TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 2, February 1999, Pages 715–734 S 0002-9947(99)02362-4

GALOIS COVERINGS OF SELFINJECTIVE ALGEBRAS BY REPETITIVE ALGEBRAS

ANDRZEJ SKOWROŃSKI AND KUNIO YAMAGATA

Abstract. In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form B̂/G where B̂ is the repetitive algebra of an artin algebra B and G is an admissible group of automorphisms of B̂. If B is of finite global dimension, then the stable module category \underline{mod}B̂ of finitely generated B̂-modules is equivalent to the derived category D^{b}(mod B) of bounded complexes of finitely generated B-modules. For a selfinjective artin algebra A, an ideal I and B = A/I, we establish a criterion for A to admit a Galois covering F : B̂ → B̂/G = A with an infinite cyclic Galois group G. As an application we prove that all selfinjective artin algebras A whose Auslander-Reiten quiver Γ_{A} has a non-periodic generalized standard translation subquiver closed under successors in Γ_{A} are socle equivalent to the algebras B̂/G, where B is a representation-infinite tilted algebra and G is an infinite cyclic group of automorphisms of B̂.

1. Introduction

In the paper, by an algebra is meant a basic, connected, artin algebra (associative, with an identity) over a commutative artin ring K. For an algebra Λ we denote by mod Λ the category of finitely generated right Λ-modules and by D : mod Λ → mod Λ^{op} the standard duality Hom_{K}(−, E) where E is a minimal injective cogenerator in mod K. An algebra Λ is called selfinjective if Λ ≃ D(Λ) in mod Λ, that is, the projective Λ-modules are injective. If Λ is selfinjective, then the left and the right socle of Λ coincide, and we denote them by socΛ. Two selfinjective algebras Λ and R are said to be socle equivalent if the factor algebras Λ/socΛ and R/socR are isomorphic. Frequently, selfinjective algebras are socle equivalent to selfinjective algebras having triangular Galois coverings, and then we may reduce the study of such algebras and their representations to that for the corresponding algebras of finite global dimension. This is the case for all representation-finite selfinjective algebras over algebraically closed fields [6], [19] and certain classes of tame representation-infinite selfinjective algebras [1], [2], [9], [11], [22]. An important class of selfinjective algebras is formed by the algebras of the form B̂/G where

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors February 4, 1997.

1991 Mathematics Subject Classification. Primary 16D50, 16G10, 16G70, 16S99.

B̂ is the repetitive algebra [15] (locally bounded, without identity)

B̂ = [\begin{matrix}... & ... \\ & B_{m−1} & Q_{m−1} \\ & & B_{m} & Q_{m} \\ & & & B_{m+1} & Q_{m+1} \\ & & & & ... & ... \end{matrix}]

of an algebra B, where B_{m} = B and Q_{m} = _{B}D(B)_{B} for all m ∈ ℤ, all the remaining entries are zero, the matrices in B̂ have only finitely many nonzero elements, addition is the usual addition of matrices, and multiplication is induced from the canonical maps B ⊗_{B} D(B) → D(B), D(B) ⊗_{B} B → D(B), and the zero map D(B) ⊗_{B} D(B) → 0, and G is an admissible group of K-automorphisms of B̂. We note that if B is of finite global dimension, then the stable module category \underline{mod}B̂ of mod B̂ is equivalent, as a triangulated category, to the derived category D^{b} (mod B) of bounded complexes over mod B [13].

In the paper we are interested in the question of when a selfinjective algebra A is isomorphic (respectively, socle equivalent) to an algebra of the form B̂/G, where B is a factor algebra A/I of A by an ideal I, and G is an admissible infinite cyclic group of automorphisms of B̂. Our main results give sufficient conditions for a selfinjective algebra A to be of the above form. Moreover, we determine the structure of certain classes of selfinjective algebras whose Auslander-Reiten quiver has a prescribed form.

We shall now sketch the content of the paper. In Section 2 we fix notations and prove some preliminary results applied in the proofs of our main results. The main result proved in Section 3 is the following theorem. Let A be a basic and connected selfinjective artin algebra, I a (two-sided) ideal of A, B = A/I, and e a residual identity of B, that is, a minimal idempotent of A representing the identity of B. Denote by ν the Nakayama automorphism of A, and by ẽ the sum of idempotents forming the ν-orbits of pairwise nonisomorphic primitive summands of the idempotent e. Assume that: (1) ẽ(IeI)ẽ = 0; (2) ν^{−}(e)Ie and eIν(e) are injective cogenerators as a right eAe/eIe-module and a left eAe/eIe-module, respectively, and (3) the canonical algebra epimorphism eAe → eAe/eIe splits. Then ẽAẽ ≃ B̂/G for an admissible infinite cyclic group G of automorphisms of B̂. We also note that ẽ = 1 if IeI = 0.

In the final two sections we derive some consequences of the above theorem and the main results proved in our paper [26]. In Section 4 we prove that if A is a basic, connected, selfinjective artin algebra, I an ideal of A, B = A/I, e a residual identity of B, and such that IeI = 0, Ie is an injective cogenerator in mod B, and the ordinary quiver Q_{B} of B has no oriented cycles, then A is socle equivalent to B̂/G for some admissible infinite cyclic group G of automorphisms of B̂. Moreover, A ≃ B̂/G if K is an algebraically closed field. In Section 5, we prove that if A is a basic and connected selfinjective artin algebra whose Auslander-Reiten quiver Γ_{A} contains a non-periodic generalized standard right stable full translation subquiver which is closed under successors in Γ_{A}, then A is socle equivalent to an algebra of the form B̂/G, where B is a representation-infinite tilted algebra and G is an admissible infinite cyclic group of automorphisms of B̂. Moreover, if K is an algebraically closed field, then A is isomorphic to B̂/G.

For basic background concerning the representation theory applied here we refer to [3], [20], [23], [29].

The results of the paper were partially announced by the authors during the conferences at Cocoyoc (August 1994) and Padova (June 1994). The research was initiated during the stay of the second named author in Toruń and completed when the first named author visited Tsukuba. Both authors gratefully acknowledge the hospitality of each other’s university and support from the Polish Scientific Grant KBN No. 2P03A 020 08 and the Grant in Aid for Scientific Research (C) No. 08640008, the Ministry of Education of Japan.

2. Preliminary results

Throughout the paper, A will denote a fixed basic and connected selfinjective artin algebra over a commutative artin ring K, and {e_{i} | 1 ≤ i ≤ s} a (complete) set of primitive orthogonal idempotents of A such that 1 = e_{1} + … + e_{s}.

Let σ be an algebra automorphism of A. For a right A-module M, M_{σ} denotes the right A-module obtained from M by changing the operation of A as follows: m · a = mσ(a) for each a ∈ A and m ∈ M. Similarly, for a left A-module N, σN denotes the left A-module obtained from N by changing the operation of A as follows: a · m = σ(a)m for each a ∈ A and m ∈ M.

Let ν be the Nakayama automorphism of A with an A-bimodule isomorphism Θ_{r} : A → _{ν}D(A). Hence we have soc(e_{i}A) ≃ top (ν(e_{i})A)(= ν(e_{i})A/ν(e_{i})(rad A)) as right A-modules for all i ∈ {1, …, s}. Since {ν(e_{i})A | 1 ≤ i ≤ s} is a set of representatives of indecomposable projective right A-modules, there is a permutation of {1, …, n}, denoted again by ν, such that ν(e_{i})A ≃ e_{ν(e_{i})}A for all i ∈ {1, …, s}. Invoking the Krull-Schmidt theorem we may assume that ν(e_{i}A) = ν(e_{i})A = e_{ν(i)}A for all i ∈ {1, …, s}. Consider now the K-linear isomorphism ( )^{∗∗} : A → D(D(A)) given by a^{∗∗}(f) = f(a) for each a ∈ A and f ∈ D(A). Then a direct checking shows that Θ_{l} = D(Θ_{r})( )^{∗∗} : A → D(A)_{ν^{−}} is an A-bimodule isomorphism. Moreover, we have the following fact.

Lemma 2.1. The Nakayama automorphism ν coincides with the composite Θ_{l}^{−1}Θ_{r} : A → _{ν}A_{ν} as an A-bimodule isomorphism.

Proof. First observe that Θ_{l}(1_{A}) = D(Θ_{r})(1_{A}^{∗∗}) = 1_{A}^{∗∗}Θ_{r}. Then, for each a ∈ A, we get Θ_{l}(1_{A})(a) = 1^{∗∗}Θ_{r}(a) = 1^{∗∗}(Θ_{r}(1_{A})a) = (Θ_{r}(1_{A})a)(1_{A}) = Θ_{r}(1_{A})(a · 1_{A}) = Θ_{r}(1_{A})(a), and hence Θ_{l}(1_{A}) = Θ_{r}(1_{A}). Further, (Θ_{l}^{−1}Θ_{r})(a) = (Θ_{l}^{−1}Θ_{r})(a1_{A}) = Θ_{l}^{−1}Θ_{r}(a1_{A}) = Θ_{l}^{−1}(ν(a)Θ_{r}(1_{A})) = ν(a)Θ_{l}^{−1}Θ_{r}(1_{A}) = ν(a), for any a ∈ A. This implies that ν = Θ_{l}^{−1}Θ_{r}.

From now on we assume that I is a (two-sided) ideal of A, B = A/I and e is an idempotent of A such that e + I is an identity of B. We may assume that e = e_{1} + … + e_{t} for some t ≤ s, and {e_{i} | 1 ≤ i ≤ t} is the subset of {e_{i} | 1 ≤ i ≤ s} consisting of all idempotents e_{i} which are not in I. Then such an idempotent e is uniquely determined by I up to an inner automorphism of A, and we call it a residual identity of B. Note that B ≃ eAe/eIe and 1 − e ∈ I. An idempotent e′ is called a summand of an idempotent f, and we write e′ < f, if e′f = fe′ = e′, and is said to be primitive if the right ideal e′A generated by e′ is indecomposable. The residue class e′ + I of e′ in B = A/I will also be denoted by e′. The ν-orbit idempotent of e, denoted by ẽ, is the sum of all e_{i} of the form e_{i} = e_{ν^{m}(j)}, for some 1 ≤ j ≤ t and an integer m.

Lemma 2.2. The algebra ẽAẽ is selfinjective.

Proof. From definition of ẽ we know that, for each idempotent e_{i} with e_{i} < ẽ, soc(e_{i}A) = soc(e_{i}A)ẽ is a simple socle of e_{i}Aẽ. Moreover, the socle of ẽAẽ, as a right ẽAẽ-module, is a direct sum of pairwise nonisomorphic simple modules. Then the top of the left ẽAẽ-module D(ẽAẽ) is a direct sum of pairwise nonisomorphic simple modules, and consequently we have an epimorphism ẽAẽ → D(ẽAẽ) of left ẽAẽ-modules. Since ẽAẽ and D(ẽAẽ) have the same length as K-modules, we get ẽAẽ ≃ D(ẽAẽ) as left ẽAẽ-modules, and then also as right ẽAẽ-modules. Therefore ẽAẽ is selfinjective.

The next lemma shows when ẽ = 1.

Lemma 2.3. Assume ẽ_{j}AeAẽ_{j} = 0 for any e_{j} < 1 − ẽ. Then ẽ = 1.

Proof. Suppose ẽ ≠ 1. Since A is connected, there exist j ∈ {1, …, s} and some integer m such that e_{j}Ae_{ν^{−m}(i)} ≠ 0 for some e_{i} < e. Applying ν^{m} we get e_{ν^{m}(j)}Ae_{i} ≠ 0. But soc(e_{ν^{m}(j)}Ae_{i}A) ≃ top e_{ν^{m+1}(j)}A. Thus e_{ν^{m}(j)}Ae_{i}Ae_{ν^{m+1}(j)} ≠ 0, and hence ẽ_{j}Ae_{i}Aẽ_{j} ≠ 0. Since e_{i} < e and e_{j} < 1 − ẽ, this contradicts our assumption.

Corollary 2.4. Assume IeI = 0. Then ẽ = 1.

Proof. Clearly 1 − ẽ ∈ I. Then IeI = 0 implies (1 − ẽ)AeA(1 − ẽ) = 0, and consequently ẽ = 1 by Lemma 2.3.

The following lemma will be useful in our investigations.

Lemma 2.5. The following conditions are equivalent:

(i) ẽ(IeI) = 0. (ii) (IeI)ẽ = 0. (iii) ẽ(IeI)ẽ = 0. (iv) ẽIe = ν^{−}(e)Ie and ν^{−}(e)IeIe = 0.

(v) ẽIe = ν^{−}(e)Ie and ν^{−}(e)IeI = 0.

Proof. The implications (ii)⇒(iii) and (v)⇒(i) are obvious. For (i)⇒(ii), suppose that IeIẽ ≠ 0. Then ẽ(IeI)ẽ ≠ 0, as a left A-module, because ẽ(soc_{A}(IeIẽ)) ≠ 0. Hence ẽ(IeI) ≠ 0. For (iv)⇒(v) observe that if ν^{−}(e)IeI ≠ 0, then (ν^{−}(e)IeI)e ≠ 0. Therefore, it remains to show that (iii) implies (iv). Assume ẽ(IeI)ẽ = 0. Since e < ẽ and ν^{−}(e) < ẽ we then get ν^{−}(e)IeIe = ν^{−}(e)(IeI)e = 0. In order to prove the first claim it suffices to show that (ẽ − ν^{−}(e))Ie = 0. Suppose that (ẽ − ν^{−}(e))Ie ≠ 0. Then (ẽ − ν^{−}(e))IeA(ẽ − e) ≠ 0 because ν(ẽ − ν^{−}(e)) = ẽ − e and soc((ẽ − ν^{−}(e))IeA)(ẽ − e) ≠ 0. Since ẽ − e ∈ I, this implies that (ẽ − ν^{−}(e))IeI(ẽ − e) ≠ 0, which contradicts our assumption ẽ(IeI)ẽ = 0. This finishes the proof.

For a subset S of A, let l_{S}(I) be the left annihilator {a ∈ S | aI = 0} of I in S and r_{S}(I) the right annihilator {a ∈ S | Ia = 0} of I in S. Then r_{A}l_{A}(I) = I and l_{A}r_{A}(I) = I, and in particular we have l_{A}(I) ≠ 0 and r_{A}(I) ≠ 0. The following lemma proved in [26, Lemma 1.1] shows that a minimal cogenerator in mod B, or in mod B^{op}, can be obtained as an ideal in A.

Lemma 2.6. The right ideal ν^{−}(e)l_{A}(I) is a minimal injective cogenerator in mod B, and the left ideal r_{A}(I)ν(e) is a minimal injective cogenerator in mod B^{op}.

In the next three lemmas we assume that the ideal I satisfies the following two conditions:

(1) ẽ(IeI)ẽ = 0.

(2) ν^{−}(e)Ie is an injective cogenerator in mod eAe/eIe and eIν(e) is an injective cogenerator in mod(eAe/eIe)^{op}.

Lemma 2.7. For each integer m, ν^{m}(ν^{−}(e)Ie) and ν^{m}(eIν(e)) have A-bimodule structures, and there are isomorphisms

\begin{aligned}Θ′_{r} : ν^{−}(e)Ie & → _{ν}D(B), & Θ′_{l} : eIν(e) & → D(B)_{ν^{−}}, \\ Θ_{r}^{(m)} : ν^{m}(ν^{−}(e)Ie) & → _{ν}D(ν^{m}(B)), & Θ_{l}^{(m)} : ν^{m}(eIν(e)) & → D(ν^{m}(B))_{ν^{−}} \end{aligned}

of A-bimodules such that, for the canonical epimorphism ρ : A → A/I = B, the following diagrams are commutative:

\begin{matrix}ν^{−}(e)Ie & ⟶^{Θ_{r}^{−}} & _{ν}D(B) \\ ↓^{∩} & & ↓^{_{ν}D(ϱ)} \\ A & ⟶^{Θ_{r}} & _{ν}D(A) \end{matrix} \begin{matrix}eIν(e) & ⟶^{Θ′_{l}} & D(B)_{ν^{−}} \\ ↓^{∩} & & ↓^{D(ϱ)_{ν^{−}}} \\ A & ⟶^{Θ_{l}} & D(A)_{ν^{−}} \end{matrix}

where ν^{m}(B) denotes the factor algebra A/ν^{m}(I) and ν^{m}(b) = ν^{m}(a) + ν^{m}(I) for b = a + I ∈ B.

Proof. Clearly B is an A-bimodule and the canonical epimorphism ϱ : A → B is an A-bimodule epimorphism. Hence D(ϱ) : D(B) → D(A) is an A-bimodule monomorphism. Let φ = Θ_{r}^{−1}_{ν}D(ϱ) : _{ν}D(B) → _{ν}D(A) → A. Then _{ν}D(B) ≃ φ(_{ν}D(B)) as A-bimodules. Since D(B)I = 0 and D(B) = eD(B)e, we conclude that φ(_{ν}D(B)) is a B-submodule of ν^{−}(e)l_{A}(I)_{B}. Further, it folows from Lemma 2.6 that ν^{−}(e)l_{A}(I)_{B} is isomorphic to D(B)_{B}, and consequently we get φ(_{ν}D(B)) = ν^{−}(e)l_{A}(I)_{B}. Moreover, by our assumption (1) and Lemma 2.5, we have ν^{−}(e)Ie ⊆ l_{ν^{−}(e)Ae}(I). On the other hand, by our assumption (2), (ν^{−}(e)Ie)_{B} is an injective cogenerator in mod B. This implies φ(_{ν}D(B)) = ν^{−}(e)Ie, because 1−e ∈ I and l_{ν^{−}(e)Ae}(I) = ν^{−}(e)l_{A}(I). In particular, ν^{−}(e)Ie is an A-bimodule and φ is an A-bimodule isomorphism. Hence Θ′_{r} = φ^{−1} : ν^{−}(e)Ie → _{ν}D(B) satisfies the required conditions. Dually, we prove that there exists Θ′_{l} : eIν(e) → D(B)_{ν^{−}} satisfying the required conditions. The isomorphisms Θ_{r}^{(m)} and Θ_{l}^{(m)} are then easily obtained by substituting ν^{m}(I) for I and ν^{m}(B) for B in the above argument, because ν^{m} : A → A is an algebra isomorphism.

Lemma 2.8. For any 1 ≤ i ≤ t and any integer m, the following inclusions hold:

ν^{m−1}(e_{i})Ie ⊆ ν^{−}(e)Ie  and  eIν^{m}(e_{i}) ⊆ eIν(e).

Moreover, the restrictions of Θ′_{r} and Θ′_{l} induce the following isomorphisms:

ν^{m−1}(e_{i})Ie ≃ ν^{m}(e_{i})D(B)e  and  eIν^{m}(e_{i}) ≃ eD(B)ν^{m−1}(e_{i}).

Proof. Fix 1 ≤ i ≤ t and an integer m. Assume first that ν^{m−1}(e_{i}) < ν^{−}(e). Then ν^{m−1}(e_{i})ν^{−}(e) = ν^{m−1}(e_{i}) and it follows from Lemma 2.7 that Θ′_{r} induces an isomorphism ν^{m−1}(e_{i})Ie ≃ ν^{m}(e_{i})D(B)e. Assume now ν^{m−1}(e_{i}) ≮ ν^{−}(e). In this case, ν^{m−1}(e_{i}) < ẽ − ν^{−}(e). Applying now Lemma 2.5 and our assumption (1) we get ν^{m−1}(e_{i})Ie = 0. On the other hand, ν^{m}(e_{i}) < ν(ẽ) − ν(ν^{−}(e)) = ẽ− e, and so we have ν^{m}(e_{i})D(B)e = 0. Thus Θ′_{r} induces the trivial isomorphism ν^{m−1}(e_{i})Ie ≃ ν^{m}(e_{i})D(B)e between the zero modules. The proof of the remaining part is dual.

Lemma 2.9. For any 1 ≤ i, j ≤ t and any integer m,

e_{ν^{m}(i)}ν^{m}(I)e_{ν^{m+1}(j)} = e_{ν^{m}(i)}ν^{m+1}(I)e_{ν^{m+1}(j)},

and the following diagram is commutative:

e_{i}D(B)e_{j}  ⟶^{Θ_{l}^{−1}}  e_{i}Ie_{ν(j)}
↓^{Θ_{r}^{−1}}  ↓_{ν^{m}}
ν^{−}(e_{i})Ie_{j} ⟶^{ν^{m+1}} e_{ν^{m}(i)}Ae_{ν^{m+1}(j)}

Proof. Let 1 ≤ i, j ≤ t and m ∈ ℤ. We know from Lemma 2.1 that ν = Θ_{l}^{−1}Θ_{r}. Moreover, by Lemma 2.7, Θ_{r} induces an isomorphism ν^{−}(e_{i})Ie_{j} ⟶^{∼} e_{i}D(B)e_{j} and Θ_{l} an isomorphism e_{i}Ie_{ν(j)} → e_{i}D(B)e_{j}. Hence

e_{i}Ie_{ν(j)} = ν(ν^{−}(e_{i})Ie_{j}) = e_{i}ν(I)e_{ν(j)}.

Applying ν^{m} we then get

e_{ν^{m}(i)}ν^{m}(I)e_{ν^{m+1}(j)} = e_{ν^{m}(i)}ν^{m+1}(I)e_{ν^{m+1}(j)}.

3. Galois coverings of selfinjective algebras

Recall that a K-category R is called locally bounded [5] if the following conditions are satisfied:

(a) distinct objects of R are not isomorphic;

(b) the algebras R(x, x) are local;

(c) for each object x of R, ∑_{y∈R} |R(x, y)| and ∑_{y∈R} |R(y, x)| are finite. Here, for a K-module V, we denote by |V | its length over K. A functor F : R → Λ between two locally bounded K-categories R and Λ is called a covering functor if the induced maps

⨁_{F (y)=a} R(x, y) → Λ(F (x), a)  and  ⨁_{F (y)=a} R(y, x) → Λ(a, F (x))

are isomorphisms for all objects x ∈ R and a ∈ Λ (see [5], [12]).

Let R be a locally bounded K-category and G a group of K-linear automorphisms of R. We assume that G acts freely on the objects of R, that is, gx ≠ x for each object x of R and g ≠ 1 in G. It follows from [12, Proposition 3.1] that the quotient R/G exists in the category of locally bounded K-categories and there is a canonical covering functor F : R → R/G. Moreover, F is universal with respect to the property F g = F for each g ∈ G, that is, each functor E : R → Λ which satisfies Eg = E for each g ∈ G admits a unique factorization E = HF. The objects of R/G are the orbits of G in the set of objects of R. A morphism f : a → b between two objects in R/G is a family f = (_{y}f_{x}) ∈ ∏_{x,y} R(x, y), where x, y range over a, b, respectively, and f satisfies the relation g(_{y}f_{x}) = _{gy}f_{gx} for all g ∈ G and all x, y. The composition ef of f : a → b and e : b → c in R/G is defined by zef_{x} = ∑_{y∈b} _{z}e_{y} _{y}f_{x} (this sum makes sense since R is locally bounded). Then we have the canonical covering functor F : R → R/G which assigns to each object x of R its G-orbit, and to a morphism ξ ∈ R(x, y) the family F ξ = (_{hy}F ξ_{gx})_{g,h∈G} such that _{hy}F ξ_{gx} = gξ if g = h and _{hy}F ξ_{gx} = 0 for g ≠ h. Frequently, we are dealing with a K-linear functor E : R → Λ such that Eg = E for all g ∈ G. Such a functor induces an isomorphism R/G ≃ Λ if and only if E is surjective on the objects and G acts transitively on the fiber E^{−1}(a) of each object a of Λ. If this is the case, the functor E : R → Λ is called a Galois covering. Clearly, the above-defined functor F : R → R/G is a Galois covering. We refer to [5], [12] for more details on covering functors.

We may consider the algebra A as a (locally bounded) K-category whose set of objects is the fixed set {e_{i} | 1 ≤ i ≤ s} of primitive orthogonal idempotents, and the K-module of morphisms Hom_{A}(e_{i}, e_{j}) from e_{i} to e_{j} is equal e_{j}Ae_{i}. Then A^{op} is the opposite K-category to A, and so Hom_{A^{op}}(e_{i}, e_{j}) = e_{i}Ae_{j} for all 1 ≤ i, j ≤ s.

Now let I be an ideal of A, B = A/I, and e = e_{1} + … + e_{t}, for some t ≤ s, a residual identity of B. Assume that I satisfies the conditions (1) and (2), introduced in Section 1, and the following condition:

(3) The canonical algebra homomorphism ϱ : eAe → eAe/eIe splits. We claim that there is a retraction ι of ϱ (an algebra homomorphism ι : eAe/eIe → eAe with ρι = 1 on eAe/eIe) such that ι(e_{i}) = e_{i} for any 1 ≤ i ≤ t. Indeed, let ι_{0} : eAe/eIe → eAe be a retraction of ϱ. Then e_{i} + eIe = ι_{0}(e_{i}) for any 1 ≤ i ≤ t. Since {ι_{0}(e_{i}) | 1 ≤ i ≤ t} is a complete set of primitive orthogonal idempotents of eAe and (eIe)^{2} = 0, by the imposed condition (1), we deduce from the Krull-Schmidt theorem that there is an invertible element u of eAe such that e_{i} = u^{−1}ι_{0}(e_{i})u for all 1 ≤ i ≤ t. Put ι(e_{i}be_{j}) = u^{−1}ι_{0}(e_{i}be_{j})u for all 1 ≤ i, j ≤ t and b ∈ eAe/eIe. Then ι : eAe/eIe → eAe is an algebra homomorphism such that ϱι = 1 on eAe/eIe and ι(e_{i}) = e_{i} for any 1 ≤ i ≤ t. Hence, under the above assumptions, we may assume that ι is a retraction of ϱ such that ι(e_{i}\overline{a}e_{j}) = e_{i}ι(\overline{a})e_{j} for all 1 ≤ i, j ≤ t and \overline{a} = a + eIe ∈ eAe/eIe. Further, by the canonical algebra homomorphisms A → B → eAe/eIe, both B and eAe/eIe can be considered as A-algebras. Hence B and eAe/eIe are isomorphic A-algebras, and from now on we shall identify them. Moreover, we shall identify B with ι(B).

Consider now the repetitive algebra

B̂ = [\begin{matrix}... & ... \\ & B_{m−1} & Q_{m−1} \\ & & B_{m} & Q_{m} \\ & & & B_{m+1} & Q_{m+1} \\ & & & & ... & ... \end{matrix}]

of B with B_{m} = B and Q_{m} = D(B) for all m ∈ ℤ. We denote by {e_{m,i} | 1 ≤ i ≤ t, m ∈ ℤ} the canonical set of primitive orthogonal idempotents of B̂ induced by {e_{i} | 1 ≤ i ≤ t}. Hence, for 1 ≤ i ≤ t and m ∈ ℤ, we have

e_{m,i}B̂ = e_{i}B ⊕ e_{i}D(B)  and  e_{m,i}B̂e_{m+1,j} = e_{i}D(B)e_{j}.

Applying the Nakayama automorphism ν of A to the idempotents e_{i}, 1 ≤ i ≤ t, we may get an idempotent ν(e_{i}) = e_{ν(i)}, with ν(i) > t, equivalently e_{ν(i)} ∈ I, and hence e_{ν(i)} is a zero element of B. Therefore, we introduce the convention that e_{n,i}B̂ = 0 = B̂e_{n,i} for i > t and n ∈ ℤ.

We may clearly consider B̂ as a locally bounded K-category whose set of objects is equal {e_{m,i} | i ≤ t, m ∈ ℤ} and HomB̂(e_{n,i}, e_{m,j}) = e_{m,j}B̂e_{n,i} for all 1 ≤ i, j ≤ t, m, n ∈ ℤ. The canonical shifting automorphism νB̂ : B̂ → B̂ with νB̂(e_{m,i}) = e_{m+1,i} for all m ∈ ℤ, 1 ≤ i ≤ t, is called the Nakayama automorphism of B̂. An automorphism φ of B̃ is said to be positive (respectively, strictly positive) if for each object e_{m,i} of B̂, there are an integer p ≥ m (respectively, p > m) and 1 ≤ j ≤ t such that φ(e_{m,i}) = e_{p,j}. Hence, a positive automorphism of B̂ is a shift of B̂ in the same direction as νB̂.

The main purpose of this section is to prove the following theorem and some its consequences.

Theorem 3.1. Let A be a basic and connected selfinjective artin algebra, I an ideal of A, B = A/I, and e a residual identity of B. Assume that the following conditions are satisfied:

(1) ẽ(IeI)ẽ = 0.

(2) ν^{−}(e)Ie is an injective cogenerator in mod eAe/eIe and eIν(e) is an injective cogenerator in mod(eAe/eIe)^{op}.

(3) The canonical algebra epimorphism ϱ : eAe → eAe/eIe splits.

Then there is a Galois covering F : B̂ → ẽAẽ with the Galois group G infinite cyclic generated by an automorphism φνB̂, where φ is a positive automorphism of B̂.

We divide the proof of the above theorem into several steps. First we define a functor F : B̂ → ẽAẽ as follows: For each object e_{m,i} of B̂ we put F (e_{m,i}) = e_{ν^{m}(i)}. Then:

F : HomB̂(e_{m,i}, e_{n,j}) → Hom_{ẽAẽ}(F (e_{m,i}), F (e_{n,j})) = Hom_{ẽAẽ}(e_{ν^{m}(i)}, e_{ν^{n}(j)})

is defined as follows:

(a) If m = n, then F is the composition

e_{m,j}B̂e_{m,i} = e_{j}Be_{i} ^{⊂}⟶^{ι} e_{j}Ae_{i} ⟶^{ν^{m}} e_{ν^{m}(j)}Ae_{ν^{m}(i)}.

(b) If n = m − 1, then F is the composition

e_{m−1,j}B̂e_{m,i} = e_{j}D(B)e_{i} ⟶_{∼}^{Θ_{r}^{−1}} e_{ν^{−}(j)}Ie_{i} ^{⊂}⟶ e_{ν^{−}(j)}Ae_{i} ⟶^{ν^{m}} e_{ν^{m−1}(j)}Ae_{ν^{m}(i)}.

(c) F is zero for n ≠ m and n ≠ m − 1. In the case when n = m − 1, we have used the isomorphism Θ_{r} : A → _{ν}D(A). But, by Lemma 2.9, the composition in (b) equals the composition

e_{m−1,j}B̂e_{m,i} = e_{j}D(B)e_{i} ⟶_{∼}^{Θ_{l}^{−1}} e_{j}Ie_{ν(i)} ^{⊂}⟶ e_{j}Ae_{ν(i)} ⟶^{ν^{m−1}} e_{ν^{m−1}(j)}Ae_{ν^{m}(i)}.

Using the fact that ι and ν are algebra homomorphisms and Θ_{r} defines a right eAehomomorphism from D(B)e to ν^{−}(e)Ie, it is easy to check that F defined above is a functor from B̂ to ẽAẽ.

Lemma 3.2. F : B̂ → ẽAẽ is a covering functor.

Proof. We have to show that for any 1 ≤ i, j ≤ t and m, n ∈ ℤ, the maps

⨁_{k∈ℤ} HomB̂(e_{m,i}, e_{k,ν^{n−k}(j)}) → Hom_{ẽAẽ}(e_{ν^{m}(i)}, e_{ν^{n}(j)})

and

⨁_{k∈ℤ} HomB̂(e_{k,ν^{m−k}(i)}, e_{n,j}) → Hom_{ẽAẽ}(e_{ν^{m}(i)}, e_{ν^{n}(j)}),

induced by F, are isomorphisms. By definition of B̂ we get

⨁_{k∈ℤ} HomB̂(e_{m,i}, e_{k,ν^{n−k}(j)})

= HomB̂(e_{m,i}, e_{m,ν^{n−m}(j)}) ⊕ HomB̂(e_{m,i}, e_{m−1,ν^{n−m+1}(j)}).

Moreover, we have the isomorphisms

e_{ν^{n−m}(j)}Be_{i} ⊕ e_{ν^{n−m+1}(j)}D(B)e_{i}
1⊕Θ_{r}^{−1}↓∼
e_{ν^{n−m}(j)}Be_{i} ⊕ e_{ν^{n−m}(j)}Ie_{i}  = e_{ν^{n−m}(j)}Ae_{i}
ν^{m}↓∼
e_{ν^{n}(j)}Ae_{ν^{m}(j)}

where, in the case when ν^{n−m}(j) > t, e_{ν^{n−m}(j)}Be_{i} = 0, by our convention, and e_{ν^{n−m}(j)}Ie_{i} = e_{ν^{n−m}(j)}Ae_{i} by the fact that 1 − e ∈ I. This proves that the first map is an isomorphism. By making use of Θ_{l} instead of Θ_{r} in the considerations above, we prove similarly that the second map is also an isomorphism.

Our next aim is to define an infinite cyclic group G of automorphisms of B̂ acting freely on the objects of B̂, and then to show that F : B̂ → ẽAẽ is a Galois covering with the Galois group G.

For each 1 ≤ i ≤ t, we put

η(i) = min{k > 0 | ν^{k}(i) ≤ t}

and define a map g : ℤ × {1, …, t} → ℤ × {1, …, t} by

g(m, i) = (m − η(i), ν^{η(i)}(i))

for all (m, i) ∈ ℤ × {1, …, t}. Observe that g is a bijection, because the action of ν on the set {1, …, s} has finite order, and, for 1 ≤ i, j ≤ t, ν^{η(i)}(i) = ν^{η(j)}(j) implies i = j. This allows to define a bijection on the objects of B̂, denoted again by g, by

g(e_{m,i}) = e_{g(m,i)}

for all (m, i) ∈ ℤ × {1, …, t}.

Lemma 3.3. For each pair (m, i) ∈ ℤ × {1, …, t}, the cyclic group generated by g acts transitively on the fiber F^{−1}(e_{ν^{m}(i)}).

Proof. Fix (m, i) ∈ ℤ × {1, …, t}. Then

F^{−1}(e_{ν^{m}(i)}) = {e_{m−k,ν^{k}(i)} | k ∈ ℤ, 1 ≤ ν^{k}(i) ≤ t}.

It suffices to show that for any k, n ∈ ℤ with k < n and 1 ≤ ν^{n}(i) ≤ t, there exists l ∈ ℤ such that g^{l}(e_{m−k,ν^{k}(i)}) = e_{m−n,ν^{n}(i)}. We define a sequence of integers

k = s_{0} < s_{1} < s_{2} < …

inductively by s_{p+1} = s_{p} + η(ν^{s_{p}}(i)). Observe that

g(e_{m−s_{p},ν^{sp} (i)}) = e_{m−s_{p+1},ν^{s_{p+1}} (i)}

for all p ≥ 0. Further, ν^{q}(i) > t for any integer q with s_{p} < q < s_{p+1}. Moreover, by our assumptions, we have k < n and ν^{n}(i) ≤ t. Hence, n = s_{l} for some l > 0, and consequently we obtain g^{l}(e_{m−k,ν^{k}(i)}) = e_{m−n,ν^{n}(i)}.

We shall now extend the bijection g on the objects of B̂ to an automorphism of B̂ such that F g = F. For this purpose, we prove several technical lemmas. Observe first that for (m, i), (n, j) ∈ ℤ × {1, …, t} we have by Lemma 3.2 the following isomorphisms:

(∗) e_{ν^{n}(j)}A_{ν^{m}(i)} ≃ ⨁_{k} HomB̂(e_{0,ν^{m}(i)}, e_{n−k,ν^{k}(i)})
≃ HomB̂(e_{0,ν^{m}(i)}, e_{0,ν^{m}(j)}) ⊕ HomB̂(e_{0,ν^{m}(i)}, e_{−1,ν^{m+1}(j)})
≃ e_{ν^{m}(j)}Be_{ν^{m}(i)} ⊕ D(e_{ν^{m}(i)}Be_{ν^{m+1}(j)})

and

(∗∗)  e_{ν^{n}(j)}Ae_{ν^{m}(i)} ≃ ⨁_{k} HomB̂(e_{m−k,ν^{k}(i)}, e_{0,ν^{n}(j)})
≃ HomB̂(e_{0,ν^{n}(i)}, e_{0,ν^{n}(j)}) ⊕ HomB̂(e_{1,ν^{n−1}(i)}, e_{0,ν^{n}(j)})
≃ e_{ν^{n}(j)}Be_{ν^{n}(i)} ⊕ D(e_{ν^{n−1}(i)}Be_{ν^{n}(j)}).

Lemma 3.4. Assume 1 ≤ i, j ≤ t and e_{j}Be_{i} is nonzero. Then η(j) = η(i) or η(j) = η(i) + 1.

Proof. Note that for any k ∈ ℤ we have the composed epimorphism

e_{ν^{k}(j)}Ae_{ν^{k}(i)} ⟶^{ν^{−k}} e_{j}Ae_{i} ⟶ e_{j}Be_{i},

and hence e_{ν^{k}(j)}Ae_{ν^{k}(i)} ≠ 0, because e_{j}Be_{i} ≠ 0 by our assumption.

Consider first the case η(j) = 1. Clearly, e_{j}Be_{i} ≠ 0 implies that e_{j}Ie_{i} is a proper subset of e_{j}Ae_{i}. Further, by Lemma 2.8, we have e_{ν(j)}D(B)e_{i} ≃ e_{j}Ie_{i}. Applying now (∗∗) for n = 1 we conclude that e_{ν(j)}Be_{ν(i)} ≠ 0, and so η(i) = 1.

Assume now that η(j) > 1. Let m = η(i) and n = η(j). Since e_{ν^{n}(j)}Ae_{ν^{n}(i)} ≠ 0, applying (∗∗), we get e_{ν^{n}(j)}Be_{ν^{n}(i)} ≠ 0 or e_{ν^{n−1}(i)}Be_{ν^{n}(j)} ≠ 0. Hence η(i) ≤ η(j). Moreover, since e_{ν^{m}(j)}Ae_{ν^{m}(i)} ≠ 0, applying (∗), we deduce that e_{ν^{m}(j)}Be_{ν^{m}(i)} ≠ 0 or e_{ν^{m}(i)}Be_{ν^{m+1}(j)} ≠ 0, and so η(j) ≤ η(i) + 1. Therefore, we get η(i) ≤ η(j) ≤ η(i) + 1, which proves our claim.

Lemma 3.5. Assume 1 ≤ i, j ≤ t and η(i) = η(j) = n. Then

e_{ν^{n}(j)}ν^{n}(B)e_{ν^{n}(i)} = e_{ν^{n}(j)}Be_{ν^{n}(i)}

and

e_{ν^{n−1}(i)}ν^{n}(I)e_{ν^{n}(j)} = e_{ν^{n−1}(i)}Ie_{ν^{n}(j)}.

Proof. First we show the second equality. It follows from Lemma 2.9 that e_{i}ν(I)e_{ν(j)} = e_{i}Ie_{ν(j)} which is the second equality for n = 1. Suppose now n > 1. Then ν^{r}(j) > t for 1 ≤ r ≤ n − 1, by definition of η(j). We then get e_{i}Ae_{ν(j)} = e_{i}Ie_{ν(j)}. Further, by Lemma 2.9, we have e_{ν^{n−1}(i)}ν^{n−1}(I)e_{ν^{n}(j)} = e_{ν^{n−1}(i)}ν^{n}(I)e_{ν^{n}(j)}. Hence, e_{ν^{n−1}(i)}ν^{n}(I)e_{ν^{n}(j)} = e_{ν^{n−1}(i)}Ae_{ν^{n}(j)}. On the other hand, ν^{n−1}(i) > t, because n = η(i), and so e_{ν^{n−1}(i)}Ie_{ν^{n}(j)} = e_{ν^{n−1}(i)}Ae_{ν^{n}(j)}. Consequently, e_{ν^{n−1}(i)}ν^{n}(I)e_{ν^{n}(j)} = e_{ν^{n−1}(i)}Ie_{ν^{n}(j)}. Now, applying ν^{−n} to the proved equality, we have e_{ν^{−}(i)}Ie_{j} = e_{ν^{−}(i)}ν^{−n}(I)e_{j}. From Lemma 2.7 we have D(e_{j}Be_{i}) = D(e_{j}ν^{−n}(B)e_{i}), and so applying D(ν^{n}) we get D(ν^{n}(e_{j}Be_{i})) = D(e_{ν^{n}(j)}Be_{ν^{n}(i)}). Hence we obtain the first required equality.

Lemma 3.6. Assume 1 ≤ i, j ≤ t, η(j) = η(i) + 1, η(i) = n. Then

e_{ν^{n}(j)}ν^{n}(B)e_{ν^{n}(i)} = e_{ν^{n}(j)}Ie_{ν^{n}(i)}

and

e_{ν^{n}(i)}ν^{n}(I)e_{ν^{n+1}(j)} = e_{ν^{n}(i)}Be_{ν^{n+1}(j)}.

Proof. Applying the isomorphisms from Lemma 2.7 as in the proof of Lemma 3.5, we deduce that the second required equality follows from the first one. Hence we have to prove only the first equality.

Since η(j) > n, we have e_{ν^{n}(j)} ∈ I, and so e_{ν^{n}(j)}Ie_{ν^{n}(i)} = e_{ν^{n}(j)}Ae_{ν^{n}(i)}. Moreover, e_{ν(j)} ∈ I because η(j) > 1. We claim that e_{j}Ie_{i} = 0. Indeed, suppose that e_{j}Ie_{i} ≠ 0. Then soc(e_{j}Ie_{i}A) ≃ top (e_{ν(j)}A) as right A-modules, and hence e_{j}Ie_{i}Ae_{ν(j)} ≠ 0. But this implies ẽ(IeI)ẽ ≠ 0 because e_{ν(j)} ∈ I, contradicting the assumption (1). Therefore, e_{j}Ie_{i} = 0 and e_{j}Ae_{i} = e_{j}Be_{i}. Applying ν^{n} we get e_{ν^{n}(j)}Ae_{ν^{n}(i)} = e_{ν^{n}(j)}ν^{n}(B)e_{ν^{n}(i)}. Since e_{ν^{n}(j)} ∈ I, this gives the required equality e_{ν^{n}(j)}ν^{n}(B)e_{ν^{n}(i)} = e_{ν^{n}(j)}Ie_{ν^{n}(i)}.

Fix now two objects e_{m,i} and e_{n,j} of B̂ and consider the K-linear homomorphisms

F_{1} : HomB̂(e_{m,i}, e_{n,j}) → F (HomB̂(e_{m,i}, e_{n,j})),
F_{2} : HomB̂(g(e_{m,i}), g(e_{n,j})) → F (HomB̂(g(e_{m,i}), g(e_{n,j})))

induced by the functor F : B̂ → ẽAẽ. It follows from definition of F that both F_{1} and F_{2} are monomorphisms.

We shall prove that the images of F_{1} and F_{2} are equal, and then

F_{2}^{−1}F_{1} : HomB̂(e_{m,i}, e_{n,j}) → HomB̂(g(e_{m,i}, e_{n,j}))

will define the required action of g on HomB̂(e_{m,i}, e_{n,j}). We have several cases to consider.

Assume first that HomB̂(e_{m,i}, e_{n,j}) ≠ 0. Then n = m or n = m − 1.

(i) Suppose n = m. Then e_{j}Be_{i} = HomB̂(e_{m,i}, e_{m,j}) ≠ 0 and hence η(j) = η(i) or η(j) = η(i) + 1, by Lemma 3.4. In the case when η(j) = η(i), by Lemma 3.5, we have e_{ν^{k}(j)}ν^{k}(B)e_{ν^{k}(i)} = e_{ν^{k}(j)}Be_{ν^{k}(i)}, where k = η(i). Then, applying ν^{m−k}, we obtain e_{ν^{m}(j)}ν^{m}(B)e_{ν^{m}(i)} = e_{ν^{m}(j)}ν^{m−k}(B)e_{ν^{m}(i)}. This shows that Im F_{1} = Im F_{2} in this case. In the case when η(j) = η(i) + 1, applying Lemma 3.6, we have e_{ν^{k}(j)}ν^{k}(B)e_{ν^{k}(i)} = e_{ν^{k}(j)}Ie_{ν^{k}(i)}, where again k = η(i). Hence e_{ν^{m}(j)}ν^{m}(B)e_{ν^{m}(i)} = e_{ν^{m}(j)}ν^{m−k}(I)e_{ν^{m}(i)}, and consequently Im F_{1} = Im F_{2}.

(ii) Suppose n = m − 1. Then e_{j}D(B)e_{i} = HomB̂(e_{m,i}, e_{m−1,j}) ≠ 0, and hence, by Lemma 3.4, η(i) = η(j) or η(i) = η(j) + 1. For η(i) = η(j), it follows from Lemma 3.5 that e_{ν^{k−1}(i)}ν^{k}(I)e_{ν^{k}(j)} = e_{ν^{k−1}(i)}Ie_{ν^{k}(i)}, and so e_{ν^{m−1}(i)}ν^{m}(I)e_{ν^{m}(j)} = e_{ν^{m−1}(i)}ν^{m−k}(I)e_{ν^{m}(i)}, where k = η(j). For η(i) = η(j) + 1, it follows from Lemma 3.6 that e_{ν^{k}(j)}ν^{k}(I)e_{ν^{k+1}(i)} = e_{ν^{k}(j)}Be_{ν^{k+1}(j)}, and so e_{ν^{m−1}(j)}ν^{m−1}(I)e_{ν^{m}(i)} = e_{ν^{m−1}(j)}ν^{m−(k+1)}(B)e_{ν^{m}(j)}, where again k = η(j). Hence, in both cases we have Im F_{1} = Im F_{2}. We have also proved that HomB̂(e_{m,i}, e_{n,j}) ≠ 0 implies HomB̂(g(e_{m,i}), g(e_{n,j})) ≠ 0. Assume next that HomB̂(e_{m,i}, e_{n,j}) = 0. We have to show that also

HomB̂(e_{g(m,i)}, e_{g(n,j)}) = HomB̂(g(e_{m,i}), g(e_{n,j})) = 0.

Suppose HomB̂(e_{g(m,i)}, e_{g(n,j)})) ≠ 0. Then HomB̂^{op}(e_{g(n,j)}, e_{g(m,i)}) ≠ 0. Observe now that the conditions (1), (2) and (3) imposed on A are left-right symmetric. For each 1 ≤ i ≤ t, put

ξ(i) = min{k > 0 | ν^{−k}(i) ≤ t}

and define h : ℤ × {1, …, t} → ℤ × {1, …, t} by

h(m, i) = (m + ξ(i), ν^{−ξ(i)}(i))

for all (m, i) ∈ ℤ × {1, …, t}. By the above remark, applied now to left B̂-modules (right B̂^{op}-modules) we deduce that HomB̂^{op}(e_{g(n,j)}, e_{g(m,i)}) ≠ 0 implies

HomB̂^{op}(e_{hg(n,j)}, e_{hg(m,i)}) = HomB̂^{op}(h(e_{g(n,j)}), h(e_{g(m,i)})) ≠ 0.

Hence we have HomB̂(e_{hg(m,i)}, e_{hg(n,j)}) ≠ 0. But ξ(ν^{η(i)}(i)) = η(i) for any 1 ≤ i ≤ t, and so hg(m, i) = (m, i) and hg(n, j) = (n, j). Therefore, we get HomB̂(e_{m,i}, e_{n,j}) ≠ 0 which contradicts our assumption. This shows that indeed Hom(e_{m,i}, e_{n,j}) = 0 implies HomB̂(g(e_{m,i}), g(e_{n,j})) = 0, and we are done.

Denote now by G the group of K-linear automorphisms of B̂ generated by g. Clearly, G is infinite cyclic, F g′ = F for any element g′ of G, and, by Lemma 3.3, G acts transitively on the fibres F^{−1}(e_{ν^{m}(i)}) of the covering F : B̂ → ẽAẽ. Therefore, F is a Galois covering with the Galois group G and ẽAẽ ≃ B̂/G. Finally, observe that g^{−1} is defined on the objects of B̂ by the formula

g^{−1}(e_{m,i}) = e_{m+η(i),ν^{−η(i)}(i)}

for all (m, i) ∈ ℤ × {1, …, t}. Since ν(i) ≥ 1 for any 1 ≤ i ≤ t, we infer that g^{−1} = φνB̂ for some positive automorphism φ of B̂ with φ(e_{m,i}) = e_{m+η(i)−1,ν^{−η(i)}(i)}, and obviously G is generated by φνB̂. This finishes the proof of Theorem 3.1.

We are now able to prove a criterion for a selfinjective artin algebra A to be of the form B̂/G. Before, we recall the following proposition proved in [23, Proposition 2.3].

Proposition 3.7. Let A be a selfinjective artin algebra, I an ideal of A, B = A/I, e a residual identity of B, and assume that IeI = 0. Then the following conditions are equivalent:

(i) Ie is an injective cogenerator in mod B. (ii) eI is an injective cogenerator in mod B^{op}. (iii) eI = r_{A}(I). (iv) Ie = l_{A}(I). Moreover, under these conditions, socA ⊆ I and eIe = l_{eAe}(I) = r_{eAe}(I).

Theorem 3.8. Let A be a basic and connected selfinjective artin algebra, I an ideal of A, B = A/I, and e a residual identity of B. Assume IeI = 0, Ie = l_{A}(I) and the canonical algebra epimorphism eAe → eAe/eIe splits. Then there is a Galois covering F : B̂ → A with the Galois group G infinite cyclic generated by an automorphism φνB̂ of B̂, where φ is a positive automorphism of B̂.

Proof. We identify again B with eAe/eIe. Since IeI = 0, it follows from Corollary 2.4 that ẽ = 1. Moreover, by the above proposition Ie is an injective cogenerator in mod B and eI is an injective cogenerator in mod B^{op}. Clearly, then Ie = ν^{−}(e)Ie and eI = eIν(e) (see also Lemma 2.6). Therefore, A satisfies the conditions (1), (2) and (3) of Theorem 3.1, and this finishes the proof.

We end this section with the following consequence of the above results.

Corollary 3.9. Let A be a basic and connected selfinjective artin algebra, I an ideal of A, and B = A/I. Assume I^{2} = 0, I is an injective cogenerator as left and right B-module, and the canonical algebra epimorphism A → A/I splits. Then there is a Galois covering F : B̂ → A with the Galois group generated by φνB̂, where (in our notations) φ is an automorphism of B̂ such that φ(e_{m,i}) = e_{m,ν^{−}(i)} for any integer m and 1 ≤ i ≤ s.

Proof. Since I is contained in the radical of A, 1 = ∑_{i=1}^{s} e_{i} is a residual identity of B = A/I. Hence, e_{ν(i)} ∈ B for any 1 ≤ i ≤ s. This implies that η(i) = 1 for any 1 ≤ i ≤ s. Hence, by definition of g, we have g(m, i) = (m − 1, ν(i)) and φ(e_{m,i}) = e_{m,ν^{−}(i)}.

Remark 3.10. Let B be any basic and connected artin algebra and A = B ⋉D(B) the trivial extension of B by D(B). Then the Nakayama automorphism ν of A is the identity and I = D(B) satisfies the conditions of the above corollary. Hence Corollary 3.9 generalizes the well-known fact that the natural functor B̂ → B ⋉ D(B) = A is a Galois covering with the Galois group generated by νB̂.

4. Selfinjective algebras with deforming ideals

In our paper [26] we investigated ring-theoretic properties of certain ideals of selfinjective artin algebras, called deforming ideals. Recall from [[26], (2.1)] that, if I is an ideal of a selfinjective artin algebra and e a residual identity of B = A/I, then I is called deforming if the ordinary quiver Q_{B} of B has no oriented cycles and l_{eAe}(I) = eIe = r_{eAe}(I). An important class of deforming ideals is formed by the ideals I satisfying the conditions: Q_{B} has no oriented cycles, IeI = 0 and Ie is an injective cogenerator in mod B (see Proposition 3.7). In this section, applying the main results of Section 3 and [26], we determine the structure of some classes of selfinjective artin algebras having deforming ideals.

Theorem 4.1. Let A be a basic and connected selfinjective artin algebra over a commutative artin ring K. Let I be an ideal of A, B = A/I, and e a residual identity of B. Assume that the ordinary quiver Q_{B} of B has no oriented cycles, IeI = 0 and Ie is an injective cogenerator in mod B. Then A is socle equivalent to an algebra B̂/G where G is an infinite cyclic group of automorphisms of B̂ generated by φνB̂, for some positive automorphism φ of B̂. Moreover, if K is an algebraically closed field, then A is isomorphic to B̂/G.

Proof. Assume first that K is an algebraically closed field. Then it follows from [26, Theorem 3.2] that the Hochschild cohomology H^{2}(eAe/eIe, eIe) vanishes, and hence the canonical algebra epimorphism eAe → eAe/eIe splits. Therefore, applying Theorem 3.8, we deduce that A = B̂/G, where G is an infinite cyclic group generated by φνB̂, for some positive automorphism φ of B̂.

If K is an arbitrary commutative artin ring (even a field), then the canonical algebra epimorphism eAe → eAe/eIe does not necessarily split (see example below). But we may replace A by a socle equivalent selfinjective algebra A[I] satisfying the conditions of Theorem 3.8. Indeed, following [26, Section 4], consider the algebra A[I] whose K-linear structure is that of B ⊕ I and the multiplication is given by

(b, x)(b′, x) = (bb′, bx′ + xb′ + xx′)

for all b, b′ ∈ B and x, x′ ∈ I. Then A[I] is a basic and connected selfinjective artin K-algebra, I = {(0, x) | x ∈ I} is an ideal of A[I], e a residual identity of B = A[I]/I, IeI = 0, Ie = l_{A[I]}(I), and clearly the canonical algebra epimorphism eA[I]e → eA[I]e/eIe splits. Hence, by Theorem 3.8, A[I] is isomorphic to an algebra B̂/G, where G is an infinite cyclic group generated by φνB̂, for some positive automorphism φ of B̂. Moreover, it is shown in [26, Theorem 4.1] that A and A[I] are socle equivalent. This finishes the proof.

Example 4.2. Let K ⊆ L be a finite field extension of a field K of characteristic 2 such that H^{2}(L, L) ≠ 0, where L is considered as a K-algebra. For example, we may take K = (ℤ/2ℤ)(x) with indeterminate x and L = K[y]/(y^{2} − x), and easily check that the algebra L[y]/(y^{2} − x)^{2} is a non-splittable extension of the K-algebra L over itself. Now take a 2-cocycle α : L × L → L corresponding to a non-splittable extension 0 → L → E → L → 0. Let Q = (Q_{0}, Q_{1}) be a finite quiver with the set of vertices Q_{0} and the set of arrows Q_{1}. Assume Q has no oriented cycles and double arrows. Denote by Q^{+} the set of all paths of length ≥ 1 in Q. Consider the path algebra H = LQ of Q over L. Then D(H) ≃ Hom_{K}(LQ, K) for D = Hom_{L}(−, L). Corresponding to each vertex i of Q, choose an idempotent e_{i} of H and corresponding to each arrow β from i to j in Q choose an element h_{β} = e_{j}h_{β}e_{i} of H. For each path p = β_{t} … β_{1} ∈ Q^{+}, we set h_{p} = h_{β_{t}} … h_{β_{1}}. Then D(H) has a basis {e_{i}^{∗}, h_{p}^{∗} | i ∈ Q_{0}, p ∈ Q^{+}} as an L-space. Let H̃ = H ⊕ D(H) be the direct sum of K-spaces and define multiplication on H̃ in the following way:

(a, u)(b, v) = (ab, av + ub + ∑_{i∈Q_{0}} α(a_{i}, b_{i})e_{i}^{∗})

for a, b ∈ H, u, v ∈ D(H), where a_{i} and b_{i} are elements of L such that

a = ∑_{i∈Q_{0}} a_{i}e_{i} + ∑_{p∈Q^{+}} r_{p}h_{p}  and  b = ∑_{i∈Q_{0}} b_{i}e_{i} + ∑_{p∈Q^{+}} s_{p}h_{p}

for r_{p}, s_{p} ∈ L. This multiplication with the usual addition of K-spaces makes H̃ a K-algebra. It is shown in [26, Proposition 6.1] that the K-algebra extension

0 ⟶ D(H) ⟶ H̃ ⟶^{ϱ} H ⟶ 0

with the canonical morphism ϱ and embedding D(H) ↪ H̃ is not splittable. Further, H̃ is a selfinjective K-algebra [27], the elements ẽ_{i} = (e_{i}, −α(1, 1)e_{i}) ∈ H ⊕ D(H) = H̃, i ∈ Q_{0}, form a complete set of primitive orthogonal idempotents of H̃, and (1, −α(1, 1) ∑_{i∈Q_{0}} e_{i}^{∗}) is the identity of H̃. Clearly, the ordinary quiver of the K-algebra H has no oriented cycles, D(H)^{2} = 0 (in H̃), and D(H) is an injective cogenerator of mod H. Hence, by the above theorem H̃ is socle equivalent to the trivial extension H ⋉ D(H). On the other hand, H̃ is not isomorphic to H ⋉ D(H). We refer also to [28] for more information on nonsplittable extensions of algebras.

5. Selfinjective algebras with non-periodic Auslander-Reiten components

The aim of this section is to determine the structure of selfinjective artin algebras whose Auslander-Reiten quiver contains non-periodic generalized standard subquivers.

For an artin algebra Λ, we denote by Γ_{Λ} the Auslander-Reiten quiver of Λ, and by τ_{Λ} and τ_{Λ}^{−} the Auslander-Reiten translations DTr and Tr D, respectively. We shall identify the vertices of Γ_{Λ} with the corresponding indecomposable Λmodules. By a component of Γ_{Λ} we mean a connected component of the quiver Γ_{Λ}. A component Γ of Γ_{Λ} is called regular if Γ contains neither a projective nor an injective module. A subquiver Σ of Γ_{A} is said to be right stable (respectively, left stable) if τ_{Λ}^{−} (respectively, τ_{Λ}) is defined on all modules in Σ. A subquiver C is called non-periodic if C does not contain τ_{A}-periodic modules, that is, modules X with X = τ_{Λ}^{m}X for some m ≥ 1. Following [24] a subquiver D of Γ_{A} is said to be generalized standard if rad _{Λ}^{∞}(X, Y) = 0 for all modules X and Y in D. Recall that rad _{Λ}^{∞}(X, Y) is the intersection of all finite powers rad^{m}(X, Y), for m ≥ 1, of the radical rad (X, Y) of Hom_{Λ}(X, Y). Finally, the annihilator r_{Λ}(E) of a subquiver E of Γ_{Λ} in Λ is the intersection of the (right) annihilators r_{Λ}(X) of all modules X in E. Clearly, r_{Λ}(E) is an ideal of Λ.

If Λ is a selfinjective artin algebra, then τ_{Λ} and τ_{Λ}^{−} are defined on all indecomposable A-modules except the projective-injective ones. Moreover, for each indecomposable projective-injective Λ-module P, we have an Auslander-Reiten sequence

0 → rad P → (rad P/socP) ⊕ P → P/socP → 0,

and hence rad P is a unique direct predecessor of P and P/socP is a unique direct successor of P in Γ_{Λ}. Hence, the Auslander-Reiten quiver Γ_{Λ/socΛ} of Λ/socΛ is obtained from Γ_{Λ} by deleting the projective-injective modules P and the arrows rad P → P and P → P/socP. For a component C of Γ_{Λ}, we denote by C′ the subquiver of Γ_{Λ/socΛ} obtained from C in this way. Obviously, if Λ is representationinfinite, then C′ is a component of Γ_{Λ/socΛ}. Observe also that if Γ_{Λ} contains a nonperiodic right stable (respectively, left stable) subquiver, then Λ is representationinfinite.

Proposition 5.1. Let Λ and A be two socle equivalent selfinjective artin algebras, and Φ : mod Λ/socΛ → mod A/socA the isomorphism of module categories induced by an algebra isomorphism Λ/socΛ → A/socA.

(i) Let Γ be a right stable (respectively, left stable) full translation subquiver of Γ_{Λ} which is closed under successors (respectively, predecessors) in Γ_{Λ}, and Σ = Φ(Γ). Then Γ is generalized standard if and only if Σ is generalized standard.

(ii) Let C be a non-periodic component of Γ_{Λ} and D a (non-periodic) component of Γ_{A} such that Φ(C′) = D′. Then C is generalized standard if and only if D is generalized standard.

Proof. (i) Since Γ is right stable (respectively, left stable), it consists of Λ/socΛmodules, and socΛ ⊆ r_{Λ}(Γ). Applying now our second assumption that Γ is closed under successors (respectively, predecessors) we deduce (see [23]) that Γ is generalized standard as a subquiver of Γ_{Λ} if and only if Γ is generalized standard as a subquiver of Γ_{Λ/socΛ}. Clearly, Σ = Φ(Γ) is also a right stable (respectively, left stable) full translation subquiver of Γ_{A} which is closed under successors (respectively, predecessors) in Γ_{A}. Hence Σ is generalized standard as a subquiver of Γ_{A} if and only if Σ is generalized standard as a subquiver of Γ_{A/socA}. Then the required equivalence follows.

(ii) Since C and D are non-periodic, it follows from [30] (see also [16]) that C and D have no oriented cycles. Then applying [17] we infer that C admits a left stable full translation subquiver Θ which is closed under predecessors and intersects the τ_{A}-orbits of all nonprojective modules of C. Dually, C admits a right stable full translation subquiver Ω which is closed under successors and intersects the τ_{A}^{−}- orbits of all nonprojective modules of C. Then we may assume that Φ(Θ) is a left stable full translation subquiver of D which is closed under predecessors and intersects the τ_{A}-orbits of all nonprojective modules of D, and Φ(Ω) is a right stable full translation subquiver of D which is closed under successors and intersects the τ_{A}^{−}-orbits of all nonprojective modules of D. We know from [23] that C is generalized standard if and only if rad _{Λ}^{∞}(X, Y) = 0 for all modules X from Ω and Y from Θ. But Θ and Ω consist entirely of Λ/socΛ-modules. Hence, as above, we deduce that rad _{Λ}^{∞}(X, Y) = rad _{Λ/socΛ}^{∞}(X, Y) for all modules X from Ω and Y from Θ. Applying again [23], we obtain that C is a generalized standard component of Γ_{A} if and only if C′ is a generalized standard component of Γ_{Λ/socΛ}. Similarly, we show that D is a generalized standard component of Γ_{A} if and only if D′ is a generalized standard component of Γ_{A/socA}. This finishes the proof.

Now let H be a basic and connected hereditary artin algebra over K, Δ the ordinary (valued) quiver of H, and n the number of vertices in Δ. Let T be a multiplicity-free tilting H-module, that is, Ext_{H}^{1}(T, T) = 0 and T is a direct sum of n pairwise nonisomorphic indecomposable H-modules (see [4], [14]). Then B = End_{H}(T) is called a tilted algebra of type Δ. The module T determines a torsion theory (F(T), G(T)) in mod H, where F(T) = {X ∈ mod H | Hom_{H}(T, X) = 0}, G(T) = {X ∈ mod H | Ext_{H}^{1}(T, H) = 0}, and a splitting torsion theory (Y(T), X (T)) in mod B, where Y(T) = {Z ∈ mod B | Tor_{1}^{B}(Z, T) = 0}, X (T) = {Z ∈ mod B | Z⊗_{B}T = 0}. By the Brenner-Butler theorem the functor Hom_{H}(T, −) induces an equivalence of G(T) and Y(T). The indecomposable direct summands of Hom_{H}(T, D(H)) belong to one connected component C_{T} of Γ_{B}, called the connecting component of Γ_{B} determined by T. This component connects the torsion-free part Y(T) of mod B and the torsion part X (T) of mod B. The component C_{T} is generalized standard, because it does not contain oriented cycles and there are no nonzero maps from the torsion modules to the torsion-free modules. Moreover, C_{T} is regular if and only if T is regular (a direct sum of indecomposable regular Hmodules). It was shown in [21] that H admits a regular tilting module if and only if Δ is neither a Dynkin nor a Euclidean (extended Dynkin) quiver and has more than two vertices. Consider now the repetitive algebra B̂ of B and an infinite cyclic group G acting freely on the objects and with finitely many orbits. Then Λ = B̂/G is a selfinjective artin K-algebra and we have a Galois covering F : B̂ → Λ with group G. Denote by F_{λ} : mod B̂ → mod Λ the push-down functor induced by F (see [5]). Assume that Δ is not a Dynkin quiver. Then, by [1], [22], [18] and [10], we know that

ΓB̂ = ⋁_{p∈ℤ}(X_{p} ∨ R_{p})

where, for each p ∈ ℤ, R_{p} is a family of components whose stable parts are tubes (if Δ is Euclidean) or of type ℤ𝔸_{∞} (if Δ is wild), and X_{p} is a component with the stable part of the form ℤΔ, νB̂(R_{p}) = R_{p+2} and νB̂(X_{p}) = X_{p+2}. Further, HomB̂(R_{p}, X_{p}) = 0 and HomB̂(R_{p} ∨ X_{p}, R_{q} ∨ X_{q}) = 0 for p, q ∈ ℤ, p < q. Since G, considered as group of automorphisms of B̂, acts freely on the indecomposable projective B̂-modules, it also acts freely on the components of ΓB̂. Moreover, we know that B̂ is locally-support finite [8], that is, for each object x of B̂ the full subcategory of B̂ consisting of the supports of indecomposable finitely generated B̂-modules having x in its support has finitely many objects. Applying [7] (see also [8]) and [12] we conclude that the push-down functor F_{λ} : mod B̂ → mod Λ is dense and preserves the Auslander-Reiten sequences. Therefore, Γ_{Λ} is obtained from ΓB̂ by identifying (via F_{λ}) X_{p} with X_{p+m} and R_{p} with R_{p+m}, for some m ≥ 1 and all p ∈ ℤ. Thus Γ_{Λ} is of the form

F_{λ}(X_{0} ∨ R_{0}) ∨ F_{λ}(X_{1} ∨ R_{1}) ∨ … ∨ F_{λ}(X_{m−1} ∨ R_{m−1}).

We have also the following facts on B̂ and Λ.

Proposition 5.2. There are two tilted algebras B_{1} = End_{H}(T_{1}), B_{2} = End_{H}(T_{2}), where T_{1} is a tilting H-module without nonzero preprojective direct summands and T_{2} is a tilting H-module without nonzero preinjective direct summands, such that B̂_{1} ≃ B̂ ≃ B̂_{2}.

Proof. See [1], for Δ Euclidean, and [18], for Δ wild.

Proposition 5.3. The following conditions are equivalent:

(i) G is generated by an element φνB̂, where φ is a positive automorphism of B̂. (ii) There exists p, 0 ≤ p ≤ m − 1, such that F_{λ}(X_{p}) contains a generalized stan-

dard right stable full translation subquiver Σ_{p} which is closed under successors. (iii) There exists q, 0 ≤ q ≤ m − 1, such that F_{λ}(X_{q}) contains a generalized

standard left stable full translation subquiver Θ_{q} which is closed under predecessors.

(iv) For each i, 0 ≤ i ≤ m − 1, F_{λ}(X_{i}) contains a generalized standard right stable full translation subquiver Σ_{i} which is closed under successors and a generalized standard left stable full translation subquiver Θ_{i} which is closed under predecessors.

Proof. See [1], [18], if Δ is Euclidean, and [10] if Δ is wild.

Proposition 5.4. Assume that F_{λ}(X_{p}), for some 0 ≤ p ≤ m − 1, is nonregular. Then the following conditions are equivalent:

(i) G is generated by an element φνB̂, where φ is a strictly positive automorphism of B̂.

(ii) The component F_{λ}(X_{p}) is generalized standard. (iii) All components F_{λ}(X_{q}), 0 ≤ q ≤ m − 1, are generalized standard.

Proof. See [1], [22], if Δ is Euclidean, and [10] if Δ is wild.

We are now able to prove our main results on the structure of selfinjective artin algebras whose Auslander-Reiten quiver contains a non-periodic generalized standard right (left) stable subquiver. We assume that A is a basic and connected selfinjective artin algebra over a commutative artin ring K.

Theorem 5.5. The following conditions are equivalent:

(i) Γ_{A} admits a non-periodic generalized standard right stable full translation subquiver which is closed under successors in Γ_{A}.

(ii) A is socle equivalent to B̂/(φνB̂), where B is a tilted algebra of the form End_{H}(T), for some hereditary artin algebra H and a tilting H-module T without nonzero preprojective direct summands, and φ is a positive automorphism of B̂.

(iii) A is socle equivalent to B̂/(φνB̂), where B is a tilted artin K-algebra not of

Dynkin type and φ is a positive automorphism of B̂. (iv) A is socle equivalent to B̂/(φνB̂), where B is a tilted algebra of the form

End_{H}(T), for some hereditary artin K-algebra H and a tilting H-module T without nonzero preinjective direct summands, and φ is a positive automorphism of B̂.

(v) Γ_{A} admits a non-periodic generalized standard left stable full translation subquiver which is closed under predecessors in Γ_{A}.

Moreover, if K is an algebraically closed field, we may replace in the above equivalences “socle equivalent” by “isomorphic”.

Proof. The equivalence of (ii), (iii) and (iv) is a direct consequence of Proposition 5.2. Moreover, the implications (ii)⇒(i) and (iv)⇒(v) are consequences of Propositions 5.1 and 5.3. We shall prove that (i) implies (ii). The proof that (v) implies (iv) is similar.

Assume now that Γ_{A} admits a non-periodic generalized standard right stable full translation subquiver Γ which is closed under successors in Γ_{A}. Since A is selfinjective, we then conclude that Γ has no projective modules and oriented cycles. Applying [17] we get that Γ contains a full translation subquiver D of the form (−ℕ)Δ, for some valued quiver Δ without oriented cycles, which is closed under successors in Γ_{A}. Since D is also generalized standard, it follows from [25, Lemma 2] that Δ is finite. Let I = r_{A}(D) be the annihilator of D in A, B = A/I and e a residual identity of B. We proved in [26, Theorem 5.1 and Proposition 5.3] that IeI = 0, Ie is an injective cogenerator in mod B, and B is a tilted algebra, having a complete slice of type Δ (in the sense of [[20], (4.2)]) formed by modules from D. Hence, B = End_{H}(T), for a hereditary artin K-algebra H of type Δ and a tilting H-module T. Moreover, D is a full translation subquiver of the connecting component C_{T} of Γ_{B} which is closed under successors in Γ_{B}. Consequently, T has no nonzero preprojective direct summands. Obviously, the ordinary quiver Q_{B} of B has no oriented cycles [20]. Applying now Theorem 4.1 we conclude that A is socle equivalent to B̂/(φνB̂), for a positive automorphism φ of B̂. Hence, (i) implies (ii).

If K is an algebraically closed field, then the required equivalences follow from the above proof and the second part of Theorem 4.1.

Corollary 5.6. (i) Γ_{A} admits a non-periodic nonregular generalized standard component if and only if A is a socle equivalent to B̂/(φνB̂), where B is a tilted algebra of the form B = End_{H}(T), for some hereditary artin K-algebra H and a nonregular tilting H-module T without nonzero preprojective (respectively, preinjective) direct summands, and φ is a strictly positive automorphism of B̂.

(ii) Γ_{A} admits a non-periodic regular generalized standard component if and only if A is socle equivalent to B̂/(φνB̂), where B is a tilted algebra of the form End_{H}(T), for some hereditary artin K-algebra H and a regular tilting H-module T, and φ is a positive automorphism of B̂.

Moreover, if K is an algebraically closed field, we may replace in the above equivalences “socle equivalent” by “isomorphic”.

Proof. It is a direct consequence of Theorem 5.5 and Propositions 5.1, 5.3 and 5.4.

Recall that an artin algebra Λ is called symmetric if Λ and D(Λ) are isomorphic as Λ-bimodules, or equivalently the Nakayama automorphism of Λ is the identity. It is well-known that the trivial extension B⋉D(B) of any algebra B by its injective cogenerator D(B) is a symmetric algebra. We get also the following consequences of our main results and definition of G in the proof of Theorem 3.1.

Corollary 5.7. Assume A is symmetric. Then Γ_{A} admits a non-periodic right stable (respectively, left stable) generalized standard full translation subquiver which is closed under successors (respectively, predecessors) in Γ_{A} if and only if A is socle equivalent to B ⋉ D(B), where B is a tilted K-algebra not of Dynkin type. Moreover, if K is an algebraically closed field, we may replace “socle equivalent” by “isomorphic”.

Corollary 5.8. Assume A is symmetric. Then the following conditions are equivalent:

(i) Γ_{A} admits a non-periodic generalized standard component. (ii) Γ_{A} admits a non-periodic regular generalized standard component. (iii) A is socle equivalent to B ⋉ D(B), where B is a tilted algebra of the form

End_{H}(T), for a hereditary artin K-algebra H and a regular tilting H-module T. Moreover, if K is an algebraically closed field, we may replace “socle equivalent” by “isomorphic”.

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Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

   E-mail address: skowron@mat.uni.torun.pl

Department of Mathematics, Tokyo University of Agriculture and Technology, Fuchu, Tokyo 183, Japan

   E-mail address: yamagata@cc.tuat.ac.jp