. The modified alge- 5 1 bra B is given by the quiver with the vertices 0, 1,..., 21 bound by α δ = 0, α σ = 0, 6 1 1 2 1 ξ α = 0, ε α = 0, ε δ = ε δ , α ρ = 0, ξ ρ = ξ ρ , ν γ = 0. Next, we apply 1 4 1 2 1 1 2 2 3 1 1 1 2 2 1 2 (ad 2 )to B with pivot the indecomposable projective B -module P , and with parameter 6 6 21 t = 1. The modified algebra B is given by the quiver with the vertices 0, 1,..., 22 bound by α δ = 0, α σ = 0, ξ α = 0, ε α = 0, ε δ = ε δ , α ρ = 0, ξ ρ = ξ ρ , 1 1 2 1 1 4 1 2 1 1 2 2 3 1 1 1 2 2 ν γ = 0, γ θ = 0, ν θ = ν θ .Now,weapply(ad1 )to C with pivot the simple regular 1 2 1 2 1 2 2 1 2 C -module S , and with parameter t = 2. The modified algebra B is given by the quiver 2 25 8 with the vertices 23, 24,..., 30 bound by ϕ ψ = 0, ϕ ψ = 0. Next, we apply (ad 1) to 1 1 4 1 B with pivot the indecomposable B -module τ S , and with parameter t = 1. The mod- 8 8 B 29 ified algebra B is given by the quiver with the vertices 23, 24,..., 32 bound by ϕ ψ = 0, 9 1 1 ϕ ψ = 0, η ϕ = 0, η ϕ = 0. Now, we apply (ad 2 )to B with pivot the indecomposable 4 1 1 2 1 3 9 projective B -module P , and with parameter t = 1. The modified algebra B is given 9 31 10 by the quiver with the vertices 23, 24,..., 33 bound by ϕ ψ = 0, ϕ ψ = 0, η ϕ = 0, 1 1 4 1 1 2 η ϕ = 0, ψ κ = 0, η κ = η ψ κ . Next, we apply (ad 4) to B × B with pivot the 1 3 2 1 2 2 1 1 1 7 10 simple B -module S , and with the finite sectional path I → I → S consisting of 10 32 13 14 15 the indecomposable B -modules, and with parameters t = 3, r = 4. The modified algebra is then equal to A. (l) Then the left quasitilted algebra A of A is the convex subcategory of A being the (l) (l) (l) (l) (l) (l) (l) product A = A × A ,where A = kQ /I is the branch coextension of C , Q is 1 2 1 1 1 1 (l) (l) a full subquiver of Q givenbythe vertices0, 1,..., 15, 17, 19, 21, 22 and I = kQ ∩ I 1 1 (l) (l) (l) (l) (l) is the ideal in kQ , A = kQ /I is the branch coextension of C , Q is a full 1 2 2 2 2 (l) (l) subquiver of Q given by the vertices 23, 24,..., 30, 33 and I = kQ ∩ I is the ideal 2 2 (l) (r) in kQ . The right quasitilted algebra A of A is the convex subcategory of A being the (r) (r) (r) (r) (r) (r) (r) product A = A × A ,where A = kQ /I is the branch extension of C , Q is 1 2 1 1 1 1 (r) a full subquiver of Q givenbythevertices0, 1,..., 8, 10, 11, 16, 17, 18, 20, 21 and I = (r) (r) (r) (r) (r) (r) kQ ∩ I is the ideal in kQ , A = kQ /I is the branch extension of C , Q is a 1 1 2 2 2 2 950 P. Malicki full subquiver of Q given by the vertices 13, 14, 15, 23, 24,..., 27, 31, 32, 34, 35,..., 38 (r) (r) (r) (l) (r) and I = kQ ∩ I is the ideal in kQ . Then, A and A are the quasitilted algebras 2 2 2 1 1 (l) (r) of wild types (4, 4, 13), (4, 4, 9), respectively. Moreover, A and A are tame. 2 2 It follows from [7, Corollary 1.4] that C is simply connected. Moreover, C is also sim- 1 2 ply connected. By the above construction we know that A is a generalized multicoil algebra obtained from C , C and does not contain exceptional configurations of modules. Fur- 1 2 A ther, by Theorem 5.1, the first Hochschild cohomology space H (A) = 0(d = 0, f = 0) A A (l) (l) (r) (r) 1 1 1 1 and H (A ) = 0, H (A ) = 0, H (A ) = 0, H (A ) = 0. Then, a direct application 1 2 1 2 (l) (l) (r) (r) of Theorem 1.2 shows that the algebras A , A , A , A and A are simply connected. 1 2 1 2 Acknowledgements I thank an anonymous referee for useful comments. Data Availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study. 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. 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Algebras and Representation Theory – Springer Journals

Published: Aug 1, 2022

Keywords: Simply connected algebra; Hochschild cohomology; Auslander–Reiten quiver; Tame algebra; Generalized multicoil algebra; Primary 16G70; Secondary 16G20

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The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

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The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

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