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Cycle-finite module categories

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited.


Introduction
Throughout the article K denotes a fixed algebraically closed field.
By an algebra we mean an associative finite dimensional K-algebra with an identity which we shall assume (without loss of generality) to be basic and connected. For an algebra A, by an A-module we mean a finite dimensional right A-module. We shall denote by modA the category of A-modules, by ind A its full subcategory formed by the indecomposable modules, by Γ A the Auslander-Reiten quiver of A, and by τ A the Auslander-Reiten translation DTr in Γ A . We shall identify an indecomposable A-module with the vertex of Γ A corresponding to it. From Drozd's Tame and Wild Theorem [30] (see also [26]) the class of algebras may be divided into two classes. One class consists of the wild algebras whose representation theory comprises the representation theories of all algebras over K (see [68,Chapter XIX]). The second class consists of the tame algebras for which the indecomposable modules occur, in each dimension d, in a finite number of discrete and a finite number of one-parameter families. Hence, a classification of the finite dimensional modules is only feasible for tame algebras. It has been shown by Crawley-Boevey [26] that, if A is a tame algebra, then, for any dimension d ≥ 1, all but finitely many isomorphism classes of indecomposable A-modules of dimension d are invariant on the action of τ A , and hence, by a result due to Hoshino [35], lie in stable tubes of rank one in Γ A . The indecomposable modules over tame algebras which do not lie in stable tubes of rank one are called discrete. A distinguished class of tame algebras is formed by the algebras of finite representation type, having only finitely many isomorphism classes of indecomposable modules, for which the representation theory is presently rather well understood (see [13], [20], [21], [22], [24]). On the other hand, the representation theory of arbitrary tame algebras is still only emerging. At present the most accessible seem to be the (tame) algebras of polynomial growth, for which there exists an integer m such that the number of one-parameter families of indecomposable modules is bounded, in each dimension d, by d m . This class of algebras has been subject of intensive research over the last 30 years.
A prominent role in the representation theory of algebras is played by cycles of modules, or more generally cycles of complexes of modules. Recall that a cycle in a module category mod A is a sequence of nonzero nonisomorphisms in ind A, and the cycle is said to be finite if the homomorphisms f 1 , . . . , f r do not belong to the infinite Jacobson radical of mod A. Following Ringel [65] a module in ind A which does not lie on cycle in ind A is called directing. It has been proved independently by Peng and Xio [53] and the third named author [74] that the Auslander-Reiten quiver Γ A of an arbitrary algebra A contains at most finitely many τ A -orbits containing directing modules. Hence, in order to obtain information on nondirecting indecomposable modules of a module category, we may study properties of cycles in mod A containing these modules. We also note that, by a result of Ringel [65] the support algebras of directing modules are tilted algebras. Following [5] an algebra A is said to be cycle-finite if all cycles in modA are finite. It has been proved by the third named author in [76] that every cycle-finite algebra A is of polynomial growth and the support algebras of the one-parametric families of indecomposable A-modules are tame concealed algebras (preprojective tilts of the path algebras of Euclidean quivers) and Ringel's tubular algebras, which are distinguished classes of cycle-finite algebras. The class of cycle-finite algebras is wide and contains the algebras of finite representation type, the tame tilted algebras [37], the tame double tilted algebras [62], the tame generalized double tilted algebras [63], the tubular algebras [65], the iterated tubular algebras [61], the tame quasi-tilted algebras [40], [78], the tame coil and multicoil algebras [5], [6], [7], the tame generalized multicoil algebras [51], and the strongly simply connected algebras of polynomial growth [77]. It has been also proved in [2], [3], [4] that the class of algebras A for which the derived category D b (mod A) of bounded complexes of A-modules is cycle-finite coincides with the class of piecewise hereditary algebras of Dynkin, Euclidean, and tubular type, and consequently these algebras are also cycle-finite. Moreover, frequently an algebra A admits a Galois covering R → R/G = A where R is a cycle-finite locally bounded category and G is an admissible group of automorphisms of R, which allows to reduce the representation theory of A to the representation theory of cycle-finite algebras being finite convex subcategories of R. For example, every selfinjective algebra A of polynomial growth admits a canonical standard form A (geometric socle deformation of A) such that A has a Galois covering R → R/G = A, where R is a cycle-finite selfinjective locally bounded category and G is an admissible infinite cyclic group of automorphisms of R, the Auslander-Reiten quiver Γ A of A is the orbit quiver Γ R /G of Γ R , and the stable Auslander-Reiten quivers of A and A are isomorphic (see [69], [80] for details). We also mention that, by the main result of [59], every algebra A which admits a cycle-finite Galois covering R → R/G = A with G torsion-free is tame.
One of the objectives of this article is to describe the structure of the category ind A of an arbitrary cycle-finite algebra A, by showing that it can be covered by the categories of indecomposable modules of tame generalized multicoil algebras and tame generalized double tilted algebras. Here, a crucial role will be played by description of support algebras of cyclic components of the Auslander-Reiten quivers of cycle-finite algebras. The second objective of the article is to exhibit geometric and homological properties of indecomposable modules over cycle-finite algebras. We are interested in the class of coherent cycle-finite algebras for which all cyclic components of the Auslander-Reiten quivers are coherent (see Section 2 for definition). Every coherent cycle-finite algebra A is triangular, and hence the (geometric) Tits quadratic form q A and the (homological) Euler form χ A of A are well defined. For a vector d in the Grothendieck group K 0 (A) of A with nonnegative coordinates, we denote by mod A (d) the affine variety of A-modules of dimension vector d and by G(d) the corresponding product of general linear groups acting on mod A (d) in such a way that the G(d)-orbits in mod A (d) correspond to the isomorphism classes of A-modules of dimension vector d. The third main aim of the article is to establish a common bound on the numbers of discrete indecomposable modules in each dimension vector over cycle-finite algebras, generalizing results proved in [84] for strongly simply connected algebras of polynomial growth.

Preliminaries
In this section we recall some concepts and results from the representation theory of algebras important for further considerations.
Let A be an algebra (which by our assumption is basic and connected). Then there is an isomorphism A ∼ = KQ/I of K-algebras, where KQ is the path algebra of the Gabriel quiver Q = Q A of A and I is an admissible ideal of KQ. Equivalently, ond Brauer-Thrall conjecture we know that A is representation-finite if and only if µ A (d) = 0 for any d ≥ 1.
The Tits form of a triangular algebra A = KQ/I is the integral quadratic form q A : Z Q 0 → Z, defined, for x = (x i ) i∈Q 0 ∈ Z Q 0 , by where Q 1 is the set of arrows in Q and r(i, j) is the cardinality of L ∩ I(i, j), for a minimal set of generators L ⊂ i, j∈Q 0 I(i, j) of the ideal I (see [18]). Moreover, the Euler form of A is the integral quadratic form χ A : Z Q 0 → Z defined in [65, 2.4] such that for any module M in mod A, we have Observe that A is of finite global dimension, because A is triangular. It is also known that if gl dim A ≤ 2 then q A = χ A (see [18]). Finally, it is known (see [54]) that, if A is tame, then q A is weakly nonnegative, that is, q A (x) ≥ 0 for all x ∈ N Q 0 . Unfortunately, the reverse implication is not true in general. However, it has been proved recently in [25] that a strongly simply connected algebra A is tame if and only if the Tits form q A is weakly nonnegative. Recall also that a triangular algebra A is called strongly simply connected [71] if the first Hochschild cohomology H 1 (C,C) of every convex subcategory C of A vanishes. We need also special types of components of the Auslander-Reiten quivers of algebras.
Recall from [28], [65] that a translation quiver Γ is called a tube if it contains a cyclical path and if its underlying topological space is homeomorphic to S 1 × R + , where S 1 is the unit circle and R + is the nonnegative real line. A tube has only two types of arrows: arrows pointing to infinity and arrows pointing to the mouth. Tubes containing neither projective vertices nor injective vertices are called stable, and are as follows. For the infinite quiver · · · · · · · · · · · · with the translation τ given by τ(i, j) = (i − 1, j) for i ∈ Z, j ∈ N. For each r ≥ 1, denote by ZA ∞ /(τ r ) the translation quiver Γ obtained from ZA ∞ by identifying each vertex (i, j) of ZA ∞ with the vertex τ r (i, j) and each arrow x → y in ZA ∞ with the arrow τ r x → τ r y. The translation quiver ZA ∞ /(τ r ) is called the stable tube of rank r. The τ-orbit of a stable tube Γ formed by all vertices having exactly one immediate predecessor (equivalently, successor) is called the mouth of Γ . A tube Γ without injective vertices (respectively, without projective vertices) is called a ray tube (respectively, coray tube).
Let A be an algebra. A component C of Γ A is called regular if C contains neither a projective module nor an injective module, and semiregular if C does not contain both a projective and an injective module. It has been shown in [41] and [87] that a regular component C of Γ A contains an oriented cycle if and only if C is a stable tube. Moreover, Liu proved in [42] that a semiregular component C of Γ A contains an oriented cycle if and only if C is a ray or coray tube. A component P of Γ A is called postprojective if P is acyclic and every module in P lies in the τ A -orbit of a projective module. Dually, a component Q of Γ A is called preinjective if Q is acyclic and every module in Q lies in the τ A -orbit of an injective module. A component Γ of Γ A is said to be coherent if the following two conditions are satisfied: (C1) For each projective module P in Γ there is an infinite sectional path Further, a component Γ of Γ A is said to be almost cyclic if all but finitely many modules of Γ lie on oriented cycles in Γ A , so contained entirely in Γ . We note that the stable tubes, ray tubes and coray tubes of Γ A are special types of almost cyclic coherent components. In general, it has been proved in [50] that a component Γ of Γ A is almost cyclic and coherent if and only if Γ is a generalized multicoil, which can be obtained from a family of stable tubes by a sequence of admissible operations (see Section 4 for algebras having such components). A component Γ of Γ A is said to be almost acyclic if all but finitely many modules of Γ are acyclic (do not lie on oriented cycles in Γ A , hence in Γ ), and acyclic if all modules of Γ are acyclic. Finally, following [73] a component C of Γ A is said to be generalized standard if rad ∞ A (X,Y ) = 0 for all modules X and Y from C . It has been proved in [73] that every generalized standard component C of Γ A is almost periodic, that is, all but finitely many τ A -orbits in C are periodic. Clearly, the postprojective and preinjective components are acyclic, and the Auslander-Reiten quivers of representation-finite algebras are almost acyclic. Moreover, these components are generalized standard (see [74]). General results on almost acyclic components and related algebras have been proved by Reiten and third named author in [62], [63] (see Section 5). For a component C of Γ A , we denote by ann A (C ) the annihilator of C in A, that is, the intersection of the annihilators ann A (X) = {a ∈ A | Xa = 0} of all modules X in C . We note that C is a component of Γ A/ ann A (C ) . Moreover, if ann A (C ) = 0, C is said to be a faithful component of Γ A . By the support of a subquiver Γ of Γ A we mean the full subcategory suppΓ of A given by the supports supp M of all modules M in Γ , and, if suppΓ = A then Γ is said to be sincere. We note that a faithful component C of Γ A is sincere.

Semiregular components of cycle-finite algebras
In this section we recall the shapes of the Auslander-Reiten quivers of representationinfinite tilted algebras of Euclidean type and tubular algebras, as well as results from [76] on semiregular components of the Auslander-Reiten quivers of cycle-finite algebras, important for further considerations.
By a tame concealed algebra we mean a tilted algebra C = End H (T ), where H is the path algebra K∆ of a quiver ∆ of Euclidean type (the underlying graph ∆ of ∆ of type A m (m ≥ 1), D n (n ≥ 4), or E p (6 ≤ p ≤ 8)) and T is a (multiplicity-free) postprojective tilting H-modules. The tame concealed algebras have been described by quivers and relations by Bongartz [19] and Happel-Vossieck [34]. Recall also that the Auslander-Reiten quiver Γ C of a tame concealed algebra C is of the form where P C is a postprojective component containing all indecomposable projective C-modules, Q C is a preinjective component containing all indecomposable injective C-modules, and T C is a P 1 (K)-family T C λ , λ ∈ P 1 (K), of pairwise orthogonal generalized standard stable tubes, all but a finite number of them of rank one. The ordering from the left to right indicates that there are nonzero homomorphisms only from any of these classes to itself and to the classes to its right. We refer to [65,Chapter 4] and [67] for more details on the module categories of tame concealed algebras.
Recall also that, if B is a representation-infinite tilted algebra of Euclidean type ∆ , then one of the following holds: (a) B is a domestic tubular extension of a tame concealed algebra C and , of pairwise orthogonal generalized standard ray tubes, obtained from the P 1 (K)-family T C of stable tubes of Γ C by ray insertions, and Q B is a preinjective component containing all indecomposable injective B-modules and a section of type ∆ ; (b) B is a domestic tubular coextension of a tame concealed algebra C and where P B is the postprojective component containing all indecomposable projective B-modules and a section of type ∆ , T B is a P 1 (K)-family T B λ , λ ∈ P 1 (K), of pairwise orthogonal generalized standard coray tubes, obtained from the P 1 (K)family T C of stable tubes of Γ C by coray insertions, and Q B = Q C is the preinjective component of Γ C . We refer to [65,Chapter 4] and [68, Chapters XV-XVII] for more details on the module categories of representation-infinite tilted algebras of Euclidean type.
By a tubular algebra we mean a tubular extension (equivalently tubular coextension) of a tame concealed algebra of tubular type (2, 2, 2, 2), (3, 3, 3), (2,4,4), or (2, 3, 6), as defined by Ringel in [65,Chapter 5]. Recall that a tubular algebra B admits two different tame concealed convex subcategories C 0 and C ∞ such that B is a tubular extension of C 0 , and a tubular coextension of C ∞ , and the Auslander-Reiten quiver Γ B of B is of the form where P B 0 = P C 0 is the postprojective component of Γ C 0 , T B 0 is a P 1 (K)-family of pairwise orthogonal generalized standard ray tubes, obtained from the P 1 (K)family T C 0 of stable tubes of Γ C 0 by ray insertions, ∞ is a P 1 (K)-family of pairwise orthogonal generalized standard coray tubes, obtained from the P 1 (K)-family T C ∞ of stable tubes of Γ C ∞ by coray insertions, and, for each q ∈ Q + (the set of positive rational numbers) T B q is a P 1 (K)-family of pairwise orthogonal generalized standard stable tubes. We refer to [65,Chapter 5] for more details on the module categories of tubular algebras.
The following characterization of tame concealed and tubular algebras has been established in [

Theorem 3.2. Let A be a cycle-finite algebra. Then A is of polynomial growth.
An algebra A is said to be minimal representation-infinite if A is representationinfinite but every proper convex subcategory of A is representation-finite. Then we have the following characterization of minimal representation-infinite cycle-finite algebras established in [76,Corollary 4.4].
(ii) A does not contain a tubular algebra as a convex subcategory. (iii) All but finitely many components of Γ A are stable tubes of rank one.
In general, we have the following information on the Auslander-Reiten quiver of a cycle-finite algebra, which is a consequence of results proved in [74], [76], and results on the shapes of regular components from [41], [87].

Tame generalized multicoil algebras
In this section we introduce and exhibit basic properties of the class of tame generalized multicoil algebras, plying a prominent role in the description of infinite cyclic components of the Auslander-Reiten quivers of cycle-finite algebras. This is the class of tame algebras among the class of all algebras having a separating family of almost cyclic coherent components investigated in [51], [52]. Recall that a family C = (C i ) i∈I of components of the Auslander-Reiten quiver Γ A of an algebra A is called separating in mod A if the modules in ind A split into three disjoint classes P A , C A = C and Q A such that: (S1) C A is a sincere generalized standard family of components; (S2) Hom A (Q A , P A ) = 0, Hom A (Q A , C A ) = 0, Hom A (C A , P A ) = 0; (S3) any morphism from P A to Q A factors through the additive category add C A of C A . We then say that C A separates P A from Q A and write Γ A =P A ∪ C A ∪ Q A . We note that then P A and Q A are uniquely determined by C A (see [ It has been proved in [50, Theorem A] that a connected component Γ of an Auslander-Reiten quiver Γ A is almost cyclic and coherent if and only if Γ is a generalized multicoil, obtained from a family of stable tubes by a sequence of operations called admissible. We recall the letter and simultaneously define the corresponding enlargements of algebras.
We start with the one-point extensions and one-point coextensions of algebras. Let A be an algebra and M be a module in mod A. Then the one-point extension of A by M is the matrix algebra with the usual addition and multiplication.
Dually, one defines also the one-point coextension of A by M as the matrix algebra For K and r ≥ 1, we denote by T r (K) the r × r-lower triangular matrix algebra  Given a generalized standard component Γ of Γ A , and an indecomposable module X in Γ , the support S (X) of the functor Hom A (X, −)| Γ is the R-linear category defined as follows [7]. Let H X denote the full subcategory of Γ consisting of the indecomposable modules M in Γ such that Hom A (X, M) = 0, and I X denote the ideal of H X consisting of the morphisms f : M → N (with M, N in H X ) such that Hom A (X, f ) = 0. We define S (X) to be the quotient category H X /I X . Following the above convention, we usually identify the R-linear category S (X) with its quiver.
From now on let A be an algebra and Γ be a family of generalized standard infinite components of Γ A . For an indecomposable brick X in Γ , called the pivot, one defines five admissible operations (ad 1)-(ad 5) and their dual (ad 1 * )-(ad 5 * ) modifying the translation quiver Γ = (Γ , τ) to a new translation quiver (Γ ′ , τ ′ ) and the algebra A to a new algebra A ′ , depending on the shape of the support S (X) (see [50,Section 2] for the figures illustrating the modified translation quivers Γ ′ ).
(ad 1) Assume S (X) consists of an infinite sectional path starting at X: In this case, we let t ≥ 1 be a positive integer, and the modified translation quiver Γ ′ of Γ to be obtained by inserting in Γ the rectangle consisting of the modules and For the remaining vertices of Γ ′ , τ ′ coincides with the translation of Γ , or Γ D , respectively.
If t = 0 we define the modified algebra A ′ to be the one-point extension A ′ = A[X] and the modified translation quiver Γ ′ to be the translation quiver obtained from Γ by inserting only the sectional path consisting of the vertices X ′ i , i ≥ 0. The non-negative integer t is such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangle equals t + 1. We call t the parameter of the operation.
Since Γ is a generalized standard family of components of Γ A , we then have In case Γ is a stable tube, it is clear that any module on the mouth of Γ satisfies the condition for being a pivot for the above operation. Actually, the above operation is, in this case, the tube insertion as considered in [28].
(ad 2) Suppose that S (X) admits two sectional paths starting at X, one infinite and the other finite with at least one arrow: where t ≥ 1. In particular, X is necessarily injective. We define the modified algebra A ′ of A to be the one-point extension A ′ = A[X] and the modified translation quiver Γ ′ of Γ to be obtained by inserting in Γ the rectangle consisting of the modules For the remaining vertices of Γ ′ , τ ′ coincides with the translation τ of Γ .
The integer t ≥ 1 is such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangle equals t + 1. We call t the parameter of the operation.
Since Γ is a generalized standard family of components of Γ A , we then have (ad 3) Assume S (X) is the mesh-category of two parallel sectional paths: where t ≥ 2. In particular, X t−1 is necessarily injective. Moreover, we consider the translation quiver Γ of Γ obtained by deleting the arrows We assume that the union Γ of connected components of Γ containing the vertices τ −1 is a finite translation quiver. Then Γ is a disjoint union of Γ and a cofinite full translation subquiver Γ * , containing the pivot X. We define the modified algebra A ′ of A to be the one-point extension A ′ = A[X] and the modified translation quiver Γ ′ of Γ to be obtained from Γ * by inserting the rectangle consisting of the modules For the remaining vertices of Γ ′ , τ ′ coincides with the translation τ of Γ * . We note that X ′ t−1 is injective. The integer t ≥ 2 is such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangle equals t + 1. We call t the parameter of the operation.
Since Γ is a generalized standard family of components of Γ A , we then have (ad 4) Suppose that S (X) consists an infinite sectional path, starting at X with t ≥ 1, be a finite sectional path in Γ A . Let r be a positive integer. Moreover, we consider the translation quiver Γ of Γ obtained by deleting the arrows We assume that the union Γ of connected components of Γ containing the vertices is a finite translation quiver. Then Γ is a disjoint union of Γ and a cofinite full translation subquiver Γ * , containing the pivot X. For r = 0 we define the modified algebra A ′ of A to be the one-point extension and the modified translation quiver Γ ′ of Γ to be obtained from Γ * by inserting the rectangle consisting of the modules and For the remaining vertices of Γ ′ , τ ′ coincides with the translation of Γ * . For We define the modified algebra A ′ of A to be the triangular matrix algebra of the form: with r + 2 columns and rows and the modified translation quiver Γ ′ of Γ to be obtained from Γ * by inserting the rectangles consisting of the modules For the remaining vertices of Γ ′ , τ ′ coincides with the translation of Γ * , or Γ G , respectively.
We note that the quiver Q A ′ of A ′ is obtained from the quiver of the double one- The integers t ≥ 1 and r ≥ 0 are such that the number of infinite sectional paths parallel to X 0 → X 1 → X 2 → · · · in the inserted rectangles equals t + r + 1. We call t + r the parameter of the operation.
Since Γ is a generalized standard family of components of Γ A , we then have (fad 1) Assume S (X) consists of a finite sectional path starting at X: We define the modified algebra A ′ of A to be the one-point extension and the modified translation quiver Γ ′ of Γ to be obtained by inserting in Γ the rectangle consisting of the modules For the remaining vertices of Γ ′ , τ ′ coincides with the translation of Γ , or Γ D , respectively. If t = 0 we define the modified algebra A ′ to be the one-point extension A ′ = A[X] and the modified translation quiver Γ ′ to be the component obtained from Γ by inserting only the sectional path consisting of the vertices Observe that for s = 0 = t the new translation quiver Γ ′ is obtained from Γ by adding the projective-injective vertex X ′ 0 and the vertex W with τ ′ W = X 0 . (fad 2) Suppose that S (X) admits two finite sectional paths starting at X, each of them with at least one arrow: where t ≥ 1 and s ≥ 1. In particular, X and X s are necessarily injective. We define the modified algebra A ′ of A to be the one-point extension A ′ = A[X] and the modified translation quiver Γ ′ of Γ to be obtained by inserting in Γ the rectangle consisting of the modules For the remaining vertices of Γ ′ , τ ′ coincides with the translation τ of Γ .
(fad 3) Assume S (X) is the mesh-category of two finite parallel sectional paths: In particular, X t−1 and X s are necessarily injective. We define the modified algebra A ′ of A to be the one-point extension A ′ = A[X] and the modified translation quiver Γ ′ of Γ to be obtained by inserting in Γ the rectangle consisting consists of a finite sectional path starting at X: and the modified translation quiver Γ ′ of Γ to be obtained by inserting in Γ the rectangle consisting of the modules For the remaining vertices of Γ ′ , τ ′ coincides with the translation of Γ .
Observe that any stable tube is trivially a generalized coil. A tube (in the sense of [28]) is a generalized coil having the property that each admissible operation in the sequence defining it is of the form (ad 1) or (ad 1 * ). If we apply only operations of type (ad 1) (respectively, of type (ad 1 * )) then such a generalized coil is called a ray tube (respectively, a coray tube). Observe that a generalized coil without injective (respectively, projective) vertices is a ray tube (respectively, a coray tube). A quasitube (in the sense of [70]) is a generalized coil having the property that each of the admissible operations in the sequence defining it is of type (ad 1), (ad 1 * ), (ad 2) or (ad 2 * ). Finally, following [7] a coil is a generalized coil having the property that each of the admissible operations in the sequence defining it is one of the forms (ad 1), (ad 1 * ), (ad 2), (ad 2 * ), (ad 3) or (ad 3 * ). We note that any generalized multicoil Γ is a coherent translation quiver with trivial valuations and its cyclic part c Γ (the translation subquiver of Γ obtained by removing from Γ all acyclic vertices and the arrows attached to them) is infinite, connected and cofinite in Γ , and so Γ is almost cyclic.
Let C be the product C 1 × . . . ×C m of a family C 1 , . . . ,C m of tame concealed algebras and T C the disjoin union T C 1 ∪ . . . ∪ T C m of P 1 (K)-families T C 1 , . . . , T C m of pairwise orthogonal generalized standard stable tubes of Γ C 1 , . . . ,Γ C m , respectively. Following [51], we say that an algebra A is a generalized multicoil enlargement of C 1 , . . . ,C m if A is obtained from C by an iteration of admissible operations of types (ad 1)-(ad 5) and (ad 1 * )-(ad 5 * ) performed either on stable tubes of T C or on generalized multicoils obtained from stable tubes of T C by means of operations done so far. It follows from [51, Corollary B] that then A is a triangular algebra, and hence the Tits and Euler forms q A and χ A of A are defined. In fact, in [51] generalized multicoil enlargements of finite families of arbitrary concealed canonical algebras have been introduced and investigated. But in the tame case we may restrict to the generalized multicoil enlargements of tame concealed algebras. Namely, we have the following consequence of [51, Theorems A and F].
where C A is a family of generalized multicoils separating P A from Q A such that: (1) C A is obtained from the P 1 (K)-families T C 1 , . . . , T C m of stable tubes of Γ C 1 , . . . ,Γ C m by admissible operations corresponding to the admissible operations leading from C 1 , . . . ,C m to A; i is a tubular algebra; i is tilted of Euclidean type, or i is a tubular algebra.
In particular, we have the following consequence of Theorems 3.1 and 4.8.

Tame generalized double tilted algebras
In this section we introduce and describe basic properties of the class of tame generalized double tilted algebras, which is the class of tame algebras in the class of generalized double tilted algebras investigated in [62], [63], [79]. 1 (−, T ) induce quasi-inverse equivalence between F (T ) and X (T ) (see [23], [33]). Moreover, (X (T ), Y (T )) is a splitting torsion pair of mod B, that is, every indecomposable module in mod B belongs either to X (T ) or Y (T ). Further, the images Hom H (T, I) of the indecomposable injective H-modules I via the functor Hom H (T, I) form a section Σ of an acyclic component C T of Γ B such that Σ is isomorphic to the opposite quiver Q op H of the quiver Q H of H, any predecessor of Σ in C T lies in Y (T ), and any proper successor of Σ in C T lies in X (T ). Therefore, the component C T of Γ B connects the torsion-free part Y (T ) with the torsion part X (T ) along the section Σ , and hence C T is called the connecting component of Γ B determined by T .
The following theorem proved independently by Liu [43] and Skowroński [72] gives a handy criterion for an algebra to be a tilted algebra. The general shape of the Auslander-Reiten quiver of a tilted algebra has been described by Kerner in [37]. We will describe only the Auslander-Reiten quivers of tame tilted algebras, which are exactly the cycle-finite tilted algebras.

Theorem 5.2. Let H = K∆ be a hereditary algebra, T a tilting H-module, and assume that the associated tilted algebra B = End H (T ) is tame. Then the connecting component C T of Γ B determined by T admits a finite (possibly empty) family of pairwise disjoint translation subquivers D
(viii) The Auslander-Reiten quiver Γ B of B has the disjoint union decomposition The following theorem follows from [18](part (i)) and [65, p.376](parts (ii) and (iii)). We refer to [17] and [29] (respectively, [55] and [56]) for a classification of representation-finite (respectively, tame representation-infinite) tame tilted algebras with sincere directing modules.

Theorem 5.3. Let A be a cycle-finite algebra, X a directing module in mod
The class of tilted algebras was extended in [62] to the class of double tilted algebras, and next in [63] to the class of generalized double tilted algebras, containing the class of all algebras of finite type, by extending the concept of a section to the concept of a multisection.
Following [63], a full connected subquiver ∆ of a component C of the Auslander-Reiten quiver Γ A of an algebra A is said to be a multisection if the following conditions are satisfied: (iv) |∆ ∩ O| = 1 for all but finitely many τ A -orbits O in C .
It has been proved in [63, Theorem 2.5] that a component C of Γ A is almost acyclic if and only if C admits a multisection ∆ . Moreover, for a multisection ∆ of a component C of Γ A , the following full subquivers of C were defined in [63]: there is a nonsectional path in C from X to a projective module P}, (ii) ∆ ′ r = {X ∈ ∆ ; there is a nonsectional path in C from an injective module I to X}, Then ∆ l is called the left part of ∆ , ∆ r the right part of ∆ , and ∆ c the core of ∆ . Following [63], an algebra B is said to be a generalized double tilted algebra if the following conditions are satisfied: (1) Γ B admits a component C with a faithful multisection ∆ . In particular, we obtain the following characterization of tame generalized double tilted algebras.  We end this section with an example of a tame generalized double tilted algebra, illustrating the above considerations.
Example 5.8. Let B = KQ/I, where Q is the quiver and I is the ideal of KQ generated by all paths of Q of length 2 (see [63,Example 4.3]). Then B is a tame generalized double tilted algebra of global dimension 4 and Γ B admits a generalized standard component C of the form with a faithful multisection ∆ formed by the indecomposable injective modules I 1 , I 2 , I 3 , I 4 , I 5 , I 6 (at the vertices 1, 2, 3, 4, 5, 6), the indecomposable projective modules P 6 , P 7 , P 8 (at the vertices 6, 7, 8), the simple modules S 5 , S 6 , S 7 (at the vertices 5, 6, 7) and the modules P 7 /S 6 , R = τ −1 B S 5 = τ B S 7 . Then the left part ∆ l of ∆ consists of the modules I 1 , I 2 , I 3 , I 4 , S 5 , P 6 and is a section of the preinjective connecting component of the tame tilted algebra B (l) being the convex subcategory of B given by the vertices 1, 2, 3, 4, 5, 6. The right part ∆ r of ∆ consists of the modules I 6 , S 7 , P 8 and is a section of the postprojective connecting component of the tame tilted algebra B (r) being the convex subcategory of B given by the vertices 6, 7, 8. Moreover, the core ∆ c of ∆ consists of the modules S 6 , P 7 , P 7 /S 6 , R, I 5 and is the cyclic part of the Auslander-Reiten quiver Γ B (c) of the representation-finite convex subcategory B (c) = supp ∆ c of B given by the vertices 5, 6, 7. We also note that every module in ind B belongs to one of its full subcategories ind B (l) , ind B (c) , or ind B (r) .

Cyclic components of cycle-finite algebras
Let A be an algebra. We denote by c Γ A the translation subquiver of Γ A , called the cyclic part of Γ A , obtained by removing from Γ A all acyclic modules and the arrows attached to them. The connected components of c Γ A are said to be cyclic components of Γ A (see [50]). The following result from [50, Proposition 5.1] will be very useful.

Proposition 6.1. Let A be an algebra and X, Y be two cyclic modules of Γ A . Then X and Y belong to the same cyclic component of Γ A if and only if there is an oriented cycle in Γ A passing through X and Y .
Moreover, we have the following property of the support algebras of cyclic components of the Auslander-Reiten quivers of cycle-finite algebras (see [49]).

Proposition 6.2. Let A be a cycle-finite algebra, Γ a cyclic component of Γ A and B = suppΓ . Then B is a convex subcategory of A.
Let A be an algebra and C be a component of Γ A . We denote by l C the left stable part of C , obtained by removing from C the τ A -orbits containing projective modules, and by r C the right stable part of C , obtained by removing from C the τ A -orbits containing injective modules. We note that if C is an infinite component of Γ A then l C or r C is not empty.
The following theorem from [49,Theorem 1] describes the supports of infinite cyclic components of the Auslander-Reiten quivers of cycle-finite algebras. The following example illustrates the above theorem.
and I is the ideal in the path algebra KQ of Q generated by the elements αβ − σ γ, ξ η − µν, πλ − ξ ηραβ , ρϕ, ψρ, jl, dc, ed, gd, hg, h f , ih. Then A is a cycle-finite and Γ A admits a component C of the form The cyclic part Γ of C is obtain by removing from C the (directing) modules S 12 , S 17 , S 18 , P 17 and the arrows attached to them. The maximal cyclic coherent part Γ cc of Γ is the full translation subquiver of C obtained by removing the modules We claim that B is a tame generalized multicoil algebra. Consider the path algebra C = K∆ of the full subquiver of Q given by the vertices 4, 5, 6, 7, 8, 9. Then C is a hereditary algebra of Euclidean type D 5 , and hence a tame concealed algebra. It is know that Γ C admits P 1 (K)-family T C λ , λ ∈ P 1 (K), of pairwise orthogonal generalized standard stable tubes, having a unique stable tube T λ of rank 3 with the mouth formed by the modules S 6 = τ C S 7 , S 7 = τ C E, E = τ C S 6 , where E is the unique indecomposable C-module with the dimension vector dim E = where P B = P B (l) , Q B = Q B (r) , and C B is the P 1 (K)-family C B λ , λ ∈ P 1 (K), of pairwise orthogonal generalized multicoils such that C B 1 = Γ * and C B λ = T C λ for all λ ∈ P 1 (K) \ {1}. Hence Γ A is of the form In particular, A is a cycle-finite algebra with (rad ∞ (mod A)) 3 = 0.
The following theorem from [49,Theorem 2] describes the supports of finite cyclic components of the Auslander-Reiten quivers of cycle-finite algebras. We note that if the core of an almost cyclic component of an Auslander-Reiten quiver Γ A is not empty that it contains a projective module and an injective module. Then we obtain the following consequence of Theorem 6.5. Corollary 6.6. Let A be a cycle-finite algebra. Then the number of finite cyclic components of Γ A is bounded by the rank of K 0 (A).
Observe also that for a cycle-finite algebra A of infinite representation type there are infinitely many (infinite) cyclic components of Γ A , since Γ A contains infinitely many stable tubes (see Theorems 3.1 and 3.3).

The structure theorems
Let A be a cycle-finite algebra and X a module in ind A. Then X is a directing module if and only if X is an acyclic vertex of Γ A . Hence X is nondirecting if and only if X belongs to a cyclic component of Γ A . Then the following structure theorem is a direct consequence of Theorems 5.3, 6.3 and Propositions 6.1, 6.2. n of Euclidean types. Moreover, all representation-infinite tilted algebras of Euclidean types and all tubular algebras are tame generalized multicoil algebras. Therefore, applying Theorem 4.8, we obtain the following completion to Theorem 7.1.

Theorem 7.2. Let A be a cycle-finite algebra.
Then there exist tame generalized multicoil quotient algebras B 1 , . . . , B q of A whose indecomposable finite dimensional modules exhaust all but finitely many isoclasses of modules in ind A.
Moreover, we have the following information on nondirecting indecomposable modules over cycle-finite algebras. Theorem 7.3. Let A be a cycle-finite algebra. Then there exist tame generalized multicoil quotient algebras B 1 , . . . , B q of A such that all but finitely many isomorphism classes of nondirecting modules in ind A belong to generalized multicoils of the Auslander -Reiten quivers Γ B 1 , . . . ,Γ B q of B 1 , . . . , B q . We exhibit an example of a cycle-finite algebra having a nongeneralized standard Auslander-Reiten component.
I the ideal in the path algebra KQ of Q generated by the elements γα, σ β , ηγ, ξ γ, δ ξ , δ η, and A = KQ/I the associated bound quiver algebra. Then A is a cycle-finite algebra whose Auslander-Reiten quiver Γ A has the disjoint union form where T λ , λ ∈ P 1 (K), is the family of stable tubes of rank 1 over the Kronecker path algebra H = K∆ given by the subquiver ∆ of Q formed by the arrows ξ and η, and C is the following glueing of the preprojective component P(H) and the Observe that the simple module S 3 is the socle of the injective module I 3 and the canonical monomorphism S 3 → I 3 belongs to rad ∞ (mod A), because it factors through any stable tube T λ . Hence C is not generalized standard. On the other hand, every cycle in ind A consists of modules of a stable tube T λ and hence is finite, be-cause the stable tubes T λ are generalized standard. Therefore, A is a cycle-finite algebra.
We end this section with an example of an algebra A with partially ordered Auslander-Reiten components of Γ A which is not cycle-finite.
Example 7.5. Let Q be the quiver I the ideal in the path algebra KQ of Q generated by the elements δ α, δ β , ξ η and ηγ, and A = KQ/I the associated bound quiver algebra. Then A is a tame algebra whose Auslander-Reiten quiver Γ A has the disjoint union form is a family of pairwise orthogonal generalized standard stable tubes, and C is a component of the form below [36,Example2.6]). Then we obtain that the components of Γ A are partially ordered in ind A. In particular, we conclude that for every cycle X 0 → X 1 → · · · → X r−1 → X r = X 0 in ind A, all modules X 0 , X 1 , . . . , X r−1 belong to the same component of Γ A . On the other hand, we have in ind A a cycle consisting of modules from C and with f 3 in rad ∞ (mod A), because there is no path in C from P 6 to I 3 . Therefore, A is not a cycle-finite algebra.

Discrete indecomposable modules
The aim of this section is to establish a common bound on the number of discrete indecomposable modules of a fixed dimension vector over a cycle-finite algebra. Proof. Without loss of generality we may assume that A is the support algebra of Γ . Let Γ be an arbitrary generalized multicoil of Γ A which is generalized standard. We shall prove our claim by induction on the number m of admissible operations which we have to do on a finite family T 1 , T 2 , . . . , T s of generalized standard stable tubes in order to obtain the generalized multicoil Γ . If m = 1, then we can only do the admissible operation (ad 1) or (ad 1 * ), so s = 1. In this case our statement follows from [46,Proposition 4.1].
Let m > 1. If the mth admissible operation is of type (ad 1), then by definition of (ad 1) we have dimV = dimW for any modules V,W ∈ {Z i j , X ′ i }, i ≥ 0, 1 ≤ j ≤ t such that V = W . Therefore, the number of indecomposable modules with the same dimension vector does not change. If it is of type (ad 1 * ) then the situation is the same. If the mth admissible operation is of type (ad 2), then in the sequence of earlier m − 1 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), and the operations done between these two must not affect the support of Hom A (X, −) restricted to the generalized multicoil containing X. Let t be the parameter of such operation (ad 1 * ) or like in definition of (fad 1 * ). Note that in general, in the sequence of earlier m − 1 admissible operations can be an operation of type (ad 5) which contains an operation (fad 4) which gives rise to the pivot X of (ad 2) but from Lemma [51, Lemma 3.10] this case can be reduced to (ad 5 * ) which contains an operation (fad 1 * ). Moreover, from [46,Lemma 3.3] we know that, for a fixed e ∈ K 0 (A), each ray and coray in Γ contains at most one module Z with dim Z = e. Therefore, we get that the number of new indecomposable modules with the same dimension vector is at most t + 1 while after applying operations (ad 1 * ) and (ad 2) or (fad 1 * ) and (ad 2) the number of new vertices in ordinary quiver of A increases by t + 2. If the mth admissible operation is of type (ad 2 * ), then the proof is dual. If the mth admissible operation is of type (ad 3), then in the sequence of earlier m − 1 admissible operations, there must be at least one operation of type (ad 1 * ) or (ad 5 * ) which contains the operation (fad 1 * ) which gives rise to the pivot X of (ad 3) and to the modules Y 1 ,Y 2 , . . . ,Y t in the support of Hom A (X, −) restricted to the generalized multicoil containing X. The operations done after must not affect this support. Again, in general, in the sequence of earlier m − 1 admissible operations can be an operation of type (ad 5) which contains an operation (fad 4) which gives rise to the pivot X of (ad 3) but from [51,Lemma 3.10] this case can be reduced to (ad 5 * ) which contains an operation (fad 1 * ). Suppose that we had r such consecutive operations of types (ad 1 * ) or (fad 1 * ), the first of which had X t as a pivot, and these admissible operations built up a branch L in A with points a, a 1 , a 2 , . . . , a t in Q A , so that X t−1 and Y t are the indecomposable injective A-modules corresponding respectively to a and a 1 , and both Y 1 and τ −1 A Y 1 are coray modules in the generalized multicoil containing the (ad 3)-pivot X. Again, from [46,Lemma 3.3] we know that, for a fixed e ∈ K 0 (A), each ray and coray in Γ contains at most one module Z with dim Z = e. Therefore, we get that the number of new indecomposable modules with the same dimension vector is at most t + 1 while after applying r consecutive operations of types (ad 1 * ) and (ad 3) or r consecutive operations of types (fad 1 * ) and (ad 3) the number of new vertices in ordinary quiver of A increases by t + 2. If the mth admissible operation is of type (ad 3 * ), then the proof is dual. If the mth admissible operation is of type (ad 4), then A is the algebra obtained from another one, say A ′ , by applying this admissible operation with pivot X and the begin Y 1 of a finite sectional path Y 1 → Y 2 → · · · → Y t . Note that this finite sectional path is the linearly oriented quiver of type A t and its support algebra Λ (given by the vertices corresponding to the simple composition factors of the modules Y 1 ,Y 2 , . . . ,Y t ) is a tilted algebra of the path algebra D of the linearly oriented quiver of type A t . From [65, (4.4)(2)] we know that Λ is a bound quiver algebra given by a truncated branch in x, where x corresponds to the unique projective-injective D-module. Moreover, the modules Y 1 ,Y 2 , . . . ,Y t are directing in Γ A ′ . Again, from [46,Lemma 3.3] we know that, for a fixed e ∈ K 0 (A), each ray and coray in Γ contains at most one module Z with dim Z = e. Therefore, we get that the number of new indecomposable modules with the same dimension vector is at most t + r + 1 while after applying operations which give rise to the finite sectional path Y 1 → Y 2 → · · · → Y t and (ad 4) the number of new vertices in ordinary quiver of A increases by t + r + 2. If the mth admissible operation is of type (ad 4 * ), then the proof is dual. There remains to consider the case where the mth admissible operation is of type (ad 5). Since in the definition of admissible operation (ad 5) we use the finite versions (fad 1), (fad 2), (fad 3), (fad 4) of the admissible operations (ad 1), (ad 2), (ad 3), (ad 4) and the admissible operation (ad 4), we conclude that the lemma follows from the above considerations. If it is of type (ad 5 * ), then the proof is dual and this finishes the proof of the lemma. ⊓ ⊔ The following proposition is essential for the proof of Theorem 8.3. Proof. It is known that an indecomposable A-module M in Γ A which does not lie on oriented cycle in Γ A is uniquely determined by [M]. Moreover, if T is a stable tube in Γ A then the support of T is a tame concealed or tubular convex subcategory of A. Hence, for any indecomposable A-module X lying in a stable tubes of rank one, we have q A (dim X) = χ A (dim X) = 0. Let d be a nonnegative vector in K 0 (A) such that there exists a nondirecting, discrete indecomposable A-module X with dim X = d. Then X belongs to a generalized multicoil Γ of Γ A . Assume first that Hom A (P, X) = 0 for some indecomposable projective module in Γ . Then it follows from the proof of [46, Proposition 3.5] that any indecomposable A-module Y with dimY = d also lies in Γ . Applying now Lemma 8.1 we conclude that the number of isomorphism classes of indecomposable A-modules Z with dim Z = dim X = d is bounded by n − 1. We get the same statement in the case when Hom A (X, I) = 0 for an indecomposable injective module I in Γ . Note that different tame concealed algebras and different tubular algebras give modules in A with different dimension vectors. Hence, it remains to consider the case when the support of X is contained in a convex subcategory, say C, which is tame concealed or tubular. Then X belongs to a P 1 (K)-family T = (T λ ) λ ∈P 1 (K) of standard stable tubes of Γ C . Moreover, if Z is a indecomposable A-module with dim Z = dim X = d then Z is a C-module and lies in one of the tubes T λ (see [65] or [75]). Denote by m the rank of K 0 (C), and by r λ the rank of the tube T λ , λ ∈ P 1 (K). Then the following equality holds ∑ λ ∈P 1 (K) (r λ − 1) = m − 2 (see [75]). Further, if Y ∈ T λ and Z ∈ T µ are two nonisomorphic modules in T with dimY = dim Z then the quasi-length of Y is divisible by r λ and the quasi-length of Z is divisible by r µ . We note that then q A (dimY ) = q C (dimY ) = χ A (dimY ) = 0 and q A (dim Z) = q C (dim Z) = χ A (dim Z) = 0, since gl dimC ≤ 2. Now a simple inspection of tubular types of tame concealed and tubular algebras shows that, if λ 1 , . . . , λ t are all indices λ ∈ P 1 (K) with r λ = 1, then r λ 1 + . . . + r λ t ≤ m + 2 ≤ n + 2.

Homological properties of indecomposable modules
Let A be an algebra and X be a nonprojective module in ind A. Then we have an almost split sequence . . ,Y s(X) indecomposable modules. We note that τ A = DTr , where D = Hom K (−, K) is the standard duality on mod A and the transpose Tr X of a module X in ind A is the cokernel of the homomorphism Hom A ( f , A) in mod A op associated to a minimal projective presentation P 1 Hence τ A is a homological operator, and s(X) is a numerical homological invariant of a nonprojective module X in ind A.
We also recall that for an indecomposable nonsimple projective-injective module P in mod A there is a canonical (up to isomorphism) almost split sequence having P as a middle term, namely 0 → radP → P ⊕ radP/socP → P/socP → 0 (see [1,Proposition IV.3.11]).
The following theorem has been proved by Bautista and Brenner in [12] (see also [44]). Theorem 9.1. Let A be an algebra of finite representation type and X a nonprojective module in ind A. Then (ii) If s(X) = 4, then one of the indecomposable middle terms Y i of an almost split sequence for X is projective-injective.
It has been conjectured by Brenner that, for any tame algebra A and every nonprojective module X in ind A, we have s(X) ≤ 5. It is still an open problem. The following theorem proved in [60,Theorem 3] by the second named author and Takane confirms the Brenners's conjecture, and extends Theorem 9.1, to cycle-finite algebras. It has been proved by Ringel [65, (2.4) (8)] that, for any directing indecomposable module X over an algebra A, we have End A (X) ∼ = K and Ext r A (X, X) = 0 for r ≥ 0. On the other hand, by [52,Theorem B], for a tame generalized multicoil algebra A and an arbitrary module X in ind A, we have dim K Ext 1 A (X, X) ≤ dim K End A (X) and Ext r A (X, X) = 0 for r ≥ 2. Hence, applying Theorems 6.3 and 6.5 (and Corollary 6.6) we obtain the following theorem. Theorem 9.3. Let A be a cycle-finite algebra. Then for all but finitely many isomorphism classes of module X in ind A we have dim K Ext 1 A (X, X) ≤ dim K End A (X) and Ext r A (X, X) = 0 for r ≥ 2. We conclude from the above theorem that, for a cycle-finite algebra A and all but finitely many isomorphism classes X in ind A, the Euler form

Theorem 9.2. Let A be a cycle-finite algebra and X a nonprojective module in ind A. Then
of X is defined and is nonnegative. For A a coherent cycle-finite algebra, it is the case for all modules X in ind A. It is known from [32, Theorem 2.3] and [63,Theorem 3.4] that if A is a quasitilted algebra or a generalized double tilted algebra then, for all but finitely many isomorphism classes of modules X in ind A, we have pd A X ≤ 1 or id A X ≤ 1. Moreover, it has been conjectured in [79] that the converse implication also holds.
We end this section by the following recent result by the third named author and Skowyrski [81] which confirms this conjecture for cycle-finite algebras. Theorem 9.4. Let A be a cycle-finite algebra such that pd A X ≤ 1 or id A X ≤ 1 for all but finitely many isomorphism classes of modules in ind A. Then A is a tame quasitilted algebra or a tame generalized double tilted algebra.

Geometric properties of indecomposable modules
The aim of this section is to present some results describing geometric properties of indecomposable modules over cycle-finite algebras.
Let A be an algebra and A ∼ = KQ/I its bound presentation. Then I is an admissible ideal in the path algebra KQ of Q generated by a finite system of forms ∑ 1≤ j≤t λ j α m j , j . . . α 1, j (called K-linear relations), where λ 1 , . . . , λ t are elements of K and α m j , j , . . . , α 1, j , 1 ≤ j ≤ t, are paths of length ≥ 2 in Q having a common source and a common end. Denote by Q 0 the set of vertices of Q, by Q 1 the set of arrows of Q, and by s, e : Q 1 → Q 0 the maps which assign to each arrow α 1 its source s(α) and its end e(α). The category modA of finite dimensional right Amodules is equivalent to the category rep K (Q, I) of all finite dimensional representations V = (V i , ϕ α ) i∈Q 0 ,α∈Q 1 of Q, where V i , i ∈ Q 0 , are finite dimensional K-vector spaces and ϕ α : V s(α) → V e(α) , α ∈ Q 1 , are K-linear maps satisfying the equations ∑ 1≤ j≤t λ j ϕ α m j , j . . . ϕ α 1, j = 0 for all K-linear relations ∑ 1≤ j≤t λ j α m j , j . . . α 1, j ∈ I (see [1]). Fix now a vector d = (d i ) i∈Q 0 ∈ K 0 (A) = Z Q 0 with nonnegative coordinates. Denote by mod A (d) the set of all representations V = (V i , ϕ α ) in rep K (Q, I) with V i = K d i for all i ∈ Q 0 . A representation V in mod A (d) is given by d e(α) × d s(α)matrices V (α) determining the maps ϕ α : K s(α) → K e(α) , α ∈ Q 1 , in the canonical bases of K d i , i ∈ Q 0 . Moreover, the matrices V (α), α ∈ Q 1 , satisfy the relations ∑ 1≤ j≤t λ j V (α m j , j ) . . .V (α 1, j ) = 0 for all K-linear relations ∑ 1≤ j≤t λ j α m j , j . . . α 1, j ∈ I. Therefore, mod A (d) is a closed subset of A(d) = ∏ α∈Q 1 K d e(α) ×d s(α) in the Zariski topology, and so mod A (d) is an affine variety. We note that mod A (d) is not necessarily irreducible. The affine (reductive) algebraic group G(d) = ∏ i∈Q 0 GL d i (K) acts on the variety mod A (d) by conjugation (gV )(α) = g e(α) V (α)g −1 s(α) for g = (g i ) ∈ G(d), V ∈ mod A (d), α ∈ Q 1 . We shall identify an In general, we have the following relations between the introduced orders. For modules M and N in the module variety mod A (d) the following implications hold Unfortunately, the reverse implications are not true in general, and it would be interesting to find out when there are true.
The following result of Zwara from [89] gives a combinatorial description of degenerations for modules over algebras of finite representation type. We also exhibit the following results from [85] and [86] on degenerations of modules from the additive categories of generalized standard Auslander-Reiten components. Observe also that, if A is a cycle-finite algebra and M < deg N, then M and N belong to the same cyclic component of Γ A . The degenerations of modules in the additive categories of generalized multicoils of Auslander-Reiten quivers of algebras were investigated in [47], [48], [83], [85]. Then, using Theorems 6.3 and 6.5, we obtain the following results. The geometry of directing modules over tame algebras has been described in [14]. In particular, we have the following consequence of [14, Theorems 1 and 2]. We refer also to [15] for the geometry of indecomposable modules over tame quasi-tilted algebras, which give information on the geometry of indecomposable nondirecting modules of semiregular tubes of cycle-finite algebras (see Theorems 3.6 and 3.7).
We end this section with the following consequence of [52, Theorems A and B] and Theorem 6.3, extending [57, Theorem A] from strongly simply connected algebras of polynomial growth to coherent cycle-finite algebras.
We note that there are indecomposable modules M over coherent cycle-finite algebras A with arbitrary large χ A (dim M) (see [58, (5.3)]).