COVERINGS OF TAME BOXES AND ALGEBRAS

YURI A. DROZD AND SERGEI A. OVSIENKO

Introduction

Coverings of algebras were first considered by Riedtmann [22] and proved their efficiency in plenty of questions of the representation theory (cf., e.g., [5, 15, 4] and others). In particular, they played a crucial role in the criterion for an algebra to be representation-finite (cf. [4]), as it had been known that an algebra is representation-finite if and only if so is its covering [15]. Certainly, one would like to apply coverings in the representation-infinite case too. It is well known that among the representation-infinite algebras there are two quite different types: tame and wild (cf. [10, 11, 7]), and the question naturally arises whether an algebra is tame if and only if so is its covering. Unfortunately, rather simple examples shows that it is not always the case (cf. [17]). Nevertheless, there was some evidence that this claim would remain valid under some conditions imposed on the covering; for instance, if one supposed the Galois group of the covering to be torsion free (for partial results in this direction cf. [8, 9]).

Our article is devoted to the proof of this conjecture both for finite dimensional algebras and for a wide class of “matrix problems” (representations of boxes or bimodules). Certainly, dealing with coverings, one should consider not only algebras but also the so called locally bound categories; they are, in some sense, “finite dimensional algebras with infinitely many objects.” The main tool is, just as in [11, 7], a reduction algorithm allowing to “simplify” representations and to prove some basic results by induction. As it often happens, to construct such an algorithm we had to enlarge the considered class of boxes (“quasitriangular boxes,” cf. Section 8), though we have no idea whether the new class could be useful anywhere but in inductive proofs.

Moreover, this procedure gives a rather complete description of the relations between the representation categories of an algebra (or a box) and of its Galois covering. Remind that in the representation-finite case all representations of the covered algebra are “direct images” of those of the covering (cf. [5]). In the tame case it is seldom so; maybe, the unique exception is the case of locally support finite coverings (cf. [8]).

Usually, there are some infinite dimensional representations of the covering which produce 1-parametre families of representations of the covered algebra. So, one has at least two sorts of representations: those obtained from finite dimensional representations of the covering and those belonging to the abovementioned families (we denote them ind_{0} Λ and ind_{1} Λ ). The main theorem can be stated as follows (cf. Theorem 9.1 and Corollary 9.7 for more details)^{1}:

Let Π : Λ̃ → Λ be a Galois covering of locally finite dimensional categories (over an algebraically closed field) with a torsion free Galois group G . Then:

1. Λ is tame if and only if so is Λ̃ .

2. If Λ and Λ̃ are tame, then:

• Every indecomposable representation of Λ belongs either to ind_{0} Λ or to ind_{1} Λ .

• ind Λ̃ is a Galois covering of ind_{0} Λ with the same Galois group G .

• The Auslander-Reiten quiver of Λ is a disjoint union of that of ind_{0} Λ (which is the orbit quiver of G on the Auslander-Reiten quiver of Λ̃ ) and that of ind_{1} Λ (which is a disjoint union of homogeneous tubes).

One can conjecture that the first claim of this theorem also remains valid when G does not contain elements of the order equal to the characteristic of the ground field, though the relations between indecomposable modules become more sofisticated in this case.

1. Categories

Through the whole paper we suppose that all our categories and algebras are categories and algebras over some fixed field k and all functors and homomorphisms are k-linear. We shall write Hom , ⊗ , dim, etc. instead of Hom_{k} , ⊗_{k} , dim_{k}, etc. For any subset S ⊆ Mor A and any two objects x, y ∈ Ob A denote S(x, y) = S ∩ A(x, y).

Remind that a category A is said to be fully additive [11] if it is additive and any idempotent in A splits (i.e., corresponds to a direct decomposition). On the other hand, call a category A basic if it satisfies the following conditions:

• all its objects are pairwise non-isomorphic;

• for each object x there are no non-trivial idempotents in A(x, x).

1Note that the proof of this theorem announced in [13] (cf. also [12]) was incomplete.

A full subcategory S ⊆ A is called a skeleton of A if it is basic and each object x ∈ A is isomorphic to a direct summand of a (finite) direct sum of some objects of S.

A fully additive category add A is called a fully additive hull of the category A if A is a full subcategory of add A and each object of add A is isomorphic to a direct summand of a direct sum of objects of A. It is known (and quite obvious) that each category A has a fully additive hull. Moreover, all such hulls are equivalent and for each functor F : A → C, where the category C is fully additive, there exists a unique (up to isomorphism) functor add A → C extending F. We denote this extension by F too. We also write A(x, y) instead of (add A)(x, y) for objects x, y from add A.

Call a category A rigid if add A is a unique direct decomposition category, i.e., any object of it can be decomposed into a direct sum of indecomposable ones and if

x_{1} ⊕ x_{2} ⊕ … ⊕ x_{n} ≃ y_{1} ⊕ y_{2} ⊕ … ⊕ y_{m}

with indecomposable objects x_{i} and y_{j} then n = m and x_{i} ≃ y_{i} (up to a permutation of indices). It is evident that if A is a rigid category then add A has a skeleton and the last one is unique up to isomorphism; thus, we may (and will) denote it by Sk A. Obviously, Sk A is also rigid. Call the objects of a skeleton of a rigid category A its vertices and denote their set by Ver A.

There are two important examples of rigid basic categories. The first one is that of local categories. Namely, a category A is said to be local if its objects are pairwise non-isomorphic and for each object x ∈ Ob A its endomorphism algebra A(x, x) is local. The well-known Adzumaya-Krull-Schmidt theorem implies that any local category is rigid (cf. e.g.

[2] ). Suppose now that A is locally finite dimensional, i.e., A(x, y) is finite dimensional vector space for each x, y. Then, in particular, A(x, x) is a finite dimensional algebra. Hence, in add A the object x splits into a direct sum of objects with local endomorphism algebras. Thus, add A has a local skeleton Sk A and A is rigid. In particular, any finite dimensional algebra is a rigid category.

For a local category A define its radical rad A as the set of all non-invertible morphisms. It is an ideal in A. Denote by rad^{∞} A the intersection ⋂_{k=1}^{∞}(rad A)^{k}. If a category A has a local skeleton one can also define its radical: by definition, a : x → y belongs to rad(x, y) if and only if its components with respect to some (hence, to any) decompositions x ≃ ⊕_{i}x_{i}, y ≃ ⊕_{j}y_{j}, where x_{i}, y_{j} ∈ Sk A, belong to rad(x_{i}, y_{j}) .

Another example of rigid categories is that of free categories. Let Γ be a graph (in the sense of [19], i.e., oriented and, maybe, with loops and multiple arrows). Define the free category kΓ generated by Γ in the following way. Its object set coincides with the set Ver Γ of vertices of Γ. The morphisms f : x → y are defined as formal (finite) sums

∑_{i} λ_{i}p_{i}

where λ_{i} ∈ k and p_{i} are some paths starting at x and ending at y. Remind that such a path is a formal product a_{1}a_{2} … a_{l}, where a_{k} : x_{k} → x_{k−1} are some arrows of Γ and x_{l} = x , x_{0} = y. The number l is called the length of this path. If x = y, it is allowed that l = 0 (the empty path at the vertex x). To define the products of morphisms, one has only to do it for paths. But in this case it can be defined as their concatenation.

Note that a free category in this sense is linear. We shall not consider free non-linear categories defined, e.g., in [19]. Surely, if Γ has only 1 vertex, the free category kΓ coincides with the free algebra generated by the set of arrows of Γ. It is known that any free category is rigid (and, of course, basic), cf., e.g., [24, 14], or [2, 6] for the case of free algebras. Moreover, the free categories generated by two graphs are isomorphic if and only if these graphs are isomorphic. A category of the form add kΓ will be called a free additive category (remark that the unique decomposition implies that it is indeed the least additive category containing kΓ). The set of arrows of the graph Γ is often called the set of free generators for kΓ (or for add kΓ).

A basic category B is said to be trivial if B(x, x) = k for any object x ∈ Ob B and B(x, y) = 0 for any two different objects x ≠ y. A category A is said to be trivial if it possesses a trivial skeleton. Surely, any trivial category A is rigid and its additive hull add A is also trivial. For any basic category B one can define its trivial part as the basic trivial category B^{∅} having the same objects as B. If A is any rigid category with a skeleton A_{0}, fix for every x ∈ Ob A an isomorphism ϕ_{x} : x →^{∼} ⨁_{k} x_{k}, where x_{k} ∈ Ob A_{0}, and define the trivial part of A to be the trivial category A^{∅} with the same objects as A, with the skeleton A_{0}^{∅} and such that all isomorphisms ϕ_{x} belong to A^{∅} too. As A is rigid, its trivial part does not depend (up to isomorphism) on the choice of the skeleton A_{0} and the decompositions ϕ_{x}.

A basic category B is said to be minimal if B(x, y) = 0 for any two distinct objects x, y, while B(x, x) either coincide with k or is isomorphic to a rational algebra, i.e., a k-algebra of the form k[t, f(t)^{−1}] for some non-zero polynomial f(t). In the latter case the morphism a : x → x corresponding to the element t will be called the loop at the vertex x and the polynomial f(t) will be denoted by f_{a}. If this polynomial is non-constant, call the vertex x and the loop a marked vertex and marked loop respectively. A category A is said to be minimal if it possesses a minimal skeleton. The marked vertices and the marked loops of this skeleton are also called the marked vertices and marked loops of the category A.

Let Vec be the category of vector spaces over k. The functors M : A → Vec are called A-modules, or, more precisely, left A-modules. If x is an object of A, the elements of M(x) are called the elements of the module M at the object x. We shall write au instead of M(a)(u) for u ∈ M(x) , a ∈ A(x, y) (then au ∈ M(y)). The category of all A-modules is denoted by A-Mod. We also define right A-modules as A^{∘}-modules, where A^{∘} is the category opposite (or dual) to A. In this case we write va for an element v ∈ N(y) of a right module N at the object y and a morphism a ∈ A(x, y) = A^{∘}(y, x). The category of right A-modules is denoted by Mod-A. Given a module M, the set ⨆_{x∈Ob A} M(x) is called the set of elements of M and denoted by El M. We may, of course, identify the categories of modules over A and over add A, and we will usually do so.

Important examples of left (right) A-modules are the modules A^{x} = A(x, ) (respectively, A_{x} = A( , x)). We call this module the principal left (right) module corresponding to the object x^{2}. The direct sums of principal modules are called free modules (if A is an algebra, i.e., a category with only 1 object, they are usual free A-modules).

If A and B are two categories, a (bilinear) bifunctor T : A^{∘}×B → Vec is called an A-B-bimodule. If t ∈ T (x, y), we say that t is an element of T with the source x and the target y and write bta instead of T (a, b)(t) for a ∈ A(x′, x) , b ∈ B(y, y′) (then bta ∈ T (x′, y′)). As before, A-B-bimodules are the same as (add A)-(add B)-bimodules. If A = B we say “A-bimodule” instead of “A-A-bimodule”. Again we can consider the set of elements of a bimodule: El T = ⨆_{x∈Ob A, y∈Ob B} T (x, y). Surely, any category A can be considered as A-bimodule mapping a pair of objects (x, y) to A(x, y). Call it the regular A-bimodule. On the other hand, having any right A-module N and any left A-module M, we can construct the bimodule N ⊗ M putting (N ⊗ M)(x, y) = N(x) ⊗ M(y). In particular, the bimodule of the form A_{x}^{y} = A_{x} ⊗ A^{y} will be called the principal bimodule with the source y and the target x. The direct sums of principal bimodules are called free bimodules. Note that the regular bimodule, as a rule, is not free. On the other hand, in contrast with algebras, there is no natural notion of the regular module over a category having more than 1 object. If V = ⨁_{j} A_{x_{j}}^{y_{j}}, then the set of its elements e_{j} = 1_{x_{j}} ⊗ 1_{y_{j}} generates V as A-bimodule. Moreover, each element of V can be uniquely written as ∑_{ij} a_{ij}e_{j}b_{ij} for a_{ij}, b_{ij} ∈ A. That is why this set is called the set of free generators for the bimodule V.

2 The usual name for these modules – “representable” – is not very convenient in the representation theory.

To any A-bimodule T one can associate a new category A[T ], called the tensor category of this bimodule, in the following way. First, construct the tensor powers T^{⊗m} of this bimodule putting

T^{⊗0} = A , T^{⊗1} = T and T^{⊗m+1} = T ⊗_{A} T^{⊗m} .

Now define A[T ] as the category whose object set coincides with that of A, but

A[T ](x, y) = ∐_{m=0}^{∞} T^{⊗m}(x, y)

with the obvious multiplication. If T is a free bimodule with a set S of free generators, call A[T ] freely generated over A (by the set S). The last set is called a set of free generators for A[T ] (over A). In particular, if A is a trivial category, then A[T ] is a free category. If A is a minimal category, call A′ = A[T ] a semi-free category with the set of semi-free generators S ∪ L, where L is the set of loops of A. Usually, the set S ∪ L is denoted by Arr A′ and called the set of arrows of A′. The marked vertices and marked loops of this category coincide, by definition, with those of the minimal category A. The set of marked objects will be denoted by Obm A and the set of marked loops by Lom A. The polynomial corresponding to a marked loop a will be denoted by mk a. Hence, the semi-free category A′ is also given by a graph Γ = Γ(A′) (whose vertices are those of A and whose set of arrows is Arr A′) equipped with the subset Lom Γ ⊆ Arr Γ of marked loops (at most one at each vertex) and the map mk : Lom Γ → k[t]. Call the triple (Γ, Lom Γ, mk) the diagram of the semi-free category A′ and the function mk its marking function. One can easily see that A′ ≃ A ∐^{A^{∅}} A_{1} , where A_{1} is the free category with the set of free generators S and ∐ denotes the amalgamation (or the free product) over the trivial subcategory A^{∅} (cf. [16]).

Define the degree of a morphism from a semi-free category with a fixed set of arrows as follows. Such a morphism is a linear combination of paths of the form p = a_{l} … a_{2}a_{1} , where each a_{i} either belong to the minimal category A or is an arrow, and if a_{k} ∈ A, then neither a_{k+1} nor a_{k−1} belongs to A. The degree of a ∈ A can be defined just as that of a rational function: the degree of the numerator minus that of the denominator. The degree of an arrow, by definition, is 1. Now, deg p is defined as ∑_{k} deg a_{k} and deg a as the maximum of degrees of the paths occurring in a.

Define the category of representations Rep(A, B) of a category A over another category B to be the functor category Func(A, add B). Any representation M : A → add B defines an A-B-bimodule _{M}B mapping a pair (x, y), where x ∈ Ob B , y ∈ Ob A, to B(x, My). On the contrary, if an A-B-bimodule T is such that the B-module T ( , y) is finitely generated and projective for any y ∈ Ob A, then one can easily construct a functor M : A → add B such that the corresponding bimodule _{M}B is isomorphic to T . Thus, we will sometimes identify M with _{M}B. In the same way, the B-A-bimodule B_{M} and the Abimodule _{M}B_{M} are defined. Sometimes we will also write B instead of any of these three bimodules if the real sense of this notation is clear from the context. Remark that, for instance, Rep(A, k) is equivalent to the full subcategory of A-Mod consisting of such modules M that M(x) is finite dimensional for any x ∈ Ob A. Of course, the categories Rep(A, B) and Rep(add A, B) are equivalent and we will often identify them.

2. Boxes

A coalgebra over a category A is, by definition, an A-bimodule V equipped with two bimodule homomorphisms: comultiplication μ : V → V ⊗_{A} V and counit ε : V → A, subject to the usual conditions (cf.

[19] ):

• (μ ⊗ 1)μ = (1 ⊗ μ)μ (co-associativity);

• (ε ⊗ 1)μ = ι_{l} and (1 ⊗ ε)μ = ι_{r} (counit properties),

where ι_{l} : V →^{∼} A ⊗_{A} V and ι_{r} : V →^{∼} V ⊗_{A} A are the natural isomorphisms. For instance, the regular A-bimodule has the natural coalgebra structure with ε = 1 and μ being the natural isomorphism. Call it the regular A-coalgebra. The kernel \overline{V} = ker ε of the counit is also called the kernel of the coalgebra.

Now a box is defined as a pair 𝔄 = (A, V) consisting of a category A and an A-coalgebra V^{3}. In particular, the pair (A, A) (the second component being the regular A-coalgebra) will be called the regular A-box^{4}. The kernel \overline{V} of the coalgebra V is also called the kernel of the box 𝔄.

3 More usual term is “bocs”, but this word seems not existing in any language.

Thus, we propose better to replace it by its existing homonym.

A representation of the box 𝔄 = (A, V) over a category C is defined as a functor M : A → add C. If M′ is another representation, then a morphism φ : M → M′ is, by definition, an homomorphism of the A-C-bimodules φ : V ⊗_{A} M → M′, where the functors are identified with bimodules as above. The set of such morphisms will be denoted by Hom_{𝔄−C}(M, M′). For another morphism ψ : M′ → M′′ their product ψφ is defined as the composition

V ⊗_{A} M ⟶^{μ⊗1} V ⊗_{A} V ⊗_{A} M ⟶^{1⊗φ} V ⊗_{A} M′ ⟶^{ψ} M′′ .

Thus, we obtain the category of representations Rep(𝔄, C) of the box 𝔄 over the category C. The unit morphism 1_{M} of this category is the composition

V ⊗_{A} M ⟶^{ε⊗1} A ⊗_{A} M ⟶^{ι_{l}^{−1}} M .

In particular, if C = Vec we call these representations 𝔄-modules, write Hom_{𝔄}(M, N) for the set of morphisms of two 𝔄-modules and 𝔄-Mod for the corresponding category. It is quite obvious that if 𝔄 = (A, A) is the regular A-box then the representation category Rep(𝔄, C) coincides with the functor category Func(A, add C). Note also that if C is an algebra (i.e. a category with 1 object), then add C can be considered as the category of finitely generated projective C-modules. In particular, add k can be identified with the category vec of finite dimensional vector spaces over k.

It is often convenient to apply the natural isomorphism

Hom_{A}(V ⊗_{A} M, M′) ≃ Hom_{A−A}(V, Hom(M, N))

and consider a morphism of boxes as a function on El V. In other words, to define a morphism M → N one has to define for every γ ∈ V(x, y) the linear mapping φ(γ) : M(x) → N(y) such that φ(aγb) = N(a)φ(γ)M(b) for all a : y → y′, b : x′ → x.

A system of generators for a box 𝔄 = (A, V) is, by definition, a union A = A_{0} ∪ A_{1}, where A_{0} is a system of generators of the category A and A_{1} that of the kernel \overline{V} of the box. If the category A has a skeleton Sk A denote by Sk V the restriction of the bimodule V on Sk A and by Sk 𝔄 the pair (Sk A, Sk V). Certainly, Sk 𝔄 is again a box, which we call the skeleton of the box 𝔄. Call A locally finitely generated if its skeleton has a system of generators A such that A(x, y) is finite for each pair of objects of Sk A. If, moreover, the set of objects of Sk A is finite, call the box 𝔄 finitely generated.

4 Remark that earlier ([23] or [11]) they used here the term “principal box”, while the term “regular” was used for other purposes, but we think that the new terminology is more convenient.

Suppose now that the category A is rigid. Denote then by rep(𝔄, C) the full subcategory of Rep(𝔄, C) consisting of the representations M such that M(x) = 0 for almost all objects x ∈ Ver A (i.e., all but a finite number of them). In particular, the category of finite dimensional 𝔄-modules 𝔄-mod = rep(𝔄, k) , is defined. The support of a representation M ∈ Rep(𝔄, C) is defined as the set

Supp(M) = { x ∈ Ver A | M(x) ≠ 0 } .

In the case when C is an algebra such that any projective C-module is free of unique rank, define the dimension of a representation M ∈ rep(𝔄, C) as the function

dim(M) : Ver A → ℕ : dim(M)(x) = rk_{C} M(x) .

In the case when Ver A = { x_{1}, x_{2}, …, x_{n} } is finite, this function can be identified with the vector (d_{1}, d_{2}, …, d_{n}), where d_{i} = rk_{C} M(x_{i}).

We call any function d : Ver A → ℕ with a finite support a dimension of representations of the box 𝔄 and denote Dim(𝔄) the set of all such dimensions. For any d ∈ Dim(𝔄) put |d| = ∑_{x∈Ver A} d(x).

Fixing a representative of free C-modules of each given rank r (say, rC), we are able to consider the set rep_{d}(𝔄, C) of all representations of 𝔄 over C of the prescribed dimension d. In particular, if 𝔄 is locally finitely generated then the set rep_{d}(𝔄) = rep_{d}(𝔄, k) can be considered as an algebraic variety over the field k.

A morphism of boxes Φ : 𝔄 → 𝔅, where 𝔄 = (A, V) and 𝔅 = (B, W), is, by definition, a pair Φ = (Φ_{0}, Φ_{1}) where Φ_{0} : A → B is a functor and Φ_{1} : V → W is an homomorphism of A-coalgebras (W is considered as A-coalgebra via its B-coalgebra structure and the functor Φ_{0}). Usually we will write Φ for both Φ_{0} and Φ_{1}. Given such morphism we can evidently construct the inverse image functor

Φ^{∗} : Rep(𝔅, C) → Rep(𝔄, C)

for every category C.

In particular, each representation M ∈ Rep(𝔄, C) can be considered as a morphism from 𝔄 to the regular box over the category add C, whose value on V is the composition M ∘ ε. In this case the inverse image functor is isomorphic to the tensor product functor M ⊗_{C} and we shall often identify them.

Call Φ an equivalence of boxes provided Φ_{0} is an equivalence of categories and Φ_{1} is an isomorphism of A-bimodules. Then Φ^{∗} is an equivalence of their categories of representations over each categoryC.

Suppose given a box 𝔄 = (A, V) and a functor F : A → B. Define a new box 𝔄^{F} = (B, V^{F}), where V^{F} = B⊗_{A}V ⊗_{A}B. The comultiplication in V^{F} is defined as the composition

V^{F} ⟶ B ⊗_{A} V ⊗_{A} V ⊗_{A} B ≃ B ⊗_{A} V ⊗_{A} A ⊗_{A} V ⊗_{A} B ⟶
⟶ B ⊗_{A} V ⊗_{A} B ⊗_{A} V ⊗_{A} B ≃ V^{F} ⊗_{B} V^{F} ,

the first arrow being induced by the comultiplication in V and the second one by the functor F. Of course, the pair (F, F′), where F′ : V → V^{F} is the natural homomorphism, is a morphism of boxes 𝔄 → 𝔄^{F}, which we also denote by F. The following evident “change-of-base theorem” shows the advantage of boxes and is the main tool in the so called “reduction processes”.

Theorem 2.1. In the above situation the functor F^{∗} : Rep(𝔄^{F} , C) → Rep(𝔄, C) is fully faithful for each category C. Hence, it induces an equivalence of Rep(𝔄^{F} , C) and the full subcategory in Rep(𝔄, C) consisting of all functors A → add C which can be factored through F.

Proof. We may suppose that C = add C. For any two representations M, N ∈ Rep(𝔄^{F} , C) we have the natural isomorphism

Hom_{B−C}(V^{F} ⊗_{B} M, N) = Hom_{B−C}(B ⊗_{A} V ⊗_{A} B ⊗_{B} M, N) ≃ ≃ Hom_{A−C}(V ⊗_{A} B ⊗_{B} M, Hom_{B}(B, N)) ≃ Hom_{A−C}(V ⊗_{A} M, N) .

But it is just what we need.

Remark. It is important to notice that even if 𝔄 = (A, A) is a regular box, the box 𝔄^{F} is, as a rule, no more regular.

Call any set ω = { ω_{x} ∈ V(x, x) | x ∈ Ob A, ε(ω_{x}) = 1_{x} } a section of the box 𝔄. If, moreover, μ(ω_{x}) = ω_{x} ⊗ ω_{x} for each x, call this section normal. A box having a normal section is also called normal.

If a section ω in Sk 𝔄 is fixed, one can define the A^{∅}-sub-bimodule V^{∅} ⊆ V. Namely, if x, y ∈ Sk A put V^{∅}(x, x) = kω_{x} and V^{∅}(x, y) = 0 if x ≠ y. Then expand this definition to the whole category A just as it has been done for A^{∅} above. If the section ω is normal, 𝔄^{∅} = (A^{∅}, V^{∅}) is even a sub-box in 𝔄.

One can easily check that given any section ω we have that ∂a = aω_{x} − ω_{y}a ∈ \overline{V} for each a ∈ A(x, y) and ∂v = μ(v) − v ⊗ ω_{x} − ω_{y} ⊗ v ∈ \overline{V} ⊗_{A} \overline{V} for each v ∈ \overline{V}(x, y). We call the mapping ∂ the differential of the box 𝔄 (with respect to the given section). We prolong ∂ to a mapping \overline{V}^{⊗2} → \overline{V}^{⊗3} which maps v ⊗ w to ∂v ⊗ w − v ⊗ ∂w; then one can verify the following rules:

∂^{2}(a) = aδ_{x} − δ_{x}a and ∂^{2}(v) = v ⊗ δ_{x} + δ_{x} ⊗ v

where δ_{x} = μ(ω_{x}) − ω_{x} ⊗ ω_{x}. In particular, if the section is normal, ∂^{2} = 0. Usually one only deals with normal boxes, but this time we need to involve some non-normal ones in the proof.

A box 𝔄 = (A, V) is said to be semi-free if A is a semi-free category, the kernel \overline{V} is a free A-bimodule and ∂a = 0 for all marked loops from A. In this case its set of arrows Arr 𝔄 (or a set of semi-free generators of 𝔄) is, by definition, the union Arr A ∪ Arr V where Arr V is the set of free generators for \overline{V}. The sets of marked objects, marked loops and the function mk for the box 𝔄 are, by definition, those for the category A. Call 𝔄 triangular if there exists a function ν : Arr 𝔄 → ℕ (called triangular height) such that, for each b ∈ Arr 𝔄 ∂b lies in the semifree sub-box generated by all elements a ∈ Arr 𝔄 with ν(a) < ν(b). A normal semi-free box 𝔄 is usually given by its bigraph Γ = Γ(𝔄) having two sorts of arrows: solid, corresponding to Arr A, and dotted, corresponding to Arr V. To define 𝔄 we also have to choose the set Lom Γ of marked loops together with the marking map Lom Γ → k[t] and to prescribe the differential ∂a for each arrow a in such way that ∂^{2} = 0 if we calculate it using the Leibniz rule. Note that while the bigraph Γ is uniquely determined by the semi-free box it is no more true for its differential. If we choose another system of semi-free generators their differentials usually change. Moreover, sometimes we need to choose a “good” system of semi-free generators with respect to the differential, i.e., such one that the formulae for the differential were the simplest (or the most convenient). In particular, the triangularity strongly depends on the choice of generators. So, when we speak about a triangular semi-free box we always mean a fixed choice. Then the definition of degree given above for semi-free categories can be evidently extended to the elements of the bimodule V putting deg v = 1 for any a ∈ Arr \overline{V} and deg ω_{x} = 0 for any object x.

If we fix a set of arrows A = Arr 𝔄 for a semi-free normal box 𝔄 we can associate to it a 2-dimensional cell complex Com(𝔄, A). Its set of vertices is that of the bigraph Γ(𝔄), the 1-dimensional cells (edges) are the elements of A, and 2-dimensional cells are constructed as follows. Let a ∈ A and ∂a = ∑_{p} λ_{p}p where p are some paths in Γ. Then for each path p such that λ_{p} ≠ 0 we attach to Com(𝔄, A) a 2-dimensional cell with the border pa^{−1} where a^{−1} denotes the same edge a but directed oppositely to the corresponding arrow. Now we can also define the fundamental group Gr(𝔄, A, x) of the pair (𝔄, A) with respect to a vertex x as that of the complex Com(𝔄, A). Simple examples show that this complex and its fundamental group depend on the choice of the system of semi-free generators A and not only on the box 𝔄 itself. If this complex is connected, then we may omit the vertex and write Gr(𝔄, A). Surely, this is the case if and only if the corresponding bigraph is connected, and then we call the box 𝔄 connected too.

A box 𝔄 = (A, V) is said to be so-trivial if A is a trivial category. If, moreover, it is regular (i.e., V = A, hence, \overline{V} = 0), it is called trivial ^{5}. For a so-trivial box one can easily reconstruct the module category 𝔄-Mod. Obviously, its indecomposable objects are of the form S_{x} where x runs through Ver A and S_{x} is determined by the formulae:

S_{x}(x) = k and S_{x}(y) = 0 for y ∈ Ver A , y ≠ x .

Any 𝔄-module can be uniquely decomposed into a direct sum of S_{x}. Moreover, Hom_{𝔄}(S_{x}, S_{y}) ≃ DV(x, y), where D denotes the duality functor for vector spaces over k: DX = Hom(X, k), while the multiplication of the homomorphisms is dual to the comultiplication in the coalgebra V.

Analogously, call a box 𝔄 = (A, V) so-minimal (respectively, minimal) provided A is a minimal category (respectively, A is minimal and the box is regular).

3. Reduction algorithms

Let 𝔄 = (A, V) be a box and 𝔄′ = (A′, V′) be its sub-box. Suppose given a functor F′ : A′ → B′. Construct the amalgamation B = A ∐^{A′} B′, or, the same, the couniversal square:

\begin{matrix}A′ & ⟶^{F′} & B′ \\ ↓ & & ↓ \\ A & ⟶^{F} & B \end{matrix}

Consider now the box 𝔄^{F} = (B, V^{F}). Then Theorem 2.1 implies the following corollary.

Corollary 3.1. In the situation above, for each category C the functor F^{∗} : Rep(𝔄^{F} , C) → Rep(𝔄, C) induces an equivalence between the category Rep(𝔄^{F} , C) and the full subcategory Rep(𝔄, C | F′) ⊆ Rep(𝔄, C) consisting of those representations M : A → add C whose restrictions on A′ can be factored through F′.

We shall use the following “elementary cases” of this procedure considered in [11] (cf. also [18, 7]).

5 Again we use the terms different from those of [23] or [11], but we are sure that ours are more convenient.

Example 3.2. 1. Suppose that A′ is freely generated by the unique solid arrow a : x → y with x ≠ y and ∂a = 0. Then take for B′ the trivial category with 3 vertices, { x′, y′, xy }, and for F′ the functor which maps:

x ↦ x′ ⊕ xy , y ↦ xy ⊕ y′ , a ↦ (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix})

(later on we usually write x and y instead of x′ and y′). They say that the box 𝔄^{F} is obtained from 𝔄 by reducing the edge a. In this case one can easily see that Rep(𝔄, Vec | F′) = Rep(𝔄, Vec). Thus, F^{∗} is an equivalence of the module categories 𝔄^{F} -Mod ≃ 𝔄-Mod.

2. Suppose now that 𝔄′ is freely generated by one solid arrow a : x → y and one dotted arrow α : x → y such that ∂a = α (never mind whether x = y or not). Then put B′ = A′/(a) and F′ : A′ → B′ being the natural projection. In this case we have that Rep(𝔄, C | F′) = Rep(𝔄, C) for any category C. Hence, F^{∗} : Rep(𝔄^{F} , C) →^{∼} Rep(𝔄, C). They say that the box 𝔄^{F} is obtained from 𝔄 by deleting the arrow a. We call such an arrow a a superfluous arrow^{6}.

3. At last, let A′ be generated by the unique solid loop (maybe, marked) a : x → x with ∂a = 0. Take for B′ the minimal category containing n+1 vertices { x_{∗}, x_{1}, x_{2}, …, x_{n} } and one marked loop a_{∗} : x_{∗} → x_{∗} with mk(a_{∗}) = (t − λ) mk(a), and put

F′(x) = x_{∗} ⊕ ⨁_{k=1}^{n} kx_{k} and F′(a) = a_{∗} ⊕ ⨁_{k=1}^{n} J_{k} ,

where J_{k} is the Jordan cell of size k with the eigenvalue λ and λ is an element of k such that mk(a)(λ) ≠ 0. Then the Jordan theorem implies that rep(𝔄, k | F′) consists of all representations M such that A^{n} = 0, where A denotes the nilpotent part of M(a − λ). In this case they say that the box 𝔄^{F} is obtained from 𝔄 by reducing the λ-part of the loop a up to degree n. (Note that even if a was not marked, a_{∗} is still marked with mk(a_{∗}) = t − λ.)

Note that in all these examples, if the box 𝔄 was semi-free and normal, so is also the box 𝔄^{F}. Moreover, if 𝔄 was triangular, 𝔄^{F} is triangular too.

Later we need one more “elementary step”.

Example 3.3. Call a line in the category A (or in the box 𝔄) any subcategory L generated by a set of its (distinct) vertices and arrows { x_{k}, a_{k} : x_{k} → x_{k+1} | k ∈ ℤ }. (We often denote this set itself by L and call it a line too.) For any interval I ⊆ ℤ (maybe, infinite), denote by L_{I} the representation of the line L such that

6 a non-regular arrow in the terminology of [18] or [11].

L_{I}(x_{k}) = \begin{cases}k & if k ∈ I \\ 0 & otherwise \end{cases}
L_{I}(a_{k}) = 1 if k, k + 1 ∈ I

The following proposition is an easy exercise in linear algebra.

Proposition 3.4. Any representation of the line L (not necessarily finite dimensional) is a direct sum of representations isomorphic to some of L_{I} .

Suppose now that ∂a_{k} = 0 for all k ∈ ℤ (such a line is said to be minimal). Put A′ = L, take for B′ the trivial category with the set of vertices { x_{ij} | i, j ∈ ℤ, 0 ≤ j − i < n }, and define the functor F′ : A′ → B′ as follows:

F′(x_{k}) = ⨁_{i≤k≤j} x_{ij} and F′(a_{k}) is given by the matrix (a_{i′j′}^{ij}) in which a_{ij}^{ij} = 1 for all possible values of i, j (all other entries of this matrix are automatically 0).

Proposition 3.4 implies that in this case the subcategory Rep(𝔄, Vec | F′) consists of all representations M of 𝔄 such that ∏_{i=0}^{n−1} a_{k+i} = 0 for each k ∈ ℤ. Again, if the box 𝔄 was semi-free, normal, triangular, so is also 𝔄^{F}. We say that 𝔄^{F} is obtained from 𝔄 by reducing the line L up to degree n.

All these algorithms “improve” the representations in the following sense. Let 𝔄 = (A, V) be a semi-free box, M ∈ rep(𝔄, C) , d = Dim M . Denote by n(x, y) the number of elements in Arr A(x, y) and put

||M|| = ||d|| = ∑_{x,y∈Ver A} n(x, y)d(x)d(y) .

Then the following proposition is immediate.

Proposition 3.5. Suppose that the box 𝔅 = 𝔄^{F} is obtained from 𝔄 by one of the following operations:

(i) reduction of an edge a : x → y; (ii) deleting an arrow a : x → y; (iii) reduction of the λ-part of a loop a : x → x; (iv) reduction of a line L = { x_{k}, a_{k} }. Let M ∈ rep(𝔄, C) , N ∈ rep(𝔅, C) be such that M ≃ F^{∗}N . Then ||N|| ≤ ||M||. Moreover, ||N|| < ||M|| if the following conditions hold (in accordance with the numeration of the cases above):

(i) M(x) ≠ 0 and M(y) ≠ 0 ; (ii) M(x) ≠ 0 and M(y) ≠ 0 ; (iii) M(a − λ) is not invertible (in particular, M(x) ≠ 0 ); (iv) M(x_{k}) ≠ 0 and M(x_{k+1} ≠ 0 for at least one k.

4. Tame and wild boxes

From now on we suppose all boxes to be rigid and locally finitely generated; a semi-free box is always supposed to be triangular. Content-Length: 7146 X-Lines: 200 Status: RO

Definition 4.1. 1. We say that a functor F : C → D

• reflects isomorphisms if F (x) ≃ F (y) implies x ≃ y for any objects x, y ∈ C;

• reflects decomposability if an object x ∈ C is decomposable (into a non-trivial direct sum) if and only if F (x) is decomposable in D.

• is strict if it reflects isomorphisms and indecomposability.

2. A morphism of boxes Φ : 𝔄 → 𝔅 is called strict if the inverse image functor Φ^{∗} is strict. (In particular, we have the notion of a strict representation of a box over a category, e.g., over an algebra).

3. A box 𝔄 is called wild if it has a strict representation over each finitely generated k-algebra. (In particular, we have the notion of a wild category (e.g., wild algebra) if we identify it with its regular box.)

Remind some standard well-known examples of wild categories and boxes (cf. [11]).

Proposition 4.2. The following categories, algebras and boxes are wild:

1. Free algebra k⟨ x, y ⟩ in 2 generators.

2. Polynomial algebra k[x, y] in 2 generators.

3. Formal power series algebra k[[x, y]] in 2 generators.

4. The free category k(x|y) generated by the graph with 2 vertices 1, 2 and two arrows x : 1 → 1 and y : 1 → 2, as well as its opposite category k(x|y)^{∘}.

5. The free category C_{5} generated by the graph with 6 vertices 0, 1, 2, 3, 4, 5 and 5 arrows a_{i} : 0 → i for i = 1, 2, …, 5, as well as the opposite category C_{5}^{∘}.

6. Any semi-free normal box 𝔚_{1} whose bigraph has one vertex and two solid arrows a, b such that ∂a = 0 and b is not superfluous.

7. Any semi-free normal box 𝔚_{2} whose diagram has two vertices 1, 2 and three solid arrows a : 1 → 1 , c : 2 → 2 , b : 1 → 2 (or b : 2 → 1) such that ∂a = 0 , ∂c = 0 and b is not superfluous.

Definition 4.3. 1. A rational algebra is, by definition, an algebra

of the form R = k[t, f(t)^{−1}] where f(t) is a non-zero polynomial.

2. A strict representation over a rational algebra will be called a rational family of representations.

3. The dimension of a rational family M is defined as the dimension dim(M).

4. For any R-module L put M(L) = M ⊗_{R} L and denote [M] = { M(L) | L ∈ R-mod }. We say that the modules isomorphic to some M(L) belong to the family M.

5. If F is any set of rational families, put [F] = ⋃_{M∈F}[M]. Again, we say that the modules isomorphic to those from [F] belong to this set of rational families. For any dimension d of representations of the box 𝔄 denote by F_{d} the set of representations from F having dimension d.

Definition 4.4. A box (in particular, a category or an algebra) 𝔄 is said to be tame if there is a set F of rational families of its representations such that:

1. F is locally finite, i.e., F_{d} is finite for any prescribed dimension d ∈ Dim(A).

2. F is almost exhaustive, i.e., for any prescribed dimension d the set [F] ∩ rep_{d}(𝔄) intersects almost all isomorphism classes of indecomposable 𝔄-modules of this dimension (i.e., all but a finite number of them).

Note that local finiteness implies that there is only finitely many M ∈ F such that [M] ∩ rep_{d}(𝔄) ≠ ∅.

It was proved in [11] (cf. also [7] ) that any (locally finitely generated, triangular) semi-free box as well as any finite dimensional algebra (indeed, any locally finite dimensional category) over an algebraically closed field is either wild or tame. It implies some simple but rather useful geometrical criteria for tameness and wildness based on the following fact concerning the properties of “geometrical dimensions” (cf. [10] ).

Remind that a constructible subset Y of an algebraic variety X is, by definition, a finite union of its locally closed subsets. The dimension of Y is defined as the maximum of dimensions of these locally closed subsets. An equivalence relation E on X is said to be constructible if it is constructible as subset of X × X. Then, of course, the equivalence class E(x) of each element x ∈ X is also constructible.

Proposition 4.5. (cf. [10].) Let X be an algebraic variety (over an algebraically closed field), E be a constructible equivalence relation on X, and Y, Z be two constructible subsets of X such that:

• dim Y ∩ E(x) ≥ m for each x ∈ X;

• dim Z ∩ E(x) ≤ n for each x ∈ X. Then dim Z − n ≤ 2 dim X − dim E ≤ dim Y − m.

Corollary 4.6. Suppose that the filed k is algebraically closed. For any semi-free box 𝔄 (or locally finite dimensional category) the following conditions are equivalent:

1. 𝔄 is not wild.

2. 𝔄 is tame.

3. For each d ∈ Dim(𝔄) there is a constructible subset Y ⊆ rep_{d}(𝔄) such that dim Y ≤ 1 and Y intersects each isomorphism class of indecomposable representations from rep_{d}(𝔄).

4. For each d ∈ Dim(𝔄) there is a constructible subset Y ⊆ rep_{d}(𝔄) such that dim Y ≤ |d| and Y intersects each isomorphism class from rep_{d}(𝔄).

5. If Z ⊆ rep_{d}(𝔄) is a constructible subset with finite intersections with each isomorphism class, then dim Z ≤ |d|.

Corollary 4.7. Suppose that 𝔄, 𝔅 are semi-free boxes (or locally finite dimensional categories) and F : 𝔄-mod → 𝔅-mod is a functor having the following properties:

1. For each d ∈ Dim(𝔄) the set

F (d) = { dim(F (M)) | M ∈ rep_{d}(𝔄) }

is finite.

2. For each d′ ∈ F (d) the subset

F^{−1}(d′, d) = { M ∈ rep_{d}(𝔄) | dim(F (M)) = d′ } ⊆ rep_{d}(𝔄)

is constructible and the induced map F^{−1}(d′, d) → rep_{d′}(𝔅) is locally regular.

3. For each L ∈ 𝔅-mod the set F^{−1}(L) = { M | F (M) ≃ L } consists

of a finite number of isomorphism classes. Then, if 𝔄 is wild, so is also 𝔅.

5. Galois coverings

Definition 5.1. An action T of a group G on a box 𝔄 = (A, V) is a mapping T : g ↦ T (g) where g ∈ G and T (g) are automorphisms (i.e., invertible morphisms of boxes) 𝔄 → 𝔄 such that T (gh) = T (g)T (h) for any g, h ∈ G^{7}. If the action T is fixed, we say that 𝔄 is a G-box and often write gx instead of T (g)x.

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Definition 5.2. The action T is said to be free if gx ≠ x for any object x ∈ Ver A and any non-unit element g ∈ G (then also ga ≠ a for any a ∈ A(x, y) or a ∈ V(x, y) with x, y ∈ Ver A).

Given a G-box 𝔄 we can define a new box G \ 𝔄 = (G \ A, G \ V) in the following way. The object set Ob(G\A) is the orbit set G\(Ob A). For two objects X, Y ∈ Ob(G \ A) put

(G \ A)(X, Y) = ⨁_{x∈X, y∈Y} A(x, y) /U_{A}

and

(G \ V)(X, Y) = ⨁_{x∈X, y∈Y} V(x, y) /U_{V}

where U_{A} and U_{V} denote the subspaces generated by all differences a−ga from the corresponding space. If a ∈ A(x, y) and b ∈ A(gy, z) , the product of the orbits Gb and Ga is defined as the orbit G(b(ga)). One can immediately check that this definition does not depend on the choice of representatives inside the orbits. Quite analogously, the bimodule structure on G \ V is defined. Moreover, the action of G can be evidently prolonged to V ⊗_{A} V in such way that G \ (V ⊗_{A} V) ≃ (G \ V) ⊗_{G\A} G \ V. Thus, we are able to define a coalgebra structure on G \ V putting

ε(Gv) = Gε(v) and μ(Gv) = Gμ(v) .

So, G \ 𝔄 is indeed a box. Call it the factor of the G-box 𝔄. The factoring morphism of boxes Π : 𝔄 → G \ 𝔄 mapping x to Gx is welldefined and the following properties are immediate.

Proposition 5.3. 1. Given any morphism of boxes Φ : 𝔄 → 𝔅 such that Φ(gx) = Φ(x) for any x ∈ Ob A ∪ Mor A ∪ El V and g ∈ G, there exists (unique up to isomorphism) a morphism of boxes Ψ : G \ 𝔄 → 𝔅 such that Φ = ΨΠ.

7We do not consider here more general notion of action with a system of factors like in [13, 12].

2. If the action is free, then for any objects x ∈ Ob A and Y ∈ Ob(G \ A) the morphism Π induces isomorphisms:

⨁_{Πy=Y} A(x, y) ≃ (G \ A)(Πx, Y) ;

⨁_{Πy=Y} A(y, x) ≃ (G \ A)(Y, Πx) ;

⨁_{Πy=Y} V(x, y) ≃ (G \ V)(Πx, Y) ;

⨁_{Πy=Y} V(y, x) ≃ (G \ V)(Y, Πx) .

Definition 5.4. Suppose given two boxes 𝔄 = (A, V), 𝔅 = ( B, W) and a morphism of boxes Φ : 𝔄 → 𝔅. Call Φ a Galois covering of the box 𝔅 with Galois group G if the following conditions hold:

1. Both A and B are rigid categories.

2. The group G acts freely on the box 𝔄.

3. The morphism Φ : 𝔄 → 𝔅 can be decomposed as ΨΠ where Π : 𝔄 → G \ 𝔄 is the factoring morphism and Ψ : G \ 𝔄 → 𝔅 is an isomorphism.

In this situation we also call the box 𝔄 a Galois covering of 𝔅 (with Galois group G).

The notion of Galois coverings of categories (cf. [5]) is a partial case of Galois coverings of boxes: one only has to consider regular boxes. As usually, the advantage of boxes is the possibility of reduction procedures. The following proposition is an immediate consequence of the universality properties of amalgamations and factors.

Proposition 5.5. Suppose given:

• a Galois covering of boxes Π : 𝔄̃ → 𝔄 with Galois group G;

• a sub-box 𝔄′ ⊆ 𝔄̃ and its pre-image 𝔄̃′ = Π^{−1}(𝔄′);

• a Galois covering of categories Θ′ : B̃′ → B′ with the same Galois group G;

• a commutative diagram of functors:

Ã′  ⟶^{F̃′}  B̃′
Π ↓  ↓ Θ′
A′  ⟶^{F′} B′

such that F̃′(gx) = gF̃′(x) for all x.

Consider the amalgamations B̃ = Ã∐Ã′ B̃′ and B = A ∐^{A′} B′ and the corresponding functors

F̃ : Ã → B̃ and F : A → B .

Then they induce a commutative diagram of morphisms of boxes:

𝔄̃ ⟶^{F̃} 𝔄̃F̃
Π ↓  ↓ Θ
𝔄 ⟶^{F} 𝔄^{F}

and Θ is again a Galois covering with the Galois group G.

Suppose that Π : 𝔄̃ → 𝔄 is a Galois covering of semi-free boxes with the Galois group G . We call this covering degree preserving if, for some choice of semi-free generators in 𝔄̃, deg ga = deg a for all morphisms of Ã and elements of Ṽ. We will only consider degree preserving coverings.

Proposition 5.6. Let Π : 𝔄̃ → 𝔄 be a degree preserving Galois covering of semi-free boxes with the Galois group G . Then:

1. There is a set of semi-free generators Σ̃ of 𝔄̃ such that GΣ̃ = Σ̃ .

2. The set of the orbits Σ = G \ Σ̃ is a set of semi-free generators of the box 𝔄.

3. For any object x ∈ Ob 𝔄 the functor Π induces an homomorphism π : Gr(𝔄, Σ, x) → G, which is an epimorphism if 𝔄̃ is connected.

Proof. Note first that any degree preserving automorphism of k(t) maps t to αt + β for some α, β ∈ k . Hence, for any g ∈ G , if a is a marked loop from Ã, so is αγ + β for some α, β ∈ k . Consider an arbitrary set of semi-free generators Σ′ of 𝔄̃ and a set of representatives X of the orbits of G on Ob Ã. Put Σ_{X} = ⋃ Ã(y, x) , where x runs through the objects from X and y through all objects from Ã. Let Σ̃ = GΣ′. For every element a one can find g ∈ G such that the source of g^{−1}a belongs to X. Hence, g^{−1}a = ∑_{i} a_{i}b_{i}, where a_{i} ∈ Σ̃ and deg b_{i} < deg a. Now an evident induction shows that Σ̃ generates 𝔄̃. Just in the same way one shows that this set of generators is semi-free.

As any element from 𝔄 is a linear combination of elements of the form Π_{∗}a with a ∈ 𝔄, the set Σ = ΠΣ̃ = ΠΣ_{X} generates 𝔄. Moreover, due to Proposition 5.3(2) any relation between these elements imply a relation between elements from Σ̃ . Hence, it is a set of semi-free generators. Now, one can construct a natural homomorphism π : Gr(𝔄, Σ, x) → G (the fundamental group of the pair (𝔄, Σ)). Indeed, let w be a circuit at the point x of the complex Com(𝔄, Σ). Evidently, we may suppose that it consists of the arrows of the corresponding bigraph and their inverses. But, for every arrow s ∈ Σ , s : x → y and any object x̃ ∈ Ob Ã with Π(x̃) = x there is a unique arrow s̃ ∈ Σ̃ such that Π(s̃) = s. Hence, there is a way w̃ in the complex Com(𝔄̃, Σ̃) starting at x̃ such that Π(w̃) = w. (We extend the mapping Π to the complex Com(𝔄̃, Σ̃) in the obvious way.) Let the vertex y ∈ Ob Ã be its end. Then Π(y) = x, hence, y = gx̃ for some (uniquely defined) g ∈ G. Moreover, one can easily see that g only depends on the homotopy class of w and if g′ corresponds to another way w′ the product g′g corresponds to the concatenation of the ways w′w. Therefore, the homomorphism π : Gr(𝔄, Σ, x) → G is well defined. If, moreover, 𝔄̃ is connected, then π is epimorphism as for any g ∈ G there is a way in Σ̃ starting at x and ending at gx.

Remark 5.7. One can conjecture that Proposition 5.6 remains valid even if we do not suppose that the action preserves degree. We cannot prove this conjecture, but we never need actions which are not degree preserving. In any case, one has the following evident complement to Proposition 5.5.

Proposition 5.8. In the situation of Proposition 5.5, suppose that 𝔄̃ is semi-free, the covering Π is degree preserving, the sub-category Ã′ is generated by a part of semi-free generators of Ã, B̃′ is also semifree, and Θ′ is degree preserving too. Then B̃ is also semi-free and Θ is degree preserving too.

It shows that all reduction procedures described above are compatible with the degree preserving coverings.

6. IND_{0} and IND_{1}

From now on we assume the field k to be algebraically closed. We denote by ind 𝔄 a skeleton of the category of modules 𝔄-mod . In other words, it is a full subcategory of 𝔄-mod consisting of some representatives of the isomorphism classes of all indecomposable modules.

Proposition 6.1. Assume that a group G acts freely on a box 𝔄̃ and 𝔄 = G \ 𝔄̃. Then if 𝔄̃ is wild, so is 𝔄.^{8}

Proof. Consider a strict representation M ∈ rep(𝔄̃, k⟨ x, y ⟩) and put \overline{M} = Π_{∗}M, S = St(Supp M). As Supp M is finite and the action is free, S is finite. Suppose that \overline{M} ⊗_{k⟨ x,y ⟩} L ≃ \overline{M} ⊗_{k⟨ x,y ⟩} L′ for some L, L′ ∈ ind k⟨ x, y ⟩. Taking inverse images we obtain: ⨁_{g∈G}(M ⊗_{k⟨ x,y ⟩} L)^{g} ≃ ⨁_{g∈G}(M ⊗_{k⟨ x,y ⟩} L′)^{g}. As all these direct summands are finite dimensional and indecomposable, we can use the unique decomposition theorem. Taking into account the supports we get that M ⊗_{k⟨ x,y ⟩} L ≃ (M ⊗_{k⟨ x,y ⟩} L′)^{g} for some g ∈ S. As M is strict the last formula defines L′ up to isomorphism. Hence, there can be only finitely many L′ ∈ ind k⟨ x, y ⟩ such that \overline{M} ⊗_{k⟨ x,y ⟩} L ≃ \overline{M} ⊗_{k⟨ x,y ⟩} L′. Then 𝔄 is wild by Corollary 4.7.

8For the case of algebras, it has been proved in [8].

Note that if G is torsion free the same consideration shows that if M is strict, \overline{M} is also strict. Moreover, the following proposition holds.

Proposition 6.2. Let Π : 𝔄̃ → 𝔄 be a Galois covering of boxes with a torsion free Galois group G . Put ind_{0} 𝔄 = Π_{∗}(ind 𝔄̃) ^{9}. Then ind_{0} 𝔄 ⊆ ind 𝔄, and Π_{∗} induces an equivalence G \ ind 𝔄̃ ≃ ind_{0} 𝔄.

The proof is quite analogous to that given for algebras by Gabriel [15], so we omit it.

Definition 6.3. A representation N ∈ Rep(𝔄̃) is called a ℤ-represen-tation if the following conditions hold:

1. N is indecomposable.

2. All spaces N(x), where x ∈ Ob A, are finite dimensional.

3. St N = { g ∈ G | N^{g} ≃ N } is an infinite cyclic group.

4. St N \ Supp N is finite.

Denote by ind_{ℤ} 𝔄̃ a set of representatives of all isomorphism classes of ℤ-representation of 𝔄̃.

As St N ≃ ℤ we can choose isomorphisms φ_{g} : N →^{∼} N^{g} in such way that φ_{gh} = φ_{g}^{h}φ_{h}. Therefore, if N is a ℤ-representation of 𝔄̃ its direct image Π_{∗}N can be considered as a representation of G \ 𝔄̃ over the group ring k[ℤ] ≃ k[T, T^{−1}]. Moreover, as the group St N acts freely on Supp N, Π_{∗}N is free as k[ℤ] -module. Hence, for each k[ℤ]-module L we can define the representation Π_{∗}N ⊗_{k[ℤ]} L of 𝔄 over k.

Definition 6.4. For any ℤ-representation of the box 𝔄̃ denote by Π_{↓}N the set { Π_{∗}N ⊗_{k[ℤ]} J | J ∈ ind k[ℤ] } ⊆ rep 𝔄.

Lemma 6.5. Let G act freely on a box 𝔄̃ and N ∈ ind_{ℤ} 𝔄̃. Then End𝔄̃ N is local.

9In [9], they denote this subcategory by ind_{1} .

Proof. We may (and will) assume that the representation N is strict, i.e., Supp N = Ver 𝔄̃. Denote by d the maximal dimension dim N(x), by S the stabilizer of N, by s the generator of S, and by D a fundamental domain for S on Supp N. For each arrow a (solid or dotted) let l(a) be the maximal length of all solid paths occurring in ∂a. As D is finite and there are only finitely many arrows starting or ending at each vertex, l = sup { l(a) | a ∈ Supp N } is finite. Just in the same way, ν = sup { ν(a) | a ∈ Supp N } is also finite, ν(a) being the triangular height of a.

Consider all arrows starting or ending in D. Find such an integer m that all of them have both sources and targets in D_{1} = ⋃_{i=−m}^{m} s^{i}D. Then any arrow having one end in D_{1} has the second one in D_{1} ∪ s^{m}D_{1} ∪ s^{−m}D_{1}. Put d_{1} = |D_{1}|.

Let α ∈ End(N) and α_{x} = α(ω_{x}). Every α_{x} is a linear mapping in the finite dimensional vector space N(x), hence, there is a nonzero polynomial annihilating α_{x}. If there is a common annihilating polynomial f(T) for all α_{x} then α is either invertible or nilpotent. Indeed, otherwise f(T) = f_{1}(T)f_{2}(T) for coprime polynomials f_{1}, f_{2}. We can suppose f(T) being of the minimal possible degree. There are such polynomials u_{i}(T) that u_{2}(T)f_{1}(T) + u_{1}(T)f_{2}(T) = 1 and deg u_{i} < deg f_{i}. Therefore, u_{2}(α_{x})f_{1}(α_{x}) are all idempotents and not all of them are 0 or 1. As it was shown in [18] then there is a non-trivial idempotent in End(N), that is impossible.

Suppose that there is no polynomial annihilating all α_{x}. As all dimensions dim N(x) are bounded, there should be infinitely many elements λ ∈ k occurring as eigenvalues of α_{x}. Find a subset Q ⊂ Supp N such that the mappings α_{x} for x ∈ Q have at least 2r + 1 distinct eigenvalues where r = 2dd_{1}(2ν − 1)(2l + 1). We can suppose that Q = ∪_{i=0}^{q}s^{i}(D_{1}) for some q. Put

D_{−} = ⋃_{i=0}^{l} s^{im}D_{1} ,

D_{+} = ⋃_{i=l}^{2l} s^{im}D_{1} ,

D_{±}^{i} = \begin{cases}s^{q+(2l+1)im}D_{±} & if i > 0 \\ s^{(2l+1)im}D_{±} & if i < 0 \end{cases}
D_{2}^{i} = D_{−}^{i} ∪ D_{+}^{i} ,
P = ⋃_{i=1}^{2ν−1}(D_{2}^{i} ∪ D_{2}^{−i}) ,

Q_{1} = Q ∪ (⋃_{i=1}^{ν}(D_{2}^{i} ∪ D_{2}^{−i}) ,
Q′ = Q_{1} \ (D_{+}^{ν} ∪ D_{−}^{−ν}) .

Then there are at most 2r distinct eigenvalues of all α_{x} with x ∈ P; hence, they do not exhaust all eigenvalues of α_{x} with x ∈ Q. Therefore, we can find a polynomial f(T) such that f(α_{x}) = 0 for x ∈ P and f(α_{x}) is non-nilpotent for some x ∈ Q. Put β = f(α)^{ν}. Then β(φ) = 0 for each dotted arrow φ having both target and source in D_{2}^{±ν}.

Define an homomorphism γ : N → N in the following way:

γ_{x} = β_{x} for x ∈ Q_{1} ;
γ_{x} = 0 otherwise ;
γ(φ) = β(φ) if both source and target of φ are in Q_{1} ;
γ(φ) = 0 otherwise .

Certainly, we have to verify that γ is indeed an homomorphism, i.e., to check for each solid arrow a : x → y with ∂a = ∑_{j} p_{j}φ_{j}q_{j} that

N(a)γ_{x} − γ_{y}N(a) = ∑_{j} N(p_{j})γ(φ_{j})N(q_{j}) . (1)

Suppose first that either x or y, or the source or the target of at least one of φ_{j} is in Q′. Then x, y ∈ Q_{1} and all φ_{j} have their targets and sources in Q_{1}. Hence, all values of γ occurring in (1) coincide with the corresponding values of β and (1) holds for γ as it holds for β. Otherwise, all values of γ occurring in (1) are zeroes as their sources and targets are either in D_{2}^{±ν} or outside Q_{1}. Hence, (1) holds trivially.

Thus, we have obtained an endomorphism of N which is neither nilpotent nor invertible and is annihilated by some polynomial. As we have seen before it is impossible.

Lemma 6.6. Suppose that a group G acts freely on a semi-free box 𝔄̃, 𝔄 = G \ 𝔄̃ and N ∈ ind_{ℤ} 𝔄̃. Then:

1. Π_{∗}N is strict, i.e., is a rational family of 𝔄̃-modules.

2. No module from ind_{0} 𝔄 belongs to the family Π_{∗}N .

3. If N′ ∈ ind_{ℤ} 𝔄̃ is such that N′ ≄ N^{g} for all g ∈ G then no 𝔄̃-module belonging to the family Π_{∗}N belongs to Π_{∗}N′.

We will prove a generalization of this lemma in Section 8 (Lemma 8.5).

7. Main Theorem for Boxes

Now we can state the main theorem of this article for the case of boxes. Remind that the ground field is supposed to be algebraically closed and all semi-free boxes are supposed triangular.

Theorem 7.1. Let Π : 𝔄̃ → 𝔄 be a degree preserving Galois covering of semi-free boxes with a torsion free Galois group G ; 𝔄̃ = (Ã, Ṽ) , 𝔄 = (A, V) . Then:

1. 𝔄 is tame if and only if so is 𝔄̃.

2. If 𝔄̃ is tame (hence, 𝔄 is also tame), then:

(a) ind 𝔄 = ind_{0} 𝔄 ⊔ ind_{1} 𝔄.

(b) If M ∈ ind_{0} 𝔄, M′ ∈ ind_{1} 𝔄 or M ∈ Π_{↓}N, M′ ∈ Π_{↓}N′ for N′ ≄ N, then Hom_{𝔄}(M, M′) ∪ Hom_{𝔄}(M′, M) ⊆ rad_{𝔄}^{∞}. Moreover, if M = Π_{∗}N ⊗_{k[ℤ]} J and M′ = Π_{∗}N ⊗_{k[ℤ]} J′ for N ∈ ind_{ℤ} 𝔄̃, then Hom_{𝔄}(M, M′) = 1 ⊗ Hom_{k[ℤ]}(J, J′) + rad_{𝔄}^{∞}(M, M′).

(c) If N ∈ ind_{ℤ} 𝔄̃, there is a representation \overline{N} ∈ rep(𝔄, k[T]) such that Π_{∗}N ≃ \overline{N} ⊗_{k[T]} k[ℤ] and \overline{N} ⊗_{k[T]} J_{0,m} ∈ ind_{0} 𝔄 for each m, where J_{0,m} = k[T]/(T^{m}). (So to speak, the representations from ind_{1} 𝔄 are deformations of those from ind_{0} 𝔄.)

Here rad_{𝔄} denotes the radical of the category rep(𝔄) . (Note that it is always locally finite dimensional.)

Proof. We know that if 𝔄̃ is wild then 𝔄 is also wild (cf. Proposition 6.1). Suppose that 𝔄̃ is tame and let M ∈ 𝔄-mod be any finite dimensional 𝔄-module. We are going to prove that M is isomorphic to a module either from ind_{0} 𝔄 or from ind_{1} 𝔄. Without loss of generality we may suppose M to be sincere, i.e., M(x) ≠ 0 for all x ∈ Ob A. Now use the induction by ||M||. If ||M|| = 0, M is a trivial representation: M ≃ S_{x} for some x ∈ Ob A. Then M ≃ Π_{∗}S_{y} ∈ ind_{0} 𝔄 where y is a preimage of x in Ob Ã. Now suppose that the claim is true for all boxes and all representations M′ with smaller values of ||M′||. Choose a semi-free triangular system of generators Σ̃ of 𝔄̃ such that Σ = G\Σ̃ is a semi-free triangular system of generators for 𝔄 (cf. Proposition 5.6). Then the system Σ contains either a minimal edge, or a minimal loop, or a superfluous arrow.

Let first a : x → y be a minimal edge. Then its preimage in Σ̃ is a set { a_{i} : x_{i} → y_{i} } where all a_{i} are also minimal edges. Let A′ be the subcategory of A generated by a and Ã′ be the subcategory of Ã generated by all a_{i}. Denote by B′ the trivial category with three objects x, y, xy, by B̃′ the trivial category with the objects x_{i}, y_{i}, x_{i}y_{i}, and consider the functors F′ : A′ → B′ and F̃′ : Ã′ → B̃′ defined as in Example 3.2 (1). Then we are in the situation of Proposition 5.5. Hence, we obtain a commutative diagram

\begin{matrix}𝔄̃ & ⟶^{F̃} & 𝔄̃F̃ \\ Π ↓ & & ↓ Θ \\ 𝔄 & ⟶^{F} & 𝔄^{F} \end{matrix} (2)

of Galois coverings with the group G. In this case the morphisms F̃ and F induce equivalences of the representation categories 𝔄̃F̃-mod ≃ 𝔄̃-mod and 𝔄^{F} -mod ≃ 𝔄-mod respectively. In particular, the box 𝔄̃F̃ is tame. Moreover, if M ≃ F^{∗}M′ then ||M′|| < ||M||, as M(x) ≠ 0 and M(y) ≠ 0 (cf. Proposition 3.5). Hence, M′ is isomorphic to a representation belonging either to ind_{0} 𝔄^{F} or to ind_{1} 𝔄^{F}. But as the diagram above is commutative, ind_{0} 𝔄 ≃ F^{∗} ind_{0} 𝔄^{F} and ind_{1} 𝔄 ≃ F^{∗} ind_{1} 𝔄^{F}. Thus, M is also isomorphic to a representation belonging either to ind_{0} 𝔄 or to ind_{1} 𝔄.

Quite the same observation is valid in the case when Σ contains a superfluous arrow or a minimal loop a such that its preimage in Σ̃ consists of loops.

Suppose now that a : x → x is a minimal loop from Σ and its preimage contains an edge a_{0} : x_{0} → x_{1} with x_{0} ≠ x_{1}. Then Πx_{0} = Πx_{1}, hence, x_{1} = gx_{0} for some g ∈ G. Consider the cyclic group C = ⟨ g ⟩ and put x_{k} = g^{k}x, a_{k} = g^{k}a : x_{k} → x_{k+1}. As g^{k} ≠ 1 for k ≠ 0, all objects x_{k} and all edges a_{k} are pairwise distinct. Therefore, the subcategory L ⊆ A generated by all these objects and arrows is a line, i.e., is of the shape:

… ⟶^{a_{−1}} x_{0} ⟶^{a_{0}} x_{1} ⟶^{a_{1}} x_{2} ⟶^{a_{2}} … ⟶^{a_{k−1}} x_{k} ⟶^{a_{k}} x_{k+1} ⟶^{a_{k+1}} … (3)

Moreover, as G acts transitively on the preimage of a, this preimage is a disjoint union of lines. In this case we say that a is a loop lifting into lines.

Suppose first that M(a) is nilpotent: M(a)^{n} = 0. Let A′ be the subcategory of A generated by a and Ã′ be the subcategory of Ã generated by all preimages of a. We know that Ã′ is a disjoint union of lines. Hence, its indecomposable representations are in 1-1 correspondence with the connected finite parts of Ã′. Let B′ be a trivial category with n objects x^{m} (m = 1, 2, … n). Define a functor F′ : A′ → B′ mapping x into the direct sum ⨁_{m=1}^{n} mx^{m} and a into the direct sum of Jordan cells ⨁_{m=1}^{n} J_{m}(0), where J_{m}(0) is considered as a morphism of mx^{m} to itself.

Let now B̃′ be a trivial category whose objects x^{I} are in 1-1 correspondence with finite connected parts I ⊆ Ã′ consisting at most of n objects. Define a functor F̃′ : Ã′ → B̃′ mapping an object y into the direct sum ⨁_{I∋y} x^{I} and a morphism b : y → gy into the morphism ⨁_{I∋y} x^{I} → ⨁_{I∋gy} x^{I} such that all its non-zero components are just 1 : x^{I} → x^{I} for I ⊃ { y, gy }. Define also a functor B̃′ → B′ mapping x^{I} to x^{m} where m = |I|.

Now we are again in the situation of Proposition 5.5. Hence, we can construct a commutative diagram of the form (2) of Galois coverings with the group G. In this case the image of F^{∗} consists of all representations of 𝔄 which are nilpotent of degree n at the loop a. In particular, this image contains M: M ≃ F^{∗}M′, where M′ ∈ ind 𝔄^{F} and ||M′|| < ||M|| as M(x) ≠ 0. Therefore, just as above, we can conclude that M is isomorphic to a module from ind_{0} 𝔄 ∪ ind_{1} 𝔄.

To get the whole claim we need the following lemma that will be proved in the next section.

Lemma 7.2. In the situation of Theorem 7.1, suppose that a is a minimal loop in A lifting to lines and M is an indecomposable 𝔄-module. If 𝔄̃ is tame then either M(a) is nilpotent or M is isomorphic to such module M′ that M′(b) = 0 for all arrows b ∈ Σ except of a.

The consideration above and this lemma imply that each M ∈ ind 𝔄 is isomorphic either to a module from ind_{0} 𝔄 ∪ ind_{1} 𝔄 or to a module M′ such that M′(b) = 0 for all arrows b ∈ Σ except of some minimal loop a lifting to lines, while M′(a) is invertible. Then we may suppose that M′(a) = J_{m}(λ), the Jordan cell of size m × m with the eigenvalue λ. Take any line L from the preimage of a and consider the 𝔄̃-module L such that L(x_{k}) = k for all objects x_{k} ∈ L, L(y) = 0 for all other objects, L(a_{k}) = 1 for all arrows a_{k} ∈ L and L(c) = 0 for all arrows c ∉ L. Evidently, L is an indecomposable ℤ-representation of 𝔄̃ and M′ ≃ L ⊗_{k[ℤ]} k[ℤ]/(T − λ)^{m}. In particular, M′ ∈ ind_{1} 𝔄.

Hence, we have proved claim 2a of the theorem. To prove claim 1 it only remains to prove the following result.

Proposition 7.3. In the situation of Theorem 7.1, denote by Z the set of rational families { Π_{↓}N | N ∈ ind_{ℤ} 𝔄̃ }. If 𝔄̃ is tame, this family is locally finite.

Proof. Let d be a dimension of representations of the box 𝔄. We prove that Z_{d} is finite using induction by m = ||d||. Again, we suppose this dimension to be sincere, i.e., d(x) ≠ 0 for all x ∈ Ver 𝔄. If m = 0 , 𝔄̃ is so-trivial, thus Z = ∅. Let our claim be true for all boxes and all dimensions d′ with ||d′|| < m. Just as above, we should consider the following cases:

(i) 𝔄 has a minimal edge; (ii) 𝔄 has a superfluous arrow; (iii) 𝔄 has a minimal loop a : x → x lifting to lines; (iv) 𝔄 only has minimal loops lifting to loops (neither minimal edges

or loops lifting to lines, nor superfluous arrows). Cases (i) and (ii) are easy: we use the same reduction process as above and get a commutative diagram (2) such that F̃^{∗} induces, in particular, an equivalence ind_{ℤ} 𝔄̃F̃ ≃ ind_{ℤ} 𝔄̃. Moreover, given d , the set D(F, d) = { Dim M | M ∈ rep(𝔄^{F}) and Dim F^{∗}M = d } is finite, and if d′ ∈ D(F, d) , then ||d′|| < m. So, the inductive supposition implies that our claim is valid.

In case (iii) Lemma 7.2, together with Lemma 6.6, implies that for every N ∈ ind_{ℤ} 𝔄̃ either N(b) = 0 for every solid arrow b such that Πb ≠ a or Π_{∗}N(a) is nilpotent. There is only one (up to G-shift) representation of the former type, namely, such that N(x_{k}) = k and N(a_{k}) = 1 for some line (x_{k}, a_{k}) with Πa_{k} = a. On the other hand, to the representations of the latter type one can apply the reduction of minimal lines up to degree d(x) and then use the induction argument as before.

In the remaining case (iv) we can repeat the arguments from [11] (cf. also [7]) to prove that if c is a minimal loop in 𝔄̃ there is a polynomial f(t) ∈ k[t] such that for any N ∈ ind_{ℤ} 𝔄̃ with Dim Π_{∗}N = d either f(N(c)) is nilpotent or N(b) = 0 for every solid arrow b ≠ c. Certainly, there are no indecomposable ℤ-representations of the latter type while to the former one we are able to apply the reduction of minimal loops c with Πc = a : x → x up to degree d(x) . Indeed, we have to reduce λ-parts of c for every root λ of the polynomial f(t) , but, nevertheless, it gives us only finitely many possibilities for reductions as well as for dimensions of the reduced representations. Thus, the same inductive argument accomplishes the proof of this proposition and, therefore, of claim 1 of Theorem 7.1.

Claim 2b follows from a bit more general statement. For a loop a : x → x denote by J_{k}(a, λ) a representation mapping x to mk , objects y ≠ x to 0 , a to J_{k}(λ) , and arrows b ≠ a to 0 .

Proposition 7.4. Let 𝔄 = (A, V) be a semi-free triangular box, a : x → x be an invertible minimal loop from A. Suppose that for every M ∈ ind 𝔄, either M(x) = 0 or M(a) = J_{k}(a, λ) for some k, λ. Then each homomorphism φ ∈ Hom_{𝔄}(M, M′) ∪ Hom_{𝔄}(M′, M) where M, M′ ∈ ind 𝔄, M′(x) ≠ 0 and φ(ω_{x}) = 0 belongs to rad_{𝔄}^{∞}. In particular, Hom_{𝔄}(M, M′) ∪ Hom_{𝔄}(M′, M) ⊆ rad_{𝔄}^{∞} if either M(a) and M′(a) have distinct eigenvalues or M(x) = 0.

Note that we need not even suppose 𝔄 to be tame!

Proof. Evidently, it is enough to factorize any morphism φ : M → M′ (or M′ → M) as ηφ′ (respectively, φ′η) where φ′ : M → M′′ (respectively, φ′ : M′′ → M ) for an indecomposable module M′′ which is non-isomorphic to M′, M′′(x) ≠ 0 and φ′(ω_{x}) = 0.

First consider the case when there are no solid arrows between x and Supp M. In this case any choice of φ(v) where v runs through all dotted arrows between Supp M and x defines an homomorphism M → M′. Suppose that M′ = J_{k}(a, λ). Consider the matrix T = (0 I_{k}), where 0 denotes the zero column, and its transposed T^{⊤}. Let M′′(a) = J_{k+1}(a, λ). Then putting η(ω_{x}) = T and η(α) = 0 for all dotted arrows α we obtain a homomorphism η : M′′ → M′. Now define φ′ : M → M′′ putting φ′(v) = T^{⊤}φ(v). Then φ = ηφ′ is the necessary factorization. The case when φ : M′ → M is treated analogously.

Now use the induction by m = ||M ⊕ M′||. Clearly, we may suppose this direct sum to be sincere. If m = 1 the module M is trivial: M = S_{y} and there are no solid arrows between x and y. Hence, we can use the above consideration. Suppose that the claim is true for all smaller values of m. If there is a minimal edge, or a minimal loop b ≠ a, or a superfluous arrow, we can diminish m using the corresponding reduction process. So the only case we have to consider is when a is the unique minimal arrow.

Choose a solid arrow b such that its differential contains the unique solid arrow a. Suppose first that b : x → y where y ≠ x. Then ∂b = ∑_{i=1}^{s} u_{i}f_{i}(a) for some Laurent polynomials f_{i} ∈ k[t, t^{−1}]. Let h(t) = gcd(f_{1}, f_{2}, …, f_{s}). If h = 1 the arrow b is superfluous. Then we can reduce it and diminish the value of m. Otherwise, choose λ ≠ 0 in k such that h(λ) = 0. Define 𝔄-module M putting M(x) = M(y) = k, M(z) = 0 for all objects except of x, y, M(a) = λ, M(b) = 1, and M(c) = 0 for all arrows except of a, b. Then M is indecomposable and does not satisfy the conditions of the proposition. The case when b : y → x with y ≠ x is treated in the same way.

Now suppose that b : x → x. Then ∂b = ∑_{i=1}^{s} f_{i}(t_{1}, t_{2}, t_{1}^{−1}, t_{2}^{−1})u_{i} for some Laurent polynomials f_{i} in two variables. Here, just as in [11], t_{1} describes the left multiplication by a and t_{2} the right one. Consider the ideal I generated by f_{1}, f_{2}, …, f_{s}. If I = (1), the arrow b is superfluous and can be reduced. Otherwise, there is a pair (λ, μ) of non-zero elements from k such that f_{i}(λ, μ) = 0 for all i. Define 𝔄-module M putting M(x) = k^{2}, M(y) = 0 for y ≠ x,

M(a) = (\begin{matrix}λ & 0 \\ 0 & μ \end{matrix}) , M(b) = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,

and M(c) = 0 for all arrows except a, b. This module is again indecomposable and does not satisfy the conditions of the proposition.

Now we can prove 2b using the induction by m = ||M ⊕ M′||. Note that ||M′|| ≥ 1 and ||M′|| = 1 if and only if M′ = J_{1}(a, λ) for some invertible minimal loop a. In this case we can use Proposition 7.4. If there is a minimal edge, or a minimal loop lifting to loops, or a superfluous edge in 𝔄, we use the corresponding reduction process diminishing m. So we may suppose that the only minimal arrows in 𝔄 are minimal loops lifting to lines. Let a : x → x be one of them. If both M(a) and M′(a) are nilpotent, we can also use the reduction process. Hence, we may suppose that either M or M′ is isomorphic to some of J_{k}(a, λ). In view to Lemma 7.2, a satisfies the conditions of Proposition 7.4, that accomplishes the proof.

To prove 2c consider N ∈ ind_{ℤ} 𝔄̃. We may suppose N to be sincere. Use the induction by m = ||Π_{∗}N||. If m = 1, then A consists of a unique vertex x and a unique loop a : x → x. Hence, Ã is indeed a line of the shape (3), N(x_{i}) = k, N(a_{i}) = 1. Therefore, Π_{∗}N(x) = k[ℤ], Π_{∗}(a) = T. In this case we can put \overline{N}(x) = k[T], \overline{N}(a) = T, and the claim becomes obvious. Suppose that it is true when ||Π_{∗}N|| < m. Again, if there is a minimal edge, a superfluous arrow or a minimal loop lifting to loops, we can make a reduction step diminishing m. If there is a minimal loop lifting to lines, then, as we have just proved, either Π_{∗}N is nilpotent on this loop or it is non-zero only on it. In the first case we can again use a reduction, while the second one coincides with the base of induction.

It is also easy now to compare the Auslander-Reiten quivers of a box and of its covering. Remind that the diagram (or the quiver) of a locally finite dimensional category C is defined as the oriented graph, whose vertices coincide with those of C and there are r(x, y) arrows with the source x and the target y where

r(x, y) = dim_{k} rad(x, y)/ rad^{2}(x, y) .

The Auslander-Reiten quiver AR 𝔄 of a box 𝔄 (in particular, of a category) is, by definition, the diagram of the category 𝔄-mod . Remind also that a homogeneous tube is, by definition, a connected component of an Auslander-Reiten quiver which contains no projective modules and has the shape:

M_{1} ⇆ M_{2} ⇆ M_{3} ⇆ …

In the situation of Theorem 7.1, denote by AR_{0} 𝔄 and AR_{1} 𝔄, respectively, the diagrams of the categories ind_{0} 𝔄 and ind_{1} 𝔄. Then this theorem immediately implies the following result concerning the Auslander-Reiten quivers.

Corollary 7.5. In the situation of Theorem 7.1:

• AR 𝔄 = AR_{0} 𝔄 ⊔ AR_{1} 𝔄 (a disjoint union);

• AR_{0} 𝔄 ≃ G \ AR 𝔄̃;

• AR_{1} 𝔄 is a disjoint union of homogeneous tubes.

Remark 7.6. As usually, any results valid for boxes can immediately be drawn to bimodules (cf. [11, 7]), which are indeed a partial case. Moreover, just as for algebras, any Galois covering of bimodules induces a degree preserving Galois covering of the corresponding boxes. So, one can just substitute in Theorem 7.1 and other results of Sections 6 and 7 “bimodules” for “boxes” crossing out the words “degree preserving.” On the contrary, at the moment we do not see any direct proof of these results which would not involve the tedious construction of quasitriangular boxes from the next section.

8. Proof of the Main Lemma

This section is devoted to the proof of Lemmas 7.2 and 6.6. To do it we need to consider a bit more general situation.

Definition 8.1. A quasi-triangular box is a box 𝔄 = (A, V) with a fixed section ω, a system of generators Σ = Σ_{0} ⊔ Σ_{1} ⊔ Σ′ and a set 𝔏 of lines from A satisfying the following conditions:

1. Σ_{0} is a semi-free system of generators for A and Σ_{1}⊔Σ′ is a system of generators of the kernel \overline{V}.

2. Σ_{1} is a free system of generators for a free sub-bimodule \overline{V}_{1} of \overline{V} and, for each arrow u ∈ Σ′(x, y), either x or y lies on a line from 𝔏.

3. The box 𝔏^{−1}𝔄 is semi-free triangular with the normal section ω and the semi-free system of generators Σ_{0} ∪ Σ_{1}.

4. The image in 𝔏^{−1}𝔄 of ∂a is zero for each arrow a belonging to any line L ∈ 𝔏.

5. Distinct lines from 𝔏 have no common objects and no object belonging to such a line is a marked vertex in A (in the sense of the definition of semi-free categories).

6. There is a function κ : 𝔏 → ℕ such that the set of its values is bounded and for each arrow a belonging to a line L ∈ 𝔏

∂a ∈ ∑_{\begin{matrix}x∈L′ \\ κL′<κL \end{matrix}}(A1_{x}\overline{V} + \overline{V}1_{x}A) .

Here 𝔏^{−1}A denotes the category obtained from A by inverting all arrows from all lines L ∈ 𝔏 and 𝔏^{−1}𝔄 = (𝔏^{−1}A, 𝔏^{−1}A ⊗_{A} V ⊗_{A} 𝔏^{−1}A).

We call Σ a quasi-triangular system of generators of 𝔄 and the lines from 𝔏 the special lines.

We will consider a special sort of actions of groups on quasi-triangular boxes.

Definition 8.2. Let a group G acts freely on a quasi-triangular box 𝔄 with a quasi-triangular system of generators Σ = Σ_{0} ∪ Σ_{1} ∪ Σ′ and the set of special lines 𝔏 as above. We say that G acts well if the following conditions hold:

1. gΣ_{0} = Σ_{0} and gΣ_{1} = Σ_{1} for each g ∈ G .

2. g𝔏 = 𝔏 for each g ∈ G .

3. The stabilizer St L acts transitively on L for each L ∈ 𝔏.

Note that if 𝔄 is semi-free and the action is degree preserving, Proposition 5.6 imply that one can always suppose that G acts well. From now on we consider the situation described in the following definition.

Definition 8.3. Let a group G act well on a quasi-triangular box 𝔄̃ with a quasi-triangular system of generators Σ̃ = Σ̃_{0} ⊔ Σ̃_{1} ⊔ Σ̃′. Put 𝔄 = G \ 𝔄̃, 𝔄̃^{∗} = 𝔏^{−1}𝔄̃, Σ_{i} = ΠΣ̃_{i} ( i = 0, 1 ), 𝔄^{∗} = (Π𝔏)^{−1}𝔄, and denote by ∂^{∗} the differentials in 𝔄̃^{∗} and 𝔄^{∗}. We expand the action of G onto the box 𝔄̃^{∗} in the obvious way and identify 𝔄^{∗} with G \ 𝔄̃^{∗}. Call the images of arrows from special lines the special loops. An arrow a ∈ Σ_{0} is called a quasi-minimal edge (respectively, quasi-superfluous arrow) if its image in 𝔄^{∗} is a minimal edge (respectively, a superfluous arrow). On the other hand, call a a quasi-minimal loop if either it is a special loop or its image in 𝔄^{∗} is a minimal loop. Note that for each object x ∈ Ob A there is at most one special loop in Σ_{0}(x, x). If there is one, call x a special vertex.

Note that 𝔄^{∗} is also a semi-free box. We are going to prove the following generalization of Lemma 7.2.

Lemma 8.4. In the situation of Definition 8.3, suppose that G is torsion free, 𝔄̃ is not wild, and a is a quasi-minimal loop from Σ_{0} lifting to lines. For each M ∈ ind 𝔄^{∗} either M(a) is nilpotent or M ≃ J_{k}(a, λ) for some k ∈ ℕ and λ ∈ k \ { 0 }.

Proof. Again we use the induction by m = ||M|| supposing M sincere. Moreover, for a fixed value of m we use the induction on the number of arrows from Σ_{0} which are not special loops. The cases when m = 0 or all arrows from Σ_{0} are special loops are trivial.

Consider the case when a is not a special loop and M(x) is not nilpotent. Then we can split M(a) into nilpotent and invertible parts: M(a) = A_{0} ⊕ A_{1}. Let A_{0}^{n} = 0. Denote by A′ the subcategory of A consisting of x and a and by Ã′ its preimage in Ã (which is a set of disjoint lines). Let B′ be the category consisting of n + 1 objects x^{0}, x^{1}, …, x^{n} and one generating morphism a^{0} : x^{0} → x^{0}. Define a functor F : A′ → B′ putting

F (x) = x^{0} ⊕ x^{1} ⊕ 2x^{2} ⊕ … ⊕ nx^{n};
F (a) = a^{0} ⊕ J_{1}(0) ⊕ J_{2}(0) ⊕ … ⊕ J_{n}(0) .

Note that we do not suppose a^{0} invertible unlike in the usual reduction of a minimal loop.

Now let B̃′ be the category consisting of the objects x^{I} and x_{i0} where I runs through all finite connected parts of Ã′ consisting of at most n objects and x_{i} runs through the objects of Ã′. The set of generating arrows of B̃′ is, by definition, { a_{i0} } with a_{i} running through all preimages of a. Define a functor F̃′ : Ã′ → B̃′ mapping the object x_{i} into the direct sum x_{i0}⊕(⨁_{I∋x_{i}} x^{I}) and the morphism a_{i} : x_{i} → gx_{i} (g ∈ G) into the direct sum a_{i0} ⊕ a′, where the morphism a′ : ⨁_{I∋x_{i}} x^{I} → ⨁_{I∋gx_{i}} x^{I} is constructed just as in the proof of Theorem 7.1.

Now we are again in the situation of Lemma 5.5. So we get a commutative diagram of the shape (2). The obtained category B contains the whole set 𝔏 of special lines from Ã. Define the set of special lines from B as 𝔏^{F} = 𝔏 ∪ 𝔏′, where 𝔏′ is the set of all lines consisting of the arrows a_{i0}. Evidently, all components of the images of arrows from Σ̃_{0} together with the arrows a_{i0} form a semi-free system of generators for B̃. The components of all arrows from Σ̃_{1} generate a free sub-bimodule

of \overline{Ṽ}^{F}. Denote these sets by Σ̃_{0}^{F} and by Σ̃_{1}^{F} correspondingly. Consider the image of ω_{i} = ω_{x_{i}} in V^{F}. It is of the form:

(\begin{matrix}η_{00} & η_{01} \\ η_{10} & η_{11} \end{matrix}) .

Here η_{αβ} ∈ V^{F} (x_{iβ}, x_{iα}), where α, β ∈ { 0, 1 } and x_{i1} = ⨁_{x_{i}∈I} x^{I}. Consider the arrow a_{i} : x_{i} → gx_{i}. As a is quasi-minimal, the image of ∂a_{i} = a_{i}ω_{i} −gω_{i}a_{i} in 𝔏^{−1}𝔄̃ is zero. The same is true for all components of the image of a_{i}ω_{i} − gω_{i}a_{i} in 𝔄̃^{F}. It gives us the following relations for the components η_{αβ}:

a_{i0}η_{00} = (gη_{00})a_{i0} + δ_{00} ,
a_{i0}η_{01} = (gη_{01})a_{i1} + δ_{01} ,
a_{i1}η_{10} = (gη_{10})a_{i0} + δ_{10} ,
a_{i1}η_{11} = (gη_{11})a_{i1} + δ_{11} ,

where the images of all δ_{αβ} are zero in 𝔏^{−1}𝔄̃^{F}. As a_{i1}^{n} = 0 the second and the third of these equations imply that the images of a_{i0}^{n}η_{01} and (gη_{10})a_{i0}^{d} are zeros in 𝔏^{−1}𝔄̃^{F}. Hence, the images of all possible η_{01} and η_{10} are zeros in (𝔏^{F})^{−1}𝔄̃^{F}. Now we can put ω_{x_{i0}} = η_{00}. Then the condition 4 of Definition 8.1 holds. Moreover, ∂a_{i} lies in the submodule generated by Σ̃′, hence, the same is true for ∂a_{i0}. As all arrows from Σ̃′ are incident to some lines from 𝔏, the condition 6 from Definition 8.1 holds if we put κL′ = k for L′ ∈ 𝔏′ with k > κ(L) for all L ∈ 𝔏.

At last, the conditions for η_{11} are just the same as in the reduction of minimal lines. Therefore, we can take for Σ̃_{1}^{F} the set of all components of elements from F (Σ̃_{1}) and of all components of all possible η_{11} having non-zero images in 𝔏^{−1}𝔄̃^{F}. For (Σ̃^{F})′ we can take the set of all components of elements from F (Σ̃′) and all components of all possible η_{01} and η_{10}. Then the conditions 1–3 of Definition 8.1 also hold.

Hence, we obtain a commutative diagram of the shape (2) of quasitriangular boxes. Moreover, the representation M belongs to the image of F^{∗}: M ≃ F^{∗}M′, where M′ ∈ (𝔄^{F})^{∗}-mod and ||M′|| < ||M|| whenever M(a) is not invertible. Otherwise n = 0, hence, there are no new objects (and arrows) in 𝔄̃, but there are new special loops in it. Therefore, we can use the inductive supposition.

Thus, we may suppose that a is a special loop. Denote by Σ′_{0} the set of all arrows from Σ_{0} which are not special loops. Choose an arrow b ∈ Σ′_{0} with the minimal triangular height in 𝔄^{∗}, b : y → z. Consider first the case when neither y nor z is a special vertex. Then b is either a quasi-minimal edge, or a quasi-superfluous arrow, or a quasiminimal (non-special) loop. In the first case its preimage in Ã is a set { b_{i} : y_{i} → z_{i} }, where all objects y_{i}, z_{i} are pairwise distinct and none of them lies on a special line. Then we can use the reduction of the edge b in 𝔄 and the reduction of the set of edges b_{i} in 𝔄̃. As none of y_{i}, z_{i} lies on special lines this reduction commutes with the “localization” 𝔏^{−1}. Hence, we can use Lemma 5.5 and obtain a commutative diagram of the form (2), where 𝔄̃^{F} is also quasi-triangular with the same set of special lines. Moreover, as b is a minimal edge in 𝔄^{∗}, F induces an equivalence F^{∗} : (𝔄^{F})^{∗}-mod ≃ 𝔄^{∗}-mod. In particular, M ≃ F^{∗}M′ and ||M′|| < ||M|| (as M is sincere), so the claim follows from the inductive supposition.

The same observations work in the case when b is a quasi-superfluous arrow, a quasi-minimal loop lifting to loops, or a quasi-minimal loop lifting to lines and such that M(b) is nilpotent. At last, if b is any quasi-minimal loop lifting to lines, we can repeat the reduction used above in the case when a was non-special. Hence, in all these cases the inductive arguments accomplish the proof.

Now consider the case when for each minimal b : y → z either y or z (maybe both) is a special vertex. Suppose first that y is a special vertex with the special loop c : y → y and z is not a special vertex. Then ∂^{∗}b can contain the unique solid arrow c. Let first ∂^{∗}b = ∑_{∈ℤ} λ_{i}u_{i}c^{i} ≠ 0 with λ_{i} ∈ k and u_{i} ∈ Σ_{1}. Put k = min { i | λ_{i} ≠ 0 }. Then we can replace u_{k} in Σ_{1} by ∑_{i≥k} λ_{i}u_{i}c^{i−k}. After this b becomes quasisuperfluous. Otherwise ∂^{∗}b = 0. Denote b_{0} : y_{0} → z_{0} some preimage of b. The vertex y_{0} lies on a special line

… ⟶^{c_{−1}} y_{0} ⟶^{c_{0}} y_{1} ⟶^{c_{1}} y_{2} ⟶^{c_{2}} … (4)

As ∂^{∗}b_{0} = 0 too, it becomes zero after inverting a finite set of arrows, say, S = { c_{i} | i = −m, … m }. Consider the subcategory of S^{−1}Ã generated by all c_{i} and b_{0}. Now ∂c_{i} = 0 and ∂b_{0} = 0 in S^{−1}𝔄̃ and this category contains a line and one more arrow b_{0}. It is well-known that it is wild. Therefore, 𝔄̃ is also wild. The same observations are valid when z is a special vertex and y is not.

Suppose now that y ≠ z are both special vertices with the special loops c ∈ Σ_{0}(y, y) and d ∈ Σ_{0}(z, z). There are special lines L_{y} (4) and L_{z}:

… ⟶^{d_{−1}} z_{0} ⟶^{d_{0}} z_{1} ⟶^{d_{1}} z_{2} ⟶^{d_{2}} …

with Π(y_{i}) = y, Π(z_{i}) = z, Π(c_{i}) = c and Π(d_{i}) = d. We suppose that κ(L_{x}) ≤ κ(L_{y}). Then ∂^{∗}b = ∑_{i} f_{i}(X, Y)u_{i}, where u_{i} ∈ Σ_{1} and f_{i} ∈ k[X, X^{−1}, Y, Y^{−1}] are some Laurent polynomials. As in [11] we put Xu = uc and Y u = du for u ∈ V(x, y). Therefore, for appropriate powers of c and d we have d^{s}∂bc^{t} = ∑_{i} u_{i}g_{i}(X, Y) + v, where g_{i} ∈ k[X, Y] and the image of v in 𝔄^{∗} is zero. Choose a preimage b_{0} of b. Suppose that b_{0} : y_{0} → z_{0}. Let first St L_{y} ∩ St L_{z} ≠ { 1 }, g be the generator of this intersection and gb_{0} : y_{m} → z_{n}. Then D_{s}∂b_{0}C_{t} = ∑_{j} α_{j}D_{k_{j}}v_{j}C_{l_{j}} + v_{0} for some v_{j} ∈ Σ̃_{1} and some element v_{0} such that its image in 𝔄̃^{∗} is zero. Here we denote for v ∈ Ṽ(y_{p}, z_{q})

D_{k}vC_{l} = d_{q+k−1} … d_{q+1}d_{q}vc_{p−1}c_{p−2} … c_{p−l} .

Moreover, if gv_{j} = v_{j′} then l_{j′} = l_{j} + m and k_{j′} = k_{j} − n. Hence, each of the polynomials g_{i} is of the form X^{k_{i}}Y^{l_{i}}h_{i}(X^{m}/Y^{n}) for some h_{i} ∈ k[T, T^{−1}]. Denote by d(T) the greatest common divisor of all h_{i}. If d = 1, the arrow b can be considered as quasi-superfluous. Suppose that deg d > 0. Then all g_{i}(X, Y) have a common divisor of the form X^{m} − λY^{n} for some non-zero λ ∈ k. Therefore, D_{s}∂b_{0}C_{t} = η_{1}C_{m} − λD_{n}η_{0} +v_{0} for some η_{0} ∈ Ṽ and η_{1} = gη_{0}. We may suppose that λ = 1. Put b_{i} = g^{i}b_{0}. Then D_{s}∂b_{i}C_{t} = η_{i+1}C_{m} − D_{n}η_{i} + v_{i}, where η_{i} = g^{i}η_{0}, v_{i} = g^{i}v_{0}.

We are going to prove that the box 𝔄̃ is wild in contradiction with the suppositions of Lemma 8.4. To do it replace 𝔄̃ by the factor ℭ = 𝔄̃/I, where I is generated by all arrows from Σ̃_{0} except of c_{i}, d_{i} and b_{0}. We will prove that even ℭ is wild. Denote by ℭ^{∗} the box obtained from ℭ by inverting all c_{i} and d_{i}. Remark that η_{i+1}C_{m} − λD_{n}η_{i} + v_{i} = 0 in ℭ if i ≠ 0. The kernel U of the box ℭ is generated by the images of the arrows from Σ̃_{1} ∪ Σ̃′. Let F be the free bimodule with the same generators and R ⊆ F be the set of relations between these generators in U. As the images of all v_{i} are zeros in ℭ^{∗}, we have that v_{i} = ∑_{j} p_{ij}r_{j} for some r_{j} ∈ R and some polynomials p_{ij} in c_{k}, d_{l}, c_{k}^{−1}, d_{l}^{−1} and b_{0}. Just in the same way ∂d_{i} = ∑_{j} q_{ij}r_{j} for some polynomials q_{ij}, while ∂c_{i} = 0 in ℭ. Consider all polynomials p_{0j} and those of p_{ij} and q_{ij} which really contain b_{0}. There is only a finite number of them. Hence, they contain only a finite number of c_{k}^{−1} and d_{l}^{−1}. Let K be the maximal absolute value of the corresponding indices k and l. Consider now the box ℭ′ obtained from ℭ by inverting c_{k} and d_{k} with |k| ≤ K. It is enough to show that ℭ′ is wild. But in ℭ′ we have:

∂d_{i} = 0 ,
D_{s}∂b_{0}C_{t} = η_{1}C_{m} − D_{n}η_{0} ,
η_{i+1}C_{m} − D_{n}η_{i} + v_{i} = 0 for i ≠ 0

where v_{i} = ∑_{ij} p′_{ij}r_{j} and p′_{ij} are polynomials in c_{k}, c_{k}^{−1}, d_{l}, d_{l}^{−1}. Therefore, there exist s_{i}, t_{i} such that D_{s_{i}}(η_{i+1}C_{m} − D_{n}η_{i})C_{t_{i}} = 0. Denote by L the maximum of the set { s, t, s_{i}, t_{i} | − K − 4 ≤ i ≤ K }.

Consider the following representation N of ℭ′ over the free algebra k⟨X, Y ⟩:

N (y_{i}) = \begin{cases}k⟨X, Y ⟩^{2} & if − K − 4 − L ≤ i ≤ K \\ 0 & otherswise \end{cases}

N (c_{i}) = \begin{cases}1_{2} & if − K − 4 − L ≤ i ≤ K − 1 \\ 0 & otherwise \end{cases}

N (z_{i}) = \begin{cases}k⟨X, Y ⟩^{5} & if − K ≤ i ≤ K + L \\ k⟨X, Y ⟩^{5−j} & \begin{aligned}if i & = K + L + j or \\ i & = −K − j for 1 ≤ j ≤ 4 \end{aligned} \\ 0 & for all other values of i \end{cases}

N (d_{i}) = \begin{cases}1_{5} & if − K ≤ i ≤ K + L − 1 \\ (\begin{matrix}1_{4+j} \\ 0 \end{matrix}) & if i = −K − j for 1 ≤ j ≤ 4 \\ (1_{4+j} 0) & if i = K + L + j − 1 for 1 ≤ j ≤ 4 \\ 0 & for all other values of i \end{cases}

N (b_{0}) = (\begin{matrix}1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & X \\ 1 & Y \end{matrix})

We claim that it is strict. Indeed, let M_{1}, M_{2} be any k⟨X, Y ⟩-modules, N_{j} = N ⊗_{k⟨X,Y ⟩} M_{j} (j = 1, 2), and φ ∈ Hom_{ℭ′}(N_{1}, N_{2}). As all N_{j}(c_{i}) are isomorphisms for −K −4 ≤ i ≤ K −1 and all N_{j}(d_{i}) are monomorphisms for −K − 4 ≤ i ≤ K + L − 1, the relations for η_{i} give us that φ(η_{i}) = 0 for all i. Moreover, as ∂c_{i} = ∂d_{i} = 0 in ℭ′, we obtain the following relations:

N_{2}(d_{i})φ_{i} = φ_{i+1}N_{1}(d_{i})

where φ_{i} = φ(ω_{z_{i}}). In particular, all φ_{i} are equal for −K ≤ i ≤ K. Putting i = −K − j with 1 ≤ j ≤ 4 we obtain that

φ_{i+1} = (\begin{matrix}φ_{i} & ξ_{i} \\ 0 & θ_{i} \end{matrix})

for some θ_{i} and ξ_{i}. Hence, φ_{−K} is an upper triangular 5 × 5 matrix. Analogous observations show that φ_{K} is a lower triangular 5×5 matrix. As they are equal, both of them as well as φ_{0} are diagonal: φ_{0} = diag(Φ_{1}, Φ_{2}, Φ_{3}, Φ_{4}, Φ_{5}). Note that

N_{j}(b_{0}) = (\begin{matrix}1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & M_{j}(X) \\ 1 & M_{j}(Y) \end{matrix}) .

As N_{2}(b_{0})φ(y_{0}) = φ_{0}N_{1}(b_{0}), we get that Φ_{1} = Φ_{2} = Φ_{3} = Φ_{4} = Φ_{5} = φ(y_{0}) . Denote this common value by Φ. Then ΦM_{1}(X) = M_{2}(X)Φ and ΦM_{1}(Y) = M_{2}(Y)Φ. Hence, N is strict.

If St L_{y} ∩ St L_{z} = { 1 } the same observation shows that either b is quasi-superfluous or ∂^{∗}b = 0 . In the latter case the box 𝔄̃ (even that obtained from 𝔄̃ by inverting all arrows from L_{y}) is wild just as in the case when only one end of b is special.

The remaining case, when b : y → y and y is a special vertex, is quite analogous (just as in [11] or [7]) and we omit it.

Just in the same way (though much easier) we generalize Lemma 6.6 as follows.

Lemma 8.5. Suppose that a group G acts well on a quasi-triangular box 𝔄̃, 𝔄 = G \ 𝔄̃, and N ∈ ind_{ℤ} 𝔄̃. Then:

1. Π_{∗}N is strict, i.e., is a rational family of 𝔄̃-modules.

2. No module from ind_{0} 𝔄 belongs to the family Π_{∗}N .

3. If N′ ∈ ind_{ℤ} 𝔄̃ is such that N′ ≄ N^{g} for all g ∈ G then no 𝔄̃-module belonging to the family Π_{∗}N belongs to Π_{∗}N′.

Proof. We denote by S the stabilizer of N in G . Again we suppose N to be sincere and use the induction on the value m = ||Π_{∗}N||; the latter being well defined as Π_{∗}N ∈ rep(𝔄, k[ℤ]) . Here the case m = 0 is evidently impossible. If m = 1 then 𝔄 consists of one vertex x and one loop a, while 𝔄̃ consists of a unique line (3). Therefore, the only possibility for N is:

N(x_{k}) = k , and N(a_{k}) = 1 .

Thus, Π_{∗}N(x) = k[ℤ] = k[T, T^{−1}] and Π_{∗}N(a) is the multiplication by T , so all claims are obvious.

Suppose that this lemma is valid for all ℤ-representations (of all boxes) with smaller values of m. Following the same reduction processes as in the proof of Lemma 8.4 we construct a morphism of quasitriangular G-boxes F : 𝔄̃ → 𝔅̃ compatible with the action of G and a ℤ-representation M of the box 𝔅̃ such that N ≃ F^{∗}M and ||Π_{∗}M|| < ||Π_{∗}N||. Then the inductive supposition accomplishes the proof. As this procedure is quite analogous to the abovementioned one, we only sketch it. The unique non-trivial case is when we have a minimal loop in 𝔄 lifting to lines. Consider the restriction of N onto one of these lines L. It splits into a direct sum of the representations L_{I} described in Proposition 3.4. As N is S-invariant and all N(x) are finite dimensional every interval I either is finite or coincides with ℤ; moreover, the numbers of elements in all finite intervals I must be bounded and every I can only occur finitely many times. Hence, we can apply the same procedure as in the proof of Lemma 8.4, in the case of a loop lifting to lines. As the result, we obtain a commutative diagram of the shape (2) and a ℤ-representation M of 𝔄̃^{F} such that N ≃ F̃^{∗}N′ and either ||ΘM|| < ||ΠN|| or the restriction of N on L only contains direct summands of the form L_{ℤ}. In the former case we can apply the inductive supposition. In the latter one N(x) is invertible for any object x ∈ L; thus, we can preserve 𝔄̃ and just consider L as a new special line, so diminishing the number of arrows in 𝔄̃ not belonging to special lines.

9. Main Theorem for Algebras

Usual arguments (cf. [11, 7]) allows to transfer the main theorem for boxes to the case of finite dimensional algebras, or, further, of locally finite dimensional categories. Remind that the ground field is always supposed to be algebraically closed.

Theorem 9.1. Let Π : Λ̃ → Λ be a Galois covering of locally finite dimensional categories with a torsion free Galois group G . Then:

1. The representation type of Λ̃ is the same as that of Λ.

2. If Λ̃ is tame (hence, Λ is also tame), then:

(a) ind Λ = ind_{0} Λ ⊔ ind_{1} Λ.

(b) If M ∈ ind_{0} Λ, M′ ∈ ind_{1} Λ or M ∈ Π_{↓}N, M′ ∈ Π_{↓}N′ for N′ ≠ N, then Hom_{Λ}(M, M′) ∪ Hom_{Λ}(M′, M) ⊆ rad_{Λ}^{∞}. Moreover, if M = Π_{∗}N ⊗_{k[ℤ]} J and M′ = Π_{∗}N ⊗_{k[ℤ]} J′ for N ∈ ind_{ℤ} Λ̃ , then Hom_{Λ}(M, M′) = 1 ⊗ Hom_{k[ℤ]}(J, J′) + rad_{Λ}^{∞}(M, M′).

Remark 9.2. We suppose that the property (2c) of Theorem 7.1 also remains valid for finite dimensional algebras, but at the moment we do not see any proof of it.

Proof. As we are only interested in finite dimensional representations of Λ , we may suppose it finite dimensional, i.e., such that Sk Λ has finitely many objects. Then Λ̃ is evidently locally bound, i.e., for any object x ∈ Sk Λ̃ the set { y ∈ Ver Λ̃ | Λ̃(x, y) ⊕ Λ̃(y, x) ≠ 0 } is finite.

(Of course, Λ̃ is not finite dimensional itself if G ≠ { 1 }.)

Remind first the following result establishing the relation between representations of locally bound categories (in particular, of algebras) and those of boxes. For any locally bound category Λ define the category Λ-Res of Λ-resolutions, whose objects are homomorphisms φ : P′ → RP′′ where P′, P′′ are projective Λ-modules, R = rad Λ , and morphisms from φ to ψ : Q′ → Q′′ are pairs (α, β) where α : P′ → Q′, β : P′′ → Q′′ such that βφ = ψα. Let Λ-res be its full subcategory consisting of such objects φ : P′ → JP′′ that both P′ and P′′ are finite dimensional. There is a natural functor Coker : Λ-Res → Λ-Mod putting an homomorphism φ to its cokernel. The following claim is well-known (cf. [11, 7]).

Proposition 9.3. The functor Coker induces an equivalence between the factor-category Λ-Res/I → Λ-Mod where I is the ideal generated by the identity morphisms of the objects of the form 0 : P → 0 and the morphisms of the form (γf, gγ) where γ is a homomorphism P → Q′. The same is true if we replace Λ-Res by Λ-res and Λ-Mod by Λ-mod.

Given a locally bound category Λ we can construct a box 𝔄_{Λ} = 𝔄 = (A, V) in the following way (cf. [11] or [7]). Let S = Λ^{∅} ⊕ Λ^{∅} where Λ^{∅} is the trivial part of Λ, R be the bimodule

(x, y) ↦ DR(y, x)

(D being the vector space duality) considered as Λ^{∅}-bimodule. Denote by x′ the object x⊕0 of S and x′′ the object 0⊕x. Consider S-bimodules R_{12}, R_{11}, R_{22} where

R_{12}(x′, y′′) = R(x, y),
R_{12}(x′, y′) = R_{12}(x′′, y′′) = R_{12}(x′′, y′) = 0,
R_{11}(x′, y′) = R(x, y),
R_{11}(x′, y′′) = R_{11}(x′′, y′′) = R_{11}(x′′, y′) = 0,
R_{22}(x′′, y′′) = R(x, y),
R_{22}(x′, y′) = R_{22}(x′, y′′) = R_{22}(x′′, y′) = 0.

Consider also homomorphisms of S-bimodules

δ_{12} : R_{12} → R_{12} ⊗_{S} R_{11} ⊕ R_{22} ⊗_{S} R_{12},
δ_{11} : R_{11} → R_{11} ⊗_{S} R_{11},
δ_{22} : R_{22} → R_{22} ⊗_{S} R_{22}

which are dual to the multiplication mappings in rad Λ. Put A = C[R_{12}] (the tensor category) and \overline{V} = A ⊗_{S} (R_{11} ⊕ R_{22}) ⊗_{S} A. Then the box 𝔄 is given by the category A, the kernel \overline{V} and the differential ∂ induced by the mappings δ_{12}, δ_{22} and −δ_{11}.

Remark that as the field k is algebraically closed the box 𝔄_{Λ} is free, normal and triangular (cf. [11]).

We are able to construct a functor P : 𝔄-Mod → Λ-Res in the following way. Let M be an 𝔄-module. Put

P′_{M} = ⨁_{x∈Ob Λ} Λ^{x} ⊗ M(x′),
P′′_{M} = ⨁_{x∈Ob Λ} Λ^{x} ⊗ M(x′′).

For each x, y ∈ Ob Λ, M induces linear mappings φ_{xy} : R_{12}(x′, y′′) → Hom(M(x′), M(y′′)) and is uniquely defined by these mappings. But we can identify Hom(R_{12}(x′, y′′), Hom(M(x′), M(y′′))) with R(y, x) ⊗ Hom(M(x′), M(y′′)). As morphisms y → x are the same as those Λ^{x} → Λ^{y}, φ_{xy} can be considered as an homomorphism Λ^{x} ⊗ M(x′) → Λ^{y} ⊗M(y′′). Now define P(M) : P′_{M} → P′′_{M} as the homomorphism with the components φ_{xy}. One can easily check that P is indeed a functor from 𝔄-Mod to Λ-Res and, moreover, is an equivalence of categories (cf. [11] or [7]).

Therefore, we obtain the following result.

Proposition 9.4. Let Λ be a locally bound category. Then the composition cok = Coker P : 𝔄_{Λ}-Mod → Λ-Mod is full and dense. Moreover, its restriction onto the full subcategory that consists of all 𝔄-modules without trivial summands of the form S_{x′} (x ∈ Ob Λ) maps indecomposable representations into indecomposable and non-isomorphic to nonisomorphic ones^{10} and its kernel is contained in the radical of the category 𝔄_{Λ}-Mod . The same is true if we replace 𝔄-Mod by 𝔄-mod and Λ-Mod by Λ-mod.

Corollary 9.5. The representation type of the box 𝔄_{Λ} coincides with that of the category Λ.

The explicit construction described above leads also to the following result.

Corollary 9.6. Suppose that a group G acts freely on a locally bound category Λ̃ and Λ = G \ Λ̃. Then this action induces a free, degree preserving action of G on 𝔄Λ̃ and an isomorphism of boxes 𝔄_{Λ} ≃ G\𝔄Λ̃. Moreover, the functors cok commutes with the direct image functors Π_{∗}.

Note that if x = Πx̃ then S_{x′} = Π_{∗}S_{}_{x̃′}. Thus, all objects from ind 𝔄Λ̃ lying in the kernel of cok are in ind_{0} 𝔄Λ̃ . Therefore, claim 1 of Theorem 9.1 follows immediately from claim 1 of Theorem 7.1. Moreover,

ind_{0} Λ = Π_{∗}(ind Λ̃) = Π_{∗} cok ind 𝔄Λ̃ = cok Π_{∗} ind 𝔄Λ̃ = cok ind_{0} 𝔄_{Λ} and obviously

10In other words, this restriction is a representation equivalence

ind Λ = cok ind 𝔄_{Λ} = cok ind_{0} 𝔄_{Λ} ⊔ cok ind_{1} 𝔄_{Λ} .

If L ∈ ind_{ℤ} 𝔄Λ̃, then St L = St cok L. As the action of G on Λ is free, cok Π_{∗}L is torsion-free (hence, free) as k[S]-module. Thus, L ∈ ind_{ℤ} Λ̃. Moreover, as tensor product is left exact, (cok Π_{∗}L) ⊗_{k[ℤ]} J ≃ cok(Π_{∗}L ⊗_{k[ℤ]} J), whence cok ind_{1} 𝔄_{Λ} = ind_{1} Λ, which proves claim 2a. Claim 2b follows now from the corresponding claim for boxes and the fullness of the functor cok.

Note also that such results as Lemma 6.5 and Proposition 7.3 also remain valid for the coverings of locally finite dimensional categories.

Again, Theorem 9.1 immediately implies the following result for the Auslander-Reiten quivers. (We denote by AR_{0} Λ and AR_{1} Λ , respectively, the diagrams of the categories ind_{0} Λ and ind_{1} Λ .)

Corollary 9.7. In the situation of Theorem 9.1:

• AR Λ = AR_{0} Λ ⊔ AR_{1} Λ (a disjoint union);

• AR_{0} Λ ≃ G \ AR Λ̃ ;

• AR_{1} Λ is a disjoint union of homogeneous tubes.

Remark 9.8. Usually the Auslander-Reiten quiver has an additional structure, the “Auslander-Reiten translation,” related to the almost split sequences (cf. [1]). Unfortunately, it only exists for rather specific classes of free boxes, mainly when the category A is finite dimensional (cf. [3]) or, the same, there are no solid cycles in the corresponding bigraph. If almost split sequences exist, they can be defined as exact sequences

0 ⟶ M \underrightarrow{(α_{1} … α_{k})^{⊤}} N_{1} ⊕ … ⊕ N_{k} \underrightarrow{(β_{1} … β_{k})} L ⟶ 0

such that:

• M, L, N_{1}, …, N_{k} are all indecomposable;

• M is not injective and L is not projective;

• α_{1}, α_{2}, …, α_{k} generate rad_{Λ}(M, ) ;

• β_{1}, β_{2}, …, β_{k} generate rad_{Λ}( , L) . The functor Π_{∗} maps projective (injective) modules to projective (injective) ones and all of them belong to ind_{0} Λ . Moreover, it maps radΛ̃ onto rad_{Λ} ∩ ind_{0} Λ view to the property (2b) of Theorem 9.1 (or 7.1). Therefore, the functor Π_{∗} maps Auslander-Reiten sequences from Λ̃-mod to Auslander-Reiten sequences in Λ-mod . So, Corollary 9.7 (as well as 7.5) remains valid with respect to this additional structure too.

Note one special case when all finite dimensional representations of the covered algebra (or box) are actually the images of those of the covering.

Proposition 9.9. (cf. [8]) Suppose that a torsion free group G acts on a semi-free box or on a locally finite dimensional category 𝔄̃, 𝔄 = G\𝔄̃. In the case of boxes we also suppose this action degree preserving. The following conditions are equivalent:

1. ind_{ℤ} 𝔄̃ = ∅.

2. ind 𝔄 = ind_{0} 𝔄 ≃ G \ 𝔄̃. These conditions always hold if 𝔄̃ is locally support finite, i.e., for every vertex x the set

{ y ∈ Ver 𝔄̃ | there is M ∈ ind 𝔄̃ such that M(x) ≠ 0, M(y) ≠ 0 }

is finite.

Note that here we also do not suppose tameness.

Proof. Certainly, we only have to prove this claim for boxes; then the case of algebras follows from Proposition 9.4 and Corollary 9.6.

2 ⇒ 1 follows from Lemma 6.6.

1 ⇒ 2 . We prove, using induction on m = ||M||, that under condition 1 any representation M ∈ ind 𝔄 is actually in ind_{0} 𝔄. It is trivial if m = 0 . Condition 1 implies that 𝔄̃ has no minimal lines. So, 𝔄 has either a minimal edge, or a superfluous arrow, or a minimal loop lifting to loops. In all these case we can reduce it parallel to all its preimages, thus, diminishing m, just as in the proof of Theorem 7.1. One only has to remember that a given representation has a finite and fixed set of eigenvalues at every minimal loop.

The same observation shows that condition 1 holds if 𝔄 is locally support finite.

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Department of Mechanics and Mathematics, Kyiv Taras Shevchenko University, Volodimirska 64, 01033 Kyiv, Ukraine

Max–Plank–Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

E-mail address: yuriy@drozd.org, drozd@mpim-bonn.mpg.de
E-mail address: ovsienko@uni-alg.kiev.ua