Universal Galois Coverings of Selfinjective Algebras by Repetitive Algebras and Hochschild Cohomology

María Julia Redondo ^{∗}

INMABB, Universidad Nacional del Sur, Bahía Blanca, Argentina.

1 INTRODUCTION

In this paper all algebras are basic, connected, finite dimensional algebras over an algebraically closed field k. For an algebra A we denote by mod A the category of finitely generated right A–modules and by D : mod A → mod A^{op} the standard duality Hom_{k}(·, k).

We are interested in studying the representation theory of selfinjective algebras. An algebra A is called selfinjective if A ≃ D(A) in mod A, that is, if the projective A–modules are injective. Examples of selfinjective algebras are provided by algebras of finite groups. Frequently, selfinjective algebras have 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 [5, 15], certain classes of tame representation–infinite selfinjective algebras [2, 3, 8, 10, 20] and certain classes of wild representation–infinite selfinjective algebras [9]. There is a close connection between selfinjective algebras and repetitive algebras [15]. We recall that if B is an algebra of finite global dimension and B̂ is the repetitive algebra of B, 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, see [13].

We are interested in a recent class of selfinjective algebras, namely the repetitive covering selfinjective algebras, studied by Skowroński and Yamagata in [21, 22]. Examples of them are provided by trivial extensions.

Definition. A basic, connected, selfinjective finite dimensional algebra over an algebraically closed field k is called a repetitive covering selfinjective algebra if there is an ideal I of A such that the ordinary quiver Q_{B} of B = A/I has no oriented cycles, IeI = 0 and Ie is an injective cogenerator in mod B, where e is a residual identity of B.

Given a repetitive covering selfinjective algebra A, Skowroński and Yamagata [22] proved that the repetitive algebra B̂ of B is a Galois covering of A with an infinite cyclic Galois group G generated by an automorphism φνB̂, where φ is a positive automorphism of B̂ and νB̂ is the Nakayama shift of B̂. In particular, if G = (νB̂) then B̂/(νB̂) is isomorphic to the trivial extension T (B) = B ⋉ D(B) of B by D(B).

∗The author is a researcher from CONICET, Argentina.

Recall that an algebra B is called triangular if its ordinary quiver Q_{B} has no oriented cylces, and is called schurian if dim_{k} Hom_{B}(P, P′) ≤ 1 for every pair of indecomposable projective B–modules. Triangular representation–finite algebras are always schurian.

The purpose of this paper is the following: finding conditions for the repetitive algebra of B to be the universal Galois covering of A, and deducing an estimation of H^{1}(A), the first Hochschild cohomology group of A with coefficients in A, under the assumptions that B is triangular and schurian.

Recall that Galois coverings are obtained by means of fundamental groups [17]. The following is our first main result.

Theorem A. Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G, B schurian. Let (Q_{A}, I_{A}) be any presentation of A associated to the presentation (Q_{B}, I_{B}) of B. Then we have the following short exact sequence of groups

1 → π_{1}(Q_{B}, I_{B}) → π_{1}(Q_{A}, I_{A}) → G → 1.

As an immediate application we have that F : B̂ → A is the universal Galois covering of A if and only if B is simply connected. In particular, if G = (νB̂), we have that B̂ → T (B) = B ⋉ D(B) is the universal Galois covering of the trivial extension T (B) of B by D(B) if and only if B is simply connected.

Concerning the first Hochschild cohomology group of a repetitive covering selfinjective algebra A, we first show that H^{1}(A) is non zero. We are now in a position to formulate our second main result.

Theorem B. Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G = (νc ; B̂), c ≠ 0, B schurian. Then

0 → Hom(π_{1}(Q_{B}, I), k^{+}) → H^{1}(A) → k^{+} → 0

is an exact sequence of abelian groups.

As an immediate consequence we have that if B is simply connected then H^{1}(A) = k. The paper is organized as follows. In Section 2 we describe all the presentations of the

repetitive algebra B̂ for any triangular schurian algebra B. In Section 3 we prove Theorem A and its applications to the universal Galois covering of repetitive covering selfinjective algebras. Finally, Section 4 is devoted to the connection between the first Hochschild cohomology group of A ≃ B̂/ < νc ; B̂ > and the fundamental group of B.

I would like to thank Prof. Ibrahim Assem for his many, very helpful comments and suggestions and Prof. María Inés Platzeck for a careful reading of the manuscript. Finally I would like to thank the referee’s suggestions for the present version of this paper.

2 PRELIMINARIES

2.1 Galois Coverings and Repetitive Covering Selfinjective Algebras

Recall that a k–category R is a category such that the set of morphisms R(e_{x}, e_{y}) from e_{x} to e_{y} has a k–vector space structure for each pair of objects e_{x}, e_{y} in R, and such that the composition of morphisms is k–bilinear. A k–category R is called locally bounded if (a) R(e_{x}, e_{x}) is local for each object e_{x} in R; (b) distinct objects are not isomorphic; and (c) ∑_{e_{y}∈R} dim_{k} R(e_{x}, e_{y}) < ∞ and ∑_{e_{y}∈R} dim_{k} R(e_{y}, e_{x}) < ∞ for each object e_{x} in R.

We may consider a basic algebra A as a k–category with the object set consisting of a fixed complete set of primitive orthogonal idempotents {e_{1}, …, e_{n}}, and sets of morphisms A(e_{x}, e_{y}) = e_{y}Ae_{x}.

Locally bounded categories can be viewed as locally finite quivers. Recall that a quiver Q is defined by a set of points Q_{0}, a set of arrows Q_{1}, and two maps s, t : Q_{1} → Q_{0} associating with each arrow its starting and terminal point, respectively. We denote by kQ the path algebra of Q. We write a path α in kQ as a composition of consecutive arrows α = α_{1} … α_{r}, where t(α_{i}) = s(α_{i+1}) for all i = 1, …, r − 1, and we set s(α) = s(α_{1}), t(α) = t(α_{r}) . If R is a locally bounded category then R = kQ_{R}/I_{R}, where: (a) Q_{R} is locally finite, that is, the number of arrows starting or ending at any vertex is finite and (b) for each vertex x ∈ (Q_{R})_{0} there is a natural number N_{x} such that I_{R} contains each path of length ≥ N_{x} starting or ending at x. The pair (Q_{R}, I_{R}) is called a presentation of R.

In particular, if R is a locally bounded k–category, the radical rad R of R is the ideal assigning to a pair of objects (e_{x}, e_{y}) the subspace rad R(e_{x}, e_{y}) of R(e_{x}, e_{y}) consisting of the non–invertible morphisms. The radical square rad^{2} R is defined by rad^{2} R(e_{x}, e_{y}) = ∑_{e_{z}∈R} rad R(e_{z}, e_{y}) rad R(e_{x}, e_{z}). The set of vertices of the quiver Q_{R} of R is {x : e_{x} in R}, and if x, y are two such vertices the number of arrows y → x is equal to dim_{k} rad R(e_{x}, e_{y})/ rad^{2} R(e_{x}, e_{y}). We get an isomorphism kQ_{R}/I_{R} →^{∼} R for some admissible ideal I_{R} by sending the arrows y → x onto representatives in rad R(e_{x}, e_{y}) of chosen basis vectors of rad R(e_{x}, e_{y})/ rad^{2} R(e_{x}, e_{y}).

Let R and A be locally bounded k–categories. A Galois covering F : R → A defined by the action of the group G is a functor satisfying:

i) G is a group of k–linear automorphisms acting freely on the objects of R (that is, ge_{x} = e_{x} implies g = 1);

ii) F g = F for every g ∈ G;

iii) F is onto on objects and G acts transitively on F^{−1}(e_{a}) for every object e_{a} in A;

iv) F is a covering functor, that is, the induced maps

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

are bijective for all e_{x} in R and e_{a} in A.

REMARK 2.1 If F : R → A is a covering functor then

⨁_{F (e_{y})=e_{b}} rad R(e_{x}, e_{y})/ rad^{2} R(e_{x}, e_{y}) →^{∼} rad A(F (e_{x}), e_{b})/ rad^{2} A(F (e_{x}), e_{b}).

In particular, if the quiver Q_{R} contains an arrow y → x, then Q_{A} contains an arrow b → a, where e_{a} = F (e_{x}). In other words, F induces a quiver morphism Q_{F} : Q_{R} → Q_{A} such that the image of the admissible ideal I_{R} of kQ_{R} is contained in I_{A}, see [4]. Moreover, if F is a Galois covering then Q_{F} is onto.

A Galois covering F : R → A given by the action of the group G is called the universal Galois covering if for any Galois covering F′ : R′ → A given by the action of the group G′, there exists a unique Galois covering H : R → R′ given by the action of a normal subgroup N of G such that G/N ≃ G′ and F = F′H.

Let (Q_{A}, I_{A}) be a presentation of A, Q_{A} without double arrows. The universal Galois covering F : kQ̃/Ĩ → kQ_{A}/I_{A} is constructed in [17] in the following way: for an arrow α in Q_{A} we denote by α^{−1} its formal inverse. A walk in Q_{A} is a formal composition α_{1}^{ϵ_{1}} … α_{t}^{ϵ_{t}} with α_{i} ∈ (Q_{A})_{1} and ϵ_{i} = \underline{+}1 for any i with 1 ≤ i ≤ t. We denote by ∼ the equivalence relation defined on the set of all walks in Q_{A} induced by the elementary relations

a) α^{−1}α ∼ e_{b} and αα^{−1} ∼ e_{a} if α : a → b is an arrow in Q_{A};

b) u_{i} ∼ u_{j} for every 1 ≤ i, j ≤ m if ρ = ∑_{i=1}^{m} λ_{i}u_{i} ∈ I_{A}(a, b) is a minimal relation, that is, m ≥ 2 and ∑_{i∈J} λ_{i}u_{i} ∉ I_{A}(a, b) for all ∅ ≠ J ⊊ {1, …, m};

c) wuw′ ∼ wvw′ if u ∼ v and the product makes sense.

Let W be the set of all walks in Q_{A} starting at a fixed vertex x_{0}. Then Q̃_{0} = W/ ∼ is the set of vertices of a quiver Q̃ with arrows [w] →^{α} [wα] for any α ∈ (Q_{A})_{1}. Let π : Q̃ → Q_{A} be the natural projection sending a vertex x̃ of Q̃ to the terminus of any representative of x̃. Let Ĩ be the admissible ideal of kQ̃ generated by the liftings through π of the zero and the minimal relations in I_{A}. The induced functor F : kQ̃/Ĩ → kQ_{A}/I_{A} is a Galois covering defined by the action of the fundamental group π_{1}(Q_{A}, I_{A}) consisting of the equivalence classes of closed walks with origin x_{0}.

REMARK 2.2 Let F : R → A be a Galois covering and Q_{F} : Q_{R} → Q_{A} the induced quiver morphism. If ρ = ∑_{i=1}^{m} λ_{i}u_{i} ∈ I_{A}(a, b) is a minimal relation then ρ belongs to the image of the admissible ideal I_{R}. More precisely, for e_{a} = F (e_{x}) consider the isomorphism

⨁_{F (e_{y})=e_{b}} R(e_{x}, e_{y}) → A(F (e_{x}), e_{b}),

and let \overline{u_{i}} = F (\overline{w_{i}}), where \overline{w_{i}} ∈ ∑_{j} R(e_{x}, e_{y_{i_{j}}}) with F (e_{y_{i_{j}}}) = e_{b} for each i = 1, …, m. Then ∑_{i=1}^{m} λ_{i}w_{i} ∈ I_{R} and w_{i} ∉ I_{R} for all 1 ≤ i ≤ m.

We refer to [4, 12, 17] for more details on covering functors.

Let A be a selfinjective algebra, {e_{1}, …, e_{n}} a complete set of primitive orthogonal idempotents of A. Then {e_{1}A, …, e_{n}A} is a complete set of non–isomorphic indecomposable projective A–modules and {D(Ae_{1}), … D(Ae_{n})} is a complete set of non–isomorphic indecomposable injective A–modules. Let ν be the Nakayama automorphism of A with θ_{r} : A → D(A) an A–bimodule isomorphism such that θ_{r}(ab) = θ_{r}(a)b = ν(a)θ_{r}(b). Since {ν(e_{1})A, …, ν(e_{n})A} is a complete set of non–isomorphic indecomposable projective A– modules, there is a permutation of {1, …, n}, denoted again by ν, such that ν(e_{i})A ≃ e_{ν(i)}A for all i ∈ {1, …, n}. By the Krull–Schmidt theorem we may assume that ν(e_{i}A) = ν(e_{i})A = e_{ν(i)}A for all i ∈ {1, …, n}.

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 ≤ n, and that {e_{1}, …, e_{t}} is the subset of {e_{1}, …, e_{n}} consisting of all idempotents which are not in I. Note that B ≃ eAe/eIe and 1 − e ∈ I. The idempotent e is uniquely determined by I up to an inner automorphism of A, and we call it a residual identity of B.

There is a retraction ι of the canonical algebra homomorphism ρ : eAe → eAe/eIe such that ι(e_{i}) = e_{i} for any i with 1 ≤ i ≤ t, see [22]. Further B and eAe/eIe can be considered as A–algebras by the canonical algebra homomorphisms A → B → eAe/eIe. Hence B, eAe/eIe and ι(B) are isomorphic A–algebras and from now on we shall identify them.

Consider the repetitive algebra

B̂ = ( \begin{matrix}... & ... & & & 0 \\ & B_{m−1} & Q_{m−1} \\ & & B_{m} & Q_{m} \\ & & & B_{m+1} & ... \\ 0 & & & & ... \end{matrix} )

of the algebra B where B_{m} = B and Q_{m} = D(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.

We may clearly consider B̂ as a locally bounded k–category with the object set consisting of the set of primitive orthogonal idempotents {e_{m,i}}_{1≤i≤t,m∈Z}, and morphisms

B̂(e_{s,j}, e_{m,i}) = e_{m,i}B̂e_{s,j} = \{ \begin{matrix}e_{i}Be_{j} & if s = m, \\ e_{i}D(B)e_{j} & if s = m + 1, \\ 0 & otherwise. \end{matrix}

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 B̂ is said to be positive if for each object e_{m,i} of B̂ there exist p, j with 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.

DEFINITION 2.3 A basic, connected, selfinjective finite dimensional algebra over an algebraically closed field k is called a repetitive covering selfinjective algebra if there is an ideal I of A such that the ordinary quiver Q_{B} of B = A/I has no oriented cycles, IeI = 0 and Ie is an injective cogenerator in mod B, where e = e_{1} + … + e_{t} is a residual identity of B.

THEOREM 2.4 [22, Theorem 4.1] Let A be a repetitive covering selfinjective algebra. Then there is a Galois covering F : B̂ → A with the Galois group G an infinite cyclic group of automorphisms of B̂ generated by φνB̂ for some positive automorphism φ of B̂.

The functor F : B̂ → A is defined as follows: F (e_{m,i}) = e_{ν^{m}(i)} and F : B̂(e_{s,j}, e_{m,i}) → A(F (e_{s,j}), F (e_{m,i})) = A(e_{ν^{s}(j)}, e_{ν^{m}(i)}) is given by

a) if s = m, F is the composition

e_{m,i}B̂e_{m,j} = e_{i}Be_{j} ↪ e_{i}Ae_{j} →^{ν^{m}} e_{ν^{m}(i)}Ae_{ν^{m}(j)},

b) if s = m + 1, F is the composition

e_{m,i}B̂e_{m+1,j} = e_{i}D(B)e_{j} →^{θ^{−1}} e_{i}Ie_{ν(j)} ↪ e_{i}Ae_{ν(j)} →^{ν^{m}} e_{ν^{m}(i)}Ae_{ν^{m+1}(j)},

c) F is zero for s ≠ m and s ≠ m + 1,

where the isomorphism θ is induced by the isomorphism θ_{l} : A → D(A) such that θ_{l}(ab) = θ_{l}(a)ν^{−1}(b) = aθ_{l}(b), see [22].

The group G is generated by an automorphism φνB̂ with φ a positive automorphism of B̂ such that g = (φνB̂)^{−1} is defined on the objects by g(e_{m,i}) = e_{m−η(i),ν^{η(i)}(i)}, where η(i) = min{k > 0 : ν^{k}(i) ≤ t}. The action of g on B̂(e_{s,j}, e_{m,i}) is defined by the map

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

where

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

are the k–linear maps induced by F, monomorphisms by definition and whose images are equal [22, page 725].

From now on, whenever we consider a Galois covering F : B̂ → A we will assume that it has the property given in Theorem 2.4.

In order to give some examples, we recall the following result proved by Skowroński and Yamagata.

PROPOSITION 2.5 [21, Proposition 2.3] Let I be 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) Ie = l_{A}(I) = {a ∈ A : aI = 0}.

EXAMPLE 2.6 The trivial extension A = B ⋉D(B) of B by D(B) is a repetitive covering selfinjective algebra, the Nakayama automorphism ν of A is the identity and I = D(B). Recall that B ⋉ D(B) is the symmetric algebra whose underlying vector space is B ⊕ D(B), and the product is defined by (a, f).(b, g) = (ab, ag + fb) for any a, b ∈ B, f, g ∈ D(B). Moreover, the natural functor B̂ → B ⋉ D(B) = A is the well–known Galois covering with the Galois group generated by νB̂.

EXAMPLE 2.7 Let A = kQ/J^{t+1} with Q the oriented cycle with n vertices and n arrows, 1 ≤ t ≤ n, and J the ideal generated by the arrows. Then A is a repetitive covering selfinjective algebra with B = kQ_{B} the hereditary algebra with

Q_{B} : 1 → 2 → … → t.

If t = n then A ≃ B ⋉ D(B). If 1 ≤ t < n take e = ∑_{i=1}^{t} e_{i} and consider the ideal I = A(1 − e)A. Then IeI = 0, and Ie is an injective cogenerator in mod B since B = A/I and Ie = l_{A}(I). Moreover, A ≃ B̂/G with G generated by the automorphism γ^{n}, where γ : B̂ → B̂ is given by γ(m, i) = (m, i + 1) if 1 ≤ i ≤ t − 1 and γ(m, t) = (m + 1, 1).

EXAMPLE 2.8 Given a triangular algebra B, let A = B̂/(νc ; B̂), c ≠ 0. Then A is a repetitive covering selfinjective algebra: if c = 1 then A ≃ B ⋉ D(B) is the trivial extension of B by D(B); if c > 1 consider the ideal I = A(1 − e)A with e = ∑_{i∈(Q_{B})_{0}} \overline{e_{0,i}}. Then IeI = 0, and Ie is an injective cogenerator in mod B since B = A/I and Ie = l_{A}(I). In a similar way, it can be seen that the following representatives of the derived equivalence classes of standard representation–finite selfinjective algebras are repetitive covering selfinjective algebras: Λ(A_{n}, s/n, 1), s ≥ n, Λ(A_{2p+1}, s, 2), Λ(D_{n}, s, 1), Λ(D_{n}, s, 2), Λ(D_{4}, s, 3), Λ(E_{n}, s, 1) and Λ(E_{6}, s, 2), see [1, 2.5].

EXAMPLE 2.9 Let A be a selfinjective algebra whose Auslander–Reiten quiver Γ_{A} admits a nonperiodic generalized standard right (left) stable full translation subquiver which is closed under successors (predecessors) in Γ_{A}. Then A is a repetitive covering selfinjective algebra, A ≃ B̂/(φνB̂) with B a tilted algebra not of Dynkin type, B = End_{H}(T) for some hereditary algebra H and a tilting H–module T without nonzero preprojective (preinjective) direct summands, φ a positive automorphism of B̂, see [22, Theorem 5.5].

The functor F : B̂ → A is the universal Galois covering of A if π_{1}(Q_{A}, I_{A}) ≃ G for any presentation (Q_{A}, I_{A}) of A. So we want to study the relationship between the fundamental groups π_{1}(Q_{A}, I_{A}) and π_{1}(Q_{B}, I_{B}). In order to do this, we need the description of the presentations of A and B.

2.2 Presentations of Repetitive Algebras

We are going to describe the presentations of the locally bounded k–category B̂ for any triangular schurian algebra B.

Let (Q_{B}, I_{B}) be a fixed presentation of B. Given γ a path in kQ_{B} we denote by \overline{γ} the corresponding element in kQ_{B}/I_{B}.

Denote r = rad(B). Since D(B)^{2} = 0 in B̂, we have

rad B̂(e_{s,j}, e_{m,i}) = \{ \begin{matrix}e_{i}re_{j} & if s = m, \\ e_{i}D(B)e_{j} & if s = m + 1, \\ 0 & otherwise, \end{matrix}

rad^{2} B̂(e_{m,j}, e_{m,i}) = ∑_{e_{l,t}∈B̂} rad B̂(e_{l,t}, e_{m,i}) rad B̂(e_{m,j}, e_{l,t})

= ∑_{e_{m,t}∈B̂} rad B̂(e_{m,t}, e_{m,i}) rad B̂(e_{m,j}, e_{m,t})

= ∑_{e_{t}∈B}(e_{i}re_{t})(e_{t}re_{j}) = e_{i}r^{2}e_{j}

and

rad^{2} B̂(e_{m+1,j}, e_{m,i}) = ∑_{e_{l,t}∈B̂} rad B̂(e_{l,t}, e_{m,i}) rad B̂(e_{m+1,j}, e_{l,t})

= ∑_{e_{m,t}∈B̂} rad B̂(e_{m,t}, e_{m,i}) rad B̂(e_{m+1,j}, e_{m,t})

+ ∑_{e_{m+1,t}∈B̂} rad B̂(e_{m+1,t}, e_{m,i}) rad B̂(e_{m+1,j}, e_{m+1,t})

= ∑_{e_{t}∈B}(e_{i}re_{t}).(e_{t}D(B)e_{j}) + (e_{i}D(B)e_{t}).(e_{t}re_{j})
= e_{i}(D(B)r + rD(B))e_{j}.

LEMMA 2.10 [11]

0 → D(B)r + rD(B) → D(B) → D(soc_{B^{e}} B) → 0

is a short exact sequence of B–bimodules, where B^{e} = B ⊗_{k} B^{op} is the enveloping algebra.

Proof. It is known that mod B^{e} is equivalent to the category of finitely generated B– bimodules and

0 → rad_{B^{e}} D(B) → D(B) → D(soc_{B^{e}} B) → 0

is a short exact sequence of B^{e}–modules, so we only have to describe rad_{B^{e}} D(B). Since rad(B^{e}) = r ⊗ B^{op} + B ⊗ r^{op}, we know that rad_{B^{e}} D(B) = rad B^{e} D(B) ≃ rD(B) + D(B)r and the proof is complete. □

DEFINITION 2.11 A path q in kQ_{B} is called maximal if \overline{q} ≠ 0 and \overline{αq} = 0 = \overline{qα} in B for any arrow α ∈ (Q_{B})_{1}.

COROLLARY 2.12 Let B be a schurian algebra. Then D(B)/(rD(B) + D(B)r) is isomorphic as B–bimodule to the dual of the subspace of B generated by all \overline{q} ∈ B with q maximal in Q_{B}.

Proof. It is well known that soc_{B^{e}}(B) = ann_{B}(rad B^{e}), so, by the previous lemma, D(B)/(D(B)r + rD(B)) ≃ D({b ∈ B : (r ⊗ B^{op} + B ⊗ r^{op})b = 0}) = D({b ∈ B : ab = 0 = ba for all a ∈ r}). But, since B is schurian, {b ∈ B : ab = 0 = ba for all a ∈ r} is the subspace generated by the set \{\overline{q} ∈ B : q a path in kQ_{B}, a\overline{q} = 0 = \overline{q}a for all a ∈ r\} = \{\overline{q} ∈ B : q a path in kQ_{B}, \overline{αq} = 0 = \overline{qα} for all α ∈ (Q_{B})_{1}\}. This subset of B consists of all \overline{q} ∈ B with q maximal in Q_{B}. □

PROPOSITION 2.13 If B is a schurian algebra with ordinary quiver Q_{B} then the ordinary quiver of B̂ is given by

(QB̂)_{0} = {(m, i)}_{m∈Z,i∈(Q_{B})_{0}}

and

(QB̂)_{1} = {α^{m} : (m, j) → (m, i)}_{α:j→i∈(Q_{B})_{1},m∈Z} ∪ {β_{p_{1}}^{m}, …, β_{p_{r}}^{m}}_{m∈Z},

where p_{1}, …, p_{r} are maximal and \{\overline{p_{1}}, …, \overline{p_{r}}\} is a basis of the subspace of B generated by all \overline{q} ∈ B such that q is maximal, and β_{p_{i}}^{m} is an arrow from (m, t(p_{i})) to (m + 1, s(p_{i})) for each i.

Proof. The vertices of the quiver QB̂ are the objects of B̂, so (QB̂)_{0} = {(m, i)}_{m∈Z,i∈(Q_{B})_{0}}. For each pair of vertices (m, i), (s, j), the number of arrows from (m, i) to (s, j) is equal to

dim_{k} rad B̂(e_{s,j}, e_{m,i})/ rad^{2} B̂(e_{s,j}, e_{m,i}).

This is zero if s ≠ m, m + 1. If s = m, it is equal to dim_{k} e_{i}(r/r^{2})e_{j}. If s = m + 1, it is equal to dim_{k}(e_{i}D(B)e_{j})/(e_{i}(D(B)r + rD(B))e_{j}) = dim_{k} e_{i}(D(B)/(D(B)r + rD(B)))e_{j} which is zero or one, depending on the existence of a maximal path in kQ_{B} from j to i (see Corollary 2.12). □

From now on, we are going to denote by P = {p_{1}, …, p_{r}} a set of maximal paths in kQ_{B} such that \{\overline{p_{1}}, …, \overline{p_{r}}\} is a basis of the subspace of B generated by all \overline{q} ∈ B such that q is maximal.

Given a basis \{\overline{q_{1}}, …, \overline{q_{s}}\} of B, we denote by \{\overline{q_{1}}^{∗}, …, \overline{q_{s}}^{∗}\} its dual basis in D(B), where \overline{q_{i}}^{∗}(\overline{q_{j}}) is 1 if i = j and 0 otherwise.

Now we will describe all the presentations (QB̂, IB̂) of B̂ associated to a fixed presentation (Q_{B}, I_{B}) of B.

Let ϕ : kQB̂ → B̂ be a surjective morphism of algebras such that the images ϕ(α) of the arrows α : (m, i) → (s, j) of QB̂ induce a basis of rad B̂(e_{s,j}, e_{m,i})/ rad^{2} B̂(e_{s,j}, e_{m,i}). The ideal IB̂ of kQB̂ such that kQB̂/IB̂ ≃ B̂ is admissible. Now, since Q_{B} has no oriented cycles and B is schurian, we have

ϕ(α^{m}) ∈ B̂(e_{m,t(α)}, e_{m,s(α)}) = e_{s(α)}Be_{t(α)} = {λα, λ ∈ k}

and

ϕ(β_{p_{i}}^{m}) ∈ B̂(e_{m+1,s(p_{i})}, e_{m,t(p_{i})}) = e_{t(p_{i})}D(B)e_{s(p_{i})}.

So ϕ(α^{m}) = λ_{m}α^{m}, 0 ≠ λ_{m} ∈ k, and ϕ(β_{p_{i}}^{m}) = f_{i}^{m}, where f̃_{i}^{m} generates the k–vector space e_{t(p_{i})} (D(B)/(rD(B) + D(B)r)) e_{s(p_{i})}. Hence the set {f_{1}^{m}, …, f_{r}^{m}} may be extended to a basis \{f_{1}^{m}, …, f_{r}^{m}, f_{r+1}^{m}, …, f_{s}^{m}\} of D(B). Therefore there exists a basis \{\overline{q_{1}^{m}}, …, \overline{q_{s}^{m}}\} of B whose dual is the given basis, that is, f_{i}^{m} = \overline{q_{i}^{m}}^{∗}. From the fact that ˜ \overline{q}^{∗} ≠ 0 in D(B)/(rD(B) + D(B)r) if and only if \overline{q} is maximal (see Corollary 2.12), it follows that \{\overline{q_{1}^{m}}, …, \overline{q_{r}^{m}}\} is a basis of the subspace of B generated by all \overline{q} ∈ B such that q is maximal. Since B is schurian, by reordering the given basis if necessary, we have \overline{q_{i}^{m}} = μ_{i}^{m}\overline{p_{i}} for all p_{i} ∈ P and some 0 ≠ μ_{i}^{m} ∈ k. So ϕ(β_{p_{i}}^{m}) = \overline{q_{i}^{m}}^{∗} = μ_{i}^{m}\overline{p_{i}}^{∗} for some 0 ≠ μ_{i}^{m} ∈ k.

In order to describe the fundamental group we need the description of the minimal relations in IB̂. Since the non–zero coefficients of these minimal relations don’t matter in the definition of the equivalence relation used to define the fundamental group (see Section 2.1), we may assume without loss of generality that ϕ(α^{m}) = \overline{α}^{m} and ϕ(β_{p_{i}}^{m}) = \overline{p_{i}}^{∗}.

NOTATION. Given any path γ = α_{1} … α_{t} in kQ_{B}, we denote by γ^{m} = α_{1}^{m} … α_{t}^{m} the corresponding paths in kQB̂ and by \overline{γ}^{m} = \overline{α_{1} … α_{t}}^{m} = ϕ(α_{1}^{m} … α_{t}^{m}) the corresponding elements in B̂.

To describe the corresponding ideal I_{ϕ} = Ker ϕ we need the following definitions.

DEFINITION 2.14 A path C in kQB̂ is called elementary if

C = α_{s}^{m} … α_{t}^{m}β_{p}^{m}α_{1}^{m+1} … α_{s−1}^{m+1}

with \overline{α_{1} … α_{t}} = μ\overline{p} for p a maximal path in P and 0 ≠ μ ∈ k.

DEFINITION 2.15 A path q in kQB̂ is said to have a supplement if there exists a path q′ in kQB̂ such that qq′ is an elementary path in kQB̂.

Recall that D(B) is a B–bimodule with structure given by (afb)(x) = f(bxa) for any a, b, x ∈ B, f ∈ D(B).

REMARK 2.16 Let C = α_{r}^{m} … α_{t}^{m}β_{p}^{m}α_{1}^{m+1} … α_{r−1}^{m+1} be an elementary path with \overline{α_{1} … α_{t}} = μ\overline{p}, p ∈ P and 0 ≠ μ ∈ k. Then ϕ(C) ≠ 0. In fact, ϕ(C) = \overline{α_{r} … α_{t}}^{m}\overline{p}^{∗}\overline{α_{1} … α_{r−1}}^{m+1} ∈ B̂(e_{m+1,t(α_{r−1})}, e_{m,s(α_{r})}) ⊆ D(B) and

(\overline{α_{r} … α_{t}}\overline{p}^{∗}\overline{α_{1} … α_{r−1}})(e_{s(α_{r})}) = \overline{p}^{∗}(\overline{α_{1} … α_{r−1}α_{r} … α_{t}}) = \overline{p}^{∗}(μ\overline{p}) = μ ≠ 0.

REMARK 2.17 If a path q in kQB̂ has a supplement q′ then ϕ(q) ≠ 0. This is immediate by the previous remark since ϕ(qq′) ≠ 0.

REMARK 2.18 If γ^{m+1} is a supplement of a path q in kQB̂ and \overline{δ} = λ\overline{γ}, 0 ≠ λ ∈ k, then δ^{m+1} is also a supplement of q. In fact, let q = α_{s}^{m} … α_{t}^{m}β_{p}^{m}α_{1}^{m+1} … α_{s′}^{m+1}, γ^{m+1} a supplement in the elementary path C = α_{s}^{m} … α_{t}^{m}β_{p}^{m}α_{1}^{m+1} … α_{s′}^{m+1}γ^{m+1}. Since

μ\overline{p} = \overline{α_{1} … α_{s′}γα_{s} … α_{t}} = \frac{1}{λ}\overline{α_{1} … α_{s′}δα_{s} … α_{t}},

then C′ = α_{s}^{m} … α_{t}^{m}β_{p}^{m}α_{1}^{m+1} … α_{s′}^{m+1}δ^{m+1} is also an elementary path, and hence δ^{m+1} is a supplement of q.

PROPOSITION 2.19 Let ϕ : kQB̂ → B̂ be a surjective morphism of algebras such that ϕ(α^{m}) = \overline{α}^{m} and ϕ(β_{p_{i}}^{m}) = \overline{p_{i}}^{∗}. Then the ideal I_{ϕ} = Ker ϕ is admissible and is generated by

i) paths α_{s}^{m} … α_{t}^{m}β_{p}^{m}α_{1}^{m+1} … α_{s}^{m+1} and β_{p}^{m−1}α_{1}^{m} … α_{t}^{m}β_{p}^{m} for any elementary path α_{1}^{m} … α_{t}^{m}β_{p}^{m};

ii) paths whose arrows do not belong to an elementary path;

iii) elements q_{1} − μq_{2} with q_{1} and q_{2} paths sharing starting and terminal points such that:

a) q_{1} = γ_{1}^{m}, q_{2} = γ_{2}^{m}, \overline{γ_{1}} = μ\overline{γ_{2}} with 0 ≠ μ ∈ k, γ_{1}, γ_{2} paths in Q_{B}, or

b) q_{1} and q_{2} have the same supplement γ^{m+1} in elementary paths for a path γ in

kQ_{B}, that is,

q_{1}γ^{m+1} = α_{s}^{m} … α_{t}^{m}β_{p_{1}}^{m}α_{1}^{m+1} … α_{s′}^{m+1}γ^{m+1},
q_{2}γ^{m+1} = δ_{r}^{m} … δ_{l}^{m}β_{p_{2}}^{m}δ_{1}^{m+1} … δ_{r′}^{m+1}γ^{m+1},

where p_{1}, p_{2} ∈ P, s′ < s and r′ < r such that

\overline{α_{1} … α_{s′}γα_{s} … α_{t}} = λ_{1}\overline{p_{1}}, \overline{δ_{1} … δ_{r′}γδ_{r} … δ_{l}} = λ_{2}\overline{p_{2}}

with 0 ≠ μ = \frac{λ_{1}}{λ_{2}} .

Proof. It follows from the definition of ϕ that Ker ϕ is admissible. Let I_{ϕ} be the ideal generated by the relations given in i), ii) and iii). We will first show that I_{ϕ} ⊆ Ker ϕ.

i)

ϕ(β_{p}^{m−1}α_{1}^{m} … α_{t}^{m}β_{p}^{m}) = \overline{p}^{∗}\overline{α_{1} … α_{t}}^{m}\overline{p}^{∗} = 0

because D(B)^{2} = 0, and

ϕ(α_{s}^{m} … α_{t}^{m}β_{p}^{m}α_{1}^{m+1} … α_{s}^{m+1}) = \overline{α_{s} … α_{t}}^{m}\overline{p}^{∗}\overline{α_{1} … α_{s}}^{m+1} = 0

because (\overline{α_{s} … α_{t}}\overline{p}^{∗}\overline{α_{1} … α_{s}})(\overline{x}) = \overline{p}^{∗}(\overline{α_{1} … α_{s}xα_{s} … α_{t}}) = 0 for every \overline{x} ∈ B, since B triangular implies α_{s}xα_{s} = 0. ii) Let q be a path from (m, i) to (s, j) whose arrows do not belong to an elementary path.

If s ≠ m, m + 1, then ϕ(q) = 0. If s = m, then q = γ^{m} for γ = α_{1} … α_{t} and so ϕ(q) = \overline{γ}^{m}. If \overline{γ} ≠ 0, there exists a maximal path p ∈ P such that \overline{γ_{1}γγ_{2}} = λ\overline{p} with 0 ≠ λ ∈ k, γ_{1}, γ_{2} paths in kQ_{B}. This implies that the arrows of q belong to the elementary path γ_{1}^{m}γ^{m}γ_{2}^{m}β_{p}^{m}, a contradiction. Then ϕ(q) = \overline{γ}^{m} = 0. If s = m + 1, then we may put q = α_{1}^{m} … α_{r−1}^{m}β_{p}^{m}α_{r+1}^{m+1} … α_{t}^{m+1}, and hence

ϕ(q) = \overline{α_{1} … α_{r−1}}^{m}\overline{p}^{∗}\overline{α_{r+1} … α_{t}}^{m+1}.

Now, let x be a path in kQ_{B}. Then

(\overline{α_{1} … α_{r−1}}\overline{p}^{∗}\overline{α_{r+1} … α_{t}})(\overline{x}) = \overline{p}^{∗}(\overline{α_{r+1} … α_{t}xα_{1} … α_{r−1}})

is non–zero precisely when the path x is such that \overline{α_{r+1} … α_{t}xα_{1} … α_{r−1}} = λ\overline{p}, 0 ≠ λ ∈ k. But in this case q would have arrows belonging to the elementary path α_{1}^{m} … α_{r−1}^{m}β_{p}^{m}α_{r+1}^{m+1} … α_{t}^{m+1}x^{m+1}, contradicting our hypothesis. So ϕ(q) = 0.

iii) a) It is immediate since ϕ(q_{1} − μq_{2}) = \overline{γ_{1}}^{m} − μ\overline{γ_{2}}^{m}.

b) Let

q_{1}γ^{m+1} = α_{s}^{m} … α_{t}^{m}β_{p_{1}}^{m}α_{1}^{m+1} … α_{s′}^{m+1}γ^{m+1},
q_{2}γ^{m+1} = δ_{r}^{m} … δ_{l}^{m}β_{p_{2}}^{m}δ_{1}^{m+1} … δ_{r′}^{m+1}γ^{m+1},

where p_{1}, p_{2} ∈ P, s′ < s, r′ < r , and γ is a path in kQ_{B} from i to j such that

\overline{α_{1} … α_{s′}γα_{s} … α_{t}} = λ_{1}\overline{p_{1}}, \overline{δ_{1} … δ_{r′}γδ_{r} … δ_{l}} = λ_{2}\overline{p_{2}}

with 0 ≠ μ = \frac{λ_{1}}{λ_{2}} . Then

ϕ(q_{1}) = \overline{α_{s} … α_{t}}^{m}\overline{p_{1}}^{∗}\overline{α_{1} … α_{s′}}^{m+1}

and

ϕ(q_{2}) = \overline{δ_{r} … δ_{l}}^{m}\overline{p_{2}}^{∗}\overline{δ_{1} … δ_{r′}}^{m+1}.

Let x be an arbitrary path in kQ_{B}. Then

(\overline{α_{s} … α_{t}}\overline{p_{1}}^{∗}\overline{α_{1} … α_{s′}})(\overline{x}) = \overline{p_{1}}^{∗}(\overline{α_{1} … α_{s′}xα_{s} … α_{t}})

and

(\overline{δ_{r} … δ_{l}}\overline{p_{2}}^{∗}\overline{δ_{1} … δ_{r′}})(\overline{x}) = \overline{p_{2}}^{∗}(\overline{δ_{1} … δ_{r′}xδ_{r} … δ_{l}}).

If s(x) ≠ s(γ) or t(x) ≠ t(γ), the preceding expressions are both equal to zero since α_{s′}xα_{s} = 0 = δ_{r′}xδ_{r}. This is also the case if \overline{x} = 0. Assume \overline{x} ≠ 0 and s(x) = s(γ), t(x) = t(γ). Since B is schurian, \overline{x} = μ\overline{γ} for some 0 ≠ μ ∈ k. So

\overline{p_{1}}^{∗}(\overline{α_{1} … α_{s′}xα_{s} … α_{t}}) = μ\overline{p_{1}}^{∗}(λ_{1}\overline{p}_{1}) = μλ_{1}

and

\overline{p_{2}}^{∗}(\overline{δ_{1} … δ_{r′}xδ_{r} … δ_{l}}) = μ\overline{p_{2}}^{∗}(λ_{2}\overline{p}_{2}) = μλ_{2},

hence ϕ(q_{1}) = \frac{λ_{1}}{λ_{2}} ϕ(q_{2}), that is, ϕ(q_{1} − \frac{λ_{1}}{λ_{2}} q_{2}) = 0.

To finish the proof we are going to compute dim_{k} kQB̂/I_{ϕ} ((s, j), (m, i)). We have dim_{k} kQB̂/I_{ϕ} ((s, j), (m, i)) = 0 if s ≠ m, m + 1 by the relations described in ii). Consider the case s = m. If q is a non–zero path in B the arrows of q belong to an elementary path. So dim_{k} kQB̂/I_{ϕ} ((m, j), (m, i)) = dim_{k} e_{i}Be_{j}, by the relations described in ii) and iii)a). Finally, let s = m + 1. By i), dim_{k} kQB̂/I_{ϕ} ((m + 1, i), (m, i)) = dim_{k} e_{i}D(B)e_{i}, and by ii) and iii)b), dim_{k} kQB̂/I_{ϕ} ((m + 1, j), (m, i)) = dim_{k} e_{i}D(B)e_{j} for all i, j with i ≠ j. So dim_{k} kQB̂/I_{ϕ} ((s, j), (m, i)) = dim_{k} B̂(e_{s,j}, e_{m,i}). Hence I_{ϕ} = Ker ϕ. □

EXAMPLE 2.20 Let B = kQ_{B} be the bound quiver algebra given by the quiver

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and the ideal IB̂ generated by

i) paths of length 4;

ii) β_{2}^{m}α_{1}^{m+1}α_{3}^{m+1}, β_{3}^{m}α_{1}^{m+1}α_{2}^{m+1};

iii) α_{2}^{m}β_{2}^{m} − α_{3}^{m}β_{3}^{m}.

EXAMPLE 2.21 Let B = kQ_{B}/I_{B} be the bound quiver algebra given by the quiver

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and by the ideal I_{B} in kQ_{B} generated by α_{1}α_{2}. So P = {α_{1}, α_{2}, α_{3}}, and denote β_{i} = β_{α_{i}} for i = 1, 2, 3. Then B̂ = kQB̂/IB̂ with QB̂ the quiver

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β_{2}^{1} β_{3}^{1} β_{3}^{−1}

and the ideal IB̂ generated by

i) paths of length 3; ii) α_{1}^{m}α_{2}^{m}, α_{2}^{m}β_{3}^{m}, α_{3}^{m}β_{2}^{m}, β_{1}^{m}α_{3}^{m+1}, β_{3}^{m}α_{1}^{m+1}, β_{2}^{m}β_{1}^{m+1};

iii) α_{1}^{m}β_{1}^{m} − α_{3}^{m}β_{3}^{m}, α_{2}^{m}β_{2}^{m} − β_{1}^{m}α_{1}^{m+1}, β_{2}^{m}α_{2}^{m+1} − β_{3}^{m}α_{3}^{m+1}.

3 THE UNIVERSAL GALOIS COVERING

Given a presentation (QB̂, IB̂) of B̂, we are going to describe the corresponding presentation (Q_{A}, I_{A}) of A using the Galois covering F : B̂ → A given in Theorem 2.4 and the induced quiver morphism Q_{F}. In fact, to describe π_{1}(Q_{A}, I_{A}) we only need the minimal relations in I_{A}. From Remark 2.1 we know that Q_{A} = Q_{F} (QB̂) and it follows from Remark 2.2 that the minimal relations in I_{A} belong to Q_{F} (IB̂).

3.1 Presentations of Selfinjective Algebras B̂/G

We begin this section with several lemmas which will be useful in order to describe the ordinary quiver Q_{A} of A.

Given i ∈ (Q_{B})_{0}, let ν^{m}(e_{i}) = e_{ν^{m}(i)}. Recall that η(i) = min{k > 0 : ν^{k}(i) ∈ (Q_{B})_{0}} and g(e_{m,i}) = e_{m−η(i),ν^{η(i)}(i)}, as defined after Theorem 2.4.

LEMMA 3.1 Given s ∈ Z, j ∈ (Q_{B})_{0}, there exists l ∈ Z such that g^{l}(e_{s,j}) = e_{m,i} with 0 ≤ m < η(i), i ∈ (Q_{B})_{0}.

Proof. It follows from the definition of g that i ∈ (Q_{B})_{0} if g^{l}(e_{s,j}) = e_{m,i} for l ∈ Z. Suppose s ≥ 0. Define s_{0} = s, j_{0} = j, s_{r+1} = s_{r} − η(j_{r}) and j_{r+1} = ν^{η(j_{r})}(j_{r}). Hence g(e_{s_{r},j_{r}}) = e_{s_{r+1},j_{r+1}}. Since s > s_{1} > s_{2} > … there exists l ∈ Z such that s_{l} ≥ 0 and s_{l+1} < 0. This implies that 0 ≤ s_{l} = s_{l+1} + η(j_{l}) < η(j_{l}). Hence g^{l}(e_{s,j}) = e_{s_{l},j_{l}} and 0 ≤ s_{l} < η(j_{l}). If s < 0, g^{−s}(e_{s,i}) = e_{s′,i′} with s′ ≥ 0, by the definition of g. Now we know that there exists l ∈ Z such that g^{l}(e_{s′,i′}) = e_{s′′,i′′} with 0 ≤ s′′ < η(i′′). Hence g^{l−s}(e_{s,i}) = e_{s′′,i′′} and the proof is complete. □

LEMMA 3.2 For i, j ∈ (Q_{B})_{0}, ν^{m}(i) = ν^{s}(j) with 0 ≤ m < η(i) and 0 ≤ s < η(j) imply that m = s and i = j.

Proof. If m = s then i = j since ν is an automorphism. Suppose m > s, then ν^{m−s}(i) = j ∈ (Q_{B})_{0}. Hence m − s ≥ η(i) by the definition of η. So m ≥ η(i), a contradiction. □

LEMMA 3.3 [22, Lemma 3.4] Assume i, j ∈ (Q_{B})_{0} and e_{j}Be_{i} ≠ 0. Then η(j) = η(i) or η(j) = η(i) + 1.

We are now in a position to describe the ordinary quiver Q_{A} of A, using the quiver morphism Q_{F} : QB̂ → Q_{A} induced by the Galois covering F : B̂ → A and the description of the ordinary quiver QB̂ given in Proposition 2.13.

PROPOSITION 3.4 Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G as in Theorem 2.4 and assume that B is schurian. Then

(Q_{A})_{0} = {ν^{m}(i)}_{i∈(Q_{B})_{0},0≤m<η(i)}

and

(Q_{A})_{1} = {Q_{F} (α^{m})}_{α∈(Q_{B})_{1},0≤m<η(s(α))} ∪ {Q_{F} (β_{p}^{m})}_{p∈P,0≤m<η(t(p))}.

Moreover, Q_{A} has no double arrows.

Proof. Let V = {ν^{m}(i)}_{i∈(Q_{B})_{0},0≤m<η(i)} . Since (Q_{A})_{0} = Q_{F} ((QB̂)_{0}) and Q_{F} (m, i) = ν^{m}(i), we have V ⊆ (Q_{A})_{0}. To see that (Q_{A})_{0} ⊆ V it is enough to use that F g = F and Lemma 3.1, because, given (s, j) ∈ (QB̂)_{0}, there exists l ∈ Z such that g^{l}(e_{s,j}) = e_{m,i} with 0 ≤ m < η(i), i ∈ (Q_{B})_{0}. Then F (e_{s,j}) = F (g^{l}(e_{s,j})) = F (e_{m,i}). So ν^{s}(j) = ν^{m}(i). The elements in V are pairwise different by Lemma 3.2. Let S_{1} = {Q_{F} (α^{m})}_{α∈(Q_{B})_{1},0≤m<η(s(α))} and S_{2} = {Q_{F} (β_{p}^{m})}_{p∈P,0≤m<η(t(p))}. We know that S_{1} ∪ S_{2} ⊆ (Q_{A})_{1} since (Q_{A})_{1} = Q_{F} ((QB̂)_{1}). Let l ∈ Z be such that g^{l}(e_{m,i}) = e_{m′,i′}, 0 ≤ m′ < η(i′). If F (B̂(e_{s,j}, e_{m,i})) = F (B̂(g^{l}(e_{s,j}), g^{l}(e_{m,i})) ≠ 0 then g^{l}(e_{s,j}) = e_{s′,j′} with s′ = m′ or s′ = m′ + 1. So, for any arrow α : (m, i) → (s, j), we have F (α) ∈ F (B̂(e_{s′,j′}, e_{m′,i′})) with s′ = m′ or s′ = m′ + 1 and 0 ≤ m′ < η(i′). Hence Q_{F} (α) ∈ S_{1} if s′ = m′ and Q_{F} (α) ∈ S_{2} if s′ = m′ + 1. Therefore (Q_{A})_{1} ⊆ S_{1} ∪ S_{2}. To finish the proof we have to show that the arrows are pairwise different. We have several cases to consider.

a) Q_{F} (α^{m}), Q_{F} (γ^{s}) ∈ S_{1}, α^{m} : (m, i) → (m, j) and γ^{s} : (s, i′) → (s, j′). Now, Q_{F} (α^{m}) = Q_{F} (γ^{s}) implies that ν^{m}(i) = ν^{s}(i′) and ν^{m}(j) = ν^{s}(j′). From Lemma 3.2 we get m = s, i = i′ and this implies that also j = j′. So α, γ are arrows in Q_{B} sharing starting and terminal points. Since B is schurian Q_{B} has no double arrows, so α = γ.

b) Q_{F} (β_{p}^{m}), Q_{F} (β_{p′}^{s}) ∈ S_{2}, β_{p}^{m} : (m, i) → (m + 1, j) and β_{p′}^{s} : (s, i′) → (s + 1, j′). Now, Q_{F} (β_{p}^{m}) = Q_{F} (β_{p′}^{s}) implies that ν^{m}(i) = ν^{s}(i′) and ν^{m+1}(j) = ν^{s+1}(j′). From Lemma 3.2 we get m = s, i = i′, and this implies that also j = j′. Now p, p′ are two maximal paths in P sharing starting and terminal points and B is schurian, so p = p′ by the definition of the set P.

c) Q_{F} (α^{m}) ∈ S_{1}, Q_{F} (β_{p}^{s}) ∈ S_{2}, α^{m} : (m, i) → (m, j) and β_{p}^{s} : (s, i′) → (s + 1, j′). In this case, Q_{F} (α^{m}) = Q_{F} (β_{p}^{s}) implies that ν^{m}(i) = ν^{s}(i′) and ν^{m}(j) = ν^{s+1}(j′). From Lemma 3.2 we get m = s, i = i′ and this implies that j = ν(j′), hence η(j′) = 1. Now p is a maximal path in kQ_{B} from j′ to i′, so e_{j′}Be_{i′} ≠ 0. From Lemma 3.3 we have η(i′) ≤ η(j′), so η(i′) = 1. But 0 ≤ s < η(i′), so s = 0. Then α^{m} = α^{0} : (0, i) → (0, j) and β_{p}^{s} = β_{p}^{0} : (0, i) → (1, j′). But Q_{F} (α^{0}) ∈ e_{i}(rad B)e_{j} and Q_{F} (β_{p}^{0}) ∈ e_{i}Ie_{ν(j′)} = e_{i}Ie_{j} by the definition of F, which is impossible [21, Proposition 1.7]. □

Now, if (QB̂, IB̂) is the presentation of B̂ described in 2.13 and 2.19, and Q_{A} is the ordinary quiver of A, we are going to describe the corresponding ideal I_{A} using the Galois covering F : B̂ → A given in Theorem 2.4 and the induced quiver morphism Q_{F} : QB̂ → Q_{A}.

From Remark 2.2 we know that the minimal relations in I_{A} belong to Q_{F} (IB̂). In fact, for any minimal relation ρ ∈ I_{A} there exists ρ′ ∈ IB̂ such that Q_{F} (ρ′) = ρ and ρ′ = ∑ λ_{i}w_{i} with w_{i} ∉ IB̂.

So we have to describe Q_{F} (ρ′) for any minimal relation ρ′ ∈ IB̂ described in Proposition 2.19 iii).

NOTATION. If q = α_{1} … α_{r} is a path in kQ_{B}, we denote by q^{m} = α_{1}^{m} … α_{r}^{m} the corresponding paths in kQB̂, and, α(m) = Q_{F} (α^{m}), β_{p}(m) = Q_{F} (β_{p}^{m}), q(m) = Q_{F} (q^{m}) the corresponding paths in kQ_{A}.

So we can now describe the relations in I_{A}.

PROPOSITION 3.5 Let (QB̂, IB̂) be the presentation of B̂ given in Proposition 2.13 and 2.19, and let Q_{F} : QB̂ → Q_{A} be the quiver morphism induced by the Galois covering F : B̂ → A given in Theorem 2.4. Then the ideal I_{A} is generated by

i) paths α_{s}(m) … α_{t}(m)β_{p}(m)α_{1}(m+1) … α_{s}(m+1) and β_{p}(m−1)α_{1}(m) … α_{t}(m)β_{p}(m) for any elementary path α_{1}^{m} … α_{t}^{m}β_{p}^{m}, p ∈ P;

ii) paths Q_{F} (q) for any path q in kQB̂ whose arrows do not belong to an elementary path;

iii) elements γ_{1}(m) − μγ_{2}(m) with 0 ≠ μ ∈ k, γ_{1}, γ_{2} paths in kQ_{B} sharing starting and terminal points such that \overline{γ}_{1} ≠ 0 ≠ \overline{γ}_{2} in B;

iv) elements Q_{F} (q_{1}) − μQ_{F} (q_{2}) with 0 ≠ μ ∈ k and q_{1}, q_{2} paths in kQB̂ having the same supplement in elementary paths in kQ_{B}. In particular, if C and C′ are elementary paths in kQB̂ sharing starting points, then Q_{F} (C) − μQ_{F} (C′) ∈ I_{A}.

Proof. The proof follows immediately from Proposition 2.19. Observe that in iii) we use that B is schurian, so \overline{γ}_{1} = μ\overline{γ}_{2} for some 0 ≠ μ ∈ k; and in iv) we use that if C and C′ are elementary paths starting at (m, i) then C and C′ have the same supplement e_{m+1,i}. □

EXAMPLE 3.6 Let A = kQ_{A}/I_{A} be the bound quiver algebra with quiver

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where the ideal I_{A} in kQ_{A} is generated by all paths of length 4, β_{2}α_{1}α_{3}, β_{3}α_{1}α_{2} and α_{2}β_{2} − α_{3}β_{3}. Then A = T (B) = B̂/(νB̂) is a repetitive covering selfinjective algebra with B the algebra considered in Example 2.20, ν = id, α_{i} = α_{i}(m) and β_{i} = β_{i}(m) for all m ∈ Z.

EXAMPLE 3.7 Let A = kQ_{A}/I_{A} be the bound quiver algebra with quiver

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ξ  θ ρ  ϵ γ ζ  δ ω

where the ideal I_{A} in kQ_{A} is generated by ξζ, ζϵ, ρω, θδ, ϵγ, ωρ, γξ, δθ, ξθ − ρϵ, ζω − θγ, ωξ − ϵδ and γρ − δζ, see [21]. Then A is a repetitive covering selfinjective algebra with B the algebra considered in Example 2.21, ν(1) = 4, ν(2) = 1, ν(3) = 2, ν(4) = 3, η(1) = 2, η(2) = η(3) = 1, ξ = α_{1}(0) = α_{2}(1), γ = α_{1}(1), ζ = α_{2}(0) = β_{3}(1), ρ = α_{3}(0) = β_{1}(1), δ = α_{3}(1), θ = β_{1}(0) = β_{2}(1), ω = β_{2}(0) and ϵ = β_{3}(0).

3.2 Fundamental Groups

In this section we are going to describe the relationship between the fundamental groups of (Q_{A}, I_{A}) and (Q_{B}, I_{B}).

As we observe after Proposition 2.13, in order to describe the fundamental groups we may assume without loss of generality that the hypothesis of Proposition 2.19 are satisfied.

Let i ∈ (Q_{B})_{0}, and let p ∈ P be a maximal path going through i, that is, p = p_{−}p_{+} with i = s(p_{+}) = t(p_{−}). For any pair (i, p) with p a maximal path going through i, we consider the following path in kQ_{A}: C_{i}^{m}(p) = Q_{F} (p_{+}^{m}β_{p}^{m}p_{−}^{m+1}).

Let ∼ be the equivalence relation on the set W of all walks in Q_{A} defined in Section 2.1. The element defined by the path C_{i}^{m}(p) in the fundamental group of (Q_{A}, I_{A}) does not

depend on the choice of p, as we prove in the following lemma.

LEMMA 3.8 For any i ∈ (Q_{B})_{0}, C_{i}^{m}(p) ∼ C_{i}^{m}(q) for any p, q ∈ P going through i.

Proof. Let p = p_{−}p_{+}, q = q_{−}q_{+} and i = s(p_{+}) = t(p_{−}) = s(q_{+}) = t(q_{−}). Then C = p_{+}^{m}β_{p}^{m}p_{−}^{m+1} and C′ = q_{+}^{m}β_{q}^{m}q_{−}^{m+1} are two elementary paths starting at (m, i). So the result follows from Proposition 3.5 iv) because Q_{F} (C) − μQ_{F} (C′) is a minimal relation in I_{A}. □

To simplify the notation, for any i ∈ (Q_{B})_{0} we fix a maximal path p ∈ P going through i and we denote by C_{i}^{m} = C_{i}^{m}(p) the corresponding path in kQ_{A}.

In the following lemmas we point out some commutativity relations that will be used in the proof of the main theorem of this section.

LEMMA 3.9 For any arrow α : i → j in Q_{B} and m an integer,

α(m)C_{j}^{m} ∼ C_{i}^{m}α(m + 1) and α(m)^{−1}C_{i}^{m} ∼ C_{j}^{m}α(m + 1)^{−1}.

Proof. Any arrow α : i → j in Q_{B} belongs to a maximal path p_{1}αp_{2} in kQ_{B}. Let p ∈ P such that \overline{p} = μ\overline{p_{1}αp_{2}}, 0 ≠ μ ∈ k. Then C_{i}^{m} ∼ Q_{F} (α^{m}p_{2}^{m}β_{p}^{m}p_{1}^{m+1}) and C_{j}^{m} ∼ Q_{F} (p_{2}^{m}β_{p}^{m}p_{1}^{m+1}α^{m+1}). So α(m)C_{j}^{m} ∼ Q_{F} (α^{m}p_{2}^{m}β_{p}^{m}p_{1}^{m+1}α^{m+1}) ∼ C_{i}^{m}α(m + 1). □

LEMMA 3.10 For any arrow α : i → j in Q_{B}, i) if η(i) = η(j) then

α(η(i) − 1)C_{j}^{η(i)−1} ∼ C_{i}^{η(i)−1}γ(0) and α(η(i) − 1)^{−1}C_{i}^{η(i)−1} ∼ C_{j}^{η(i)−1}γ(0)^{−1}

for some γ ∈ (Q_{B})_{1};

ii) if η(i) = η(j) + 1 then

α(η(i) − 1)C_{j}^{η(i)−1} ∼ C_{i}^{η(i)−1}C_{ν^{η(i)}(i)}^{0}q(1)^{−1} and α(η(i) − 1)^{−1}C_{i}^{η(i)−1} ∼ q(0)

for some q ∈ P.

Proof.

i) If η(i) = η(j), then Q_{F} (α^{η(i)}) = Q_{F} (γ^{0}) = γ(0) for some arrow γ : ν^{η(i)}(i) → ν^{η(i)}(j) in (Q_{B})_{1}; it follows from Lemma 3.9 that α(η(i) − 1)C_{j}^{η(i)−1} ∼ C_{i}^{η(i)−1}α(η(i)) =

C_{i}^{η(i)−1}γ(0) and α(η(i) − 1)^{−1}C_{i}^{η(i)−1} ∼ C_{j}^{η(i)−1}α(η(i))^{−1} = C_{j}^{η(i)−1}γ(0)^{−1};

ii) if η(i) = η(j) + 1, then Q_{F} (α^{η(i)}) = Q_{F} (β_{q}^{0}) = β_{q}(0) for some arrow β_{q}^{0} : (0, ν^{η(i)}(i)) → (1, ν^{η(j)}(j)) and q : ν^{η(j)}(j) → ν^{η(i)}(i) a maximal path in P. Since

β_{q}(0) ∼ β_{q}(0)q(1)q(1)^{−1} ∼ C_{ν^{η(i)}(i)}^{0}q(1)^{−1},

the first relation follows from Lemma 3.9. To prove the second one, let p = p_{1}αp_{2} be a maximal path in kQ_{B}. From Lemma 3.3 we get η(s(p_{1})) = η(i) and η(t(p_{2})) = η(j) since η(s(p_{1})) ≥ η(i) = η(j) + 1 ≥ η(t(p_{2})) + 1. Hence Q_{F} (p_{1}^{η(i)}) = Q_{F} (q_{1}^{0}) = q_{1}(0), Q_{F} (p_{2}^{η(j)}) = Q_{F} (q_{2}^{0}) = q_{2}(0) and Q_{F} (β_{p}^{η(j)}) = Q_{F} (δ^{0}) = δ(0) for some q_{1}, q_{2} paths in kQ_{B}, δ ∈ (Q_{B})_{1} and \overline{q} = ψ\overline{q_{2}δq_{1}} for some 0 ≠ ψ ∈ k. So

α(η(i) − 1)^{−1}C_{i}^{η(i)−1} ∼ α(η(i) − 1)^{−1}Q_{F} (α^{η(i)−1}p_{2}^{η(i)−1}β_{p}^{η(i)−1}p_{1}^{η(i)})
∼ Q_{F} (p_{2}^{η(i)−1}β_{p}^{η(i)−1}p_{1}^{η(i)}) = Q_{F} (q_{2}^{0}δ^{0}q_{1}^{0}) = q(0).
□

The following theorem shows the relationship between the fundamental groups of (Q_{A}, I_{A}) and (Q_{B}, I_{B}).

THEOREM 3.11 Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G as in Theorem 2.4 and assume that B is schurian. Let (Q_{A}, I_{A}) be any presentation of A associated to a presentation (Q_{B}, I_{B}) of B. Then we have the following short exact sequence of groups

1 → π_{1}(Q_{B}, I_{B}) → π_{1}(Q_{A}, I_{A}) → G → 1.

Proof. We are going to define a surjective morphism of groups ϕ : π_{1}(Q_{A}, I_{A}) → G whose kernel is π_{1}(Q_{B}, I_{B}). Let w = w_{1}^{ϵ_{n}} … w_{n}^{ϵ_{1}} be a walk in Q_{A}, ϵ_{i} = \underline{+}1, w_{i} ∈ (Q_{A})_{1}, and let a : π_{1}(Q_{A}, I_{A}) → Z be the group morphism given by

a([w]) = ∑_{i=1}^{n}(−1)^{ϵ_{i}}h(w_{i})

with

h(w_{i}) = \{ \begin{matrix}1 & if w_{i} = β_{p}(0), \\ 0 & otherwise. \end{matrix}

We claim that a([w]) does not depend on a choice of a representative w of the equivalence class [w]. In fact, we only have to check the relations described in Proposition 3.5 iii),iv). If γ_{1}, γ_{2} are paths in kQ_{B} sharing starting and terminal points, \overline{γ}_{1} ≠ 0 ≠ \overline{γ}_{2} in B, then a([γ_{1}(m)]) = 0 = a([γ_{2}(m)]). Let q_{1}, q_{2} be paths in kQB̂ having the same supplement γ^{m+1} in elementary paths, that is,

q_{1}γ^{m+1} = α_{s}^{m} … α_{t}^{m}β_{p_{1}}^{m}α_{1}^{m+1} … α_{s′}^{m+1}γ^{m+1},
q_{2}γ^{m+1} = δ_{r}^{m} … δ_{l}^{m}β_{p_{2}}^{m}δ_{1}^{m+1} … δ_{r′}^{m+1}γ^{m+1},

where p_{1}, p_{2} ∈ P, s′ < s, r′ < r and γ is a path in kQ_{B}. Then q_{1}(m) ∼ μq_{2}(m) for some 0 ≠ μ ∈ k, and

a([q_{1}(m)] = ∑_{i=s}^{t} h(α_{i}(m)) + h(β_{p_{1}}(m)) + ∑_{i=1}^{s′} h(α_{i}(m + 1)) = \{ \begin{matrix}1 & if m = 0, \\ 0 & otherwise. \end{matrix}

The same holds for a([q_{2}(m)]), so a([q_{1}(m)] = a([q_{2}(m)]). Thus we define ϕ : π_{1}(Q_{A}, I_{A}) → G by ϕ([w]) = g^{a([w])} with g a fixed generator of G. Clearly ϕ is a group homomorphism. We will prove next that ϕ is surjective. We fix x_{0} as the origin of the paths defining π_{1}(Q_{A}, I_{A}) (see Section 2.1). We may assume that x_{0} is a source in Q_{B}. Then there exist a maximal path p_{0} ∈ P starting at x_{0} and a walk u in Q_{B} from ν^{η(x_{0})}(x_{0}) to x_{0}. Let w = p_{0}(0)β_{p_{0}}(0)p_{0}(1)β_{p_{0}}(1) … β_{p_{0}}(η(x_{0}) − 1)u(0). Then ϕ([w]) = g, so ϕ is surjective. Next we will show that Ker ϕ ≃ π_{1}(Q_{B}, I_{B}). We first observe that

ϕ([C_{x}^{m}]) = \{ \begin{matrix}g & if m = 0, \\ 1 & otherwise. \end{matrix}

The idea now is to show that any [w] ∈ π_{1}(Q_{A}, I_{A}) can be written in the form

[w] = [C_{x_{0}}^{0}C_{x_{0}}^{1} … C_{x_{0}}^{η(x_{0})−1}C_{x_{1}}^{0}C_{x_{1}}^{1} … C_{x_{1}}^{η(x_{1})−1}C_{x_{2}}^{0} … C_{x_{r}}^{η(x_{r})−1}u(0)]

with u a walk in Q_{B} from ν^{η(x_{r})}(x_{r}) to x_{0} and x_{l} = ν^{η(x_{l−1})}(x_{l−1}) for all l = 1, …, r. If this is the case, ϕ([w]) = g^{r+1}, and therefore [w] ∈ Ker ϕ if and only if [w] = [u(0)] for some walk u in Q_{B}, as desired. So we have to prove that any [w] ∈ π_{1}(Q_{A}, I_{A}) can be written as claimed. Clearly either w is u(0) for some walk u in Q_{B} or w is a composition of walks u(m)β_{p}(m)v(m + 1) and their inverses with p ∈ P and walks u, v in Q_{B}. In the first case there is nothing to prove. In the second case we may assume that the walks that occur in w are all of the form u(m)β_{p}(m)v(m + 1), or all of the form u(m + 1)β_{p}^{−1}(m)v(m), as we prove next. We first observe that β_{p}(m) ∼ p^{−1}(m)p(m)β_{p}(m) ∼ p^{−1}(m)C_{s(p)}^{m}, and, by iterated use of Lemma 3.9, we have C_{s(u)}^{m}u(m + 1) ∼ u(m)C_{t(u)}^{m} for any walk u in Q_{B}. Then

β_{p}(m)v(m + 1)l(m + 1)^{−1}β_{q}(m)^{−1} ∼ p^{−1}(m)C_{s(p)}^{m}v(m + 1)l(m + 1)^{−1}(C_{s(q)}^{m})^{−1}q(m)
∼ p^{−1}(m)C_{s(p)}^{m}(C_{s(v)}^{m})^{−1}v(m)l(m)^{−1}q(m) ∼ (p^{−1}vl^{−1}q)(m)

where v, l are walks in Q_{B}. The last relation holds because C_{s(p)}^{m} and C_{s(v)}^{m} correspond to elementary paths sharing starting points since s(p) = s(v) (see Lemma 3.8). To finish the proof we may assume that w is a composition of walks of the form u(m)β_{p}(m)v(m + 1). The case when β_{p}^{−1} occurs is analogous. For any walk u in Q_{B} with origin x, u(m)β_{p}(m) ∼ u(m)p^{−1}(m)C_{s(p)}^{m}. From Proposition 3.4 we may assume that 0 ≤ m < η(x) − 1. Now, for any arrow α : i → j appearing in u or p we apply the commutativity relations proved in Lemmas 3.9 and 3.10, depending on m, η(i) and η(j). So we get

u(m)p^{−1}(m)C_{s(p)}^{m} ∼ \{ \begin{matrix}C_{x}^{m}z(m + 1) & with s(z) = x, \\ C_{x}^{η(x)−1}C_{ν^{η(x)}(x)}^{0}z(0) & with s(z) = ν^{η(x)}(x), or \\ z(0) & with s(z) = x \end{matrix}

for some walk z in Q_{B}. Iterating this procedure we get the desired equality

[w] = [C_{x_{0}}^{0}C_{x_{0}}^{1} … C_{x_{0}}^{η(x_{0})−1}C_{x_{1}}^{0}C_{x_{1}}^{1} … C_{x_{1}}^{η(x_{1})−1}C_{x_{2}}^{0} … C_{x_{r}}^{η(x_{r})−1}u(0)]

with u a walk in Q_{B} from ν^{η(x_{r})}(x_{r}) to x_{0}, and x_{l} = ν^{η(x_{l−1})}(x_{l−1}) for all l = 1, …, r. □

Recall that a connected triangular algebra B is called simply connected if the fundamental group π_{1}(Q_{B}, I_{B}) is trivial for any presentation (Q_{B}, I_{B}) of B.

An important consequence of the previous theorem is the following.

COROLLARY 3.12 Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G as in Theorem 2.4, and assume that B is schurian. Then F : B̂ → A is the universal Galois covering of A if and only if B is simply connected.

Proof. The Galois covering F : B̂ → A is universal if and only if G ≃ π_{1}(Q_{A}, I_{A}) for any presentation (Q_{A}, I_{A}) of A, and hence it follows from Theorem 3.11 that F is universal if and only if π_{1}(Q_{B}, I_{B}) is trivial for any presentation (Q_{B}, I_{B}) of B. □

COROLLARY 3.13 Let B be a triangular schurian algebra. Then the Galois covering B̂ → B ⋉ D(B) with the Galois group generated by νB̂ is universal if and only if B is simply connected.

Proof. The trivial extension T (B) = B ⋉ D(B) of B by D(B) is a repetitive covering selfinjective algebra, as we have seen in Example 2.6. Hence we have that B̂ → T (B) = B ⋉ D(B) is the universal Galois covering of T (B) if and only if B is simply connected.

COROLLARY 3.14 Let A = kQ/J^{t+1} with Q the oriented cycle with n vertices and n arrows, 1 ≤ t ≤ n, and J the ideal generated by the arrows. Let B = kQ_{B} be the hereditary algebra with

Q_{B} : 1 → 2 → … → t.

Then B̂ → A is the universal Galois covering with the Galois group G = (γ^{n}) (see Example 2.7).

Proof. It follows from the fact that A is a repetitive covering selfinjective algebra and B is simply connected.

COROLLARY 3.15 Let B be a triangular schurian algebra, A = B̂/(νc ; B̂), c ≠ 0. Then B̂ → A is the universal Galois covering with the Galois group (νc ; B̂) if and only if B is simply connected.

REMARK 3.16 The representatives of the derived equivalence classes of standard representation–finite selfinjective algebras Λ(A_{n}, s/n, 1), s ≥ n, Λ(A_{2p+1}, s, 2), Λ(D_{n}, s, 1), Λ(D_{n}, s, 2), Λ(D_{4}, s, 3), Λ(E_{n}, s, 1) and Λ(E_{6}, s, 2) considered in Example 2.8 are repetitive covering selfinjective algebras with B = kQ_{B} and Q_{B} a Dynkin graph (see [1, 2.5]). So Theorem 3.11 shows that their universal Galois coverings are given by B̂.

EXAMPLE 3.17 Let A = kQ_{A}/I_{A} be the bound quiver algebra considered in Example 3.6. Since B is simply connected then B̂ is the universal Galois covering of A.

EXAMPLE 3.18 Let A = kQ_{A}/I_{A} be the bound quiver algebra considered in Example 3.7. Then B̂ is not the universal Galois covering of A since B is not simply connected.

4 THE FIRST HOCHSCHILD COHOMOLOGY GROUP

Now we want to show the connection between the first Hochschild cohomology group of A and the fundamental group of any presentation (Q_{B}, I_{B}) of B. Recall that H^{1}(A) ≃ Der(A)/ Inn(A), where Der(A) = {δ ∈ Hom_{k}(A, A) : δ(xy) = xδ(y) + δ(x)y} is the k–vector space of derivations of A on A, and Inn(A) = {δ_{x} : A → A : δ_{x}(y) = yx − xy, x ∈ A} is the subspace of inner derivations.

Let A be a repetitive covering selfinjective algebra, B = A/I triangular, e a residual identity of B. Then it is known that

0 → eIe → eAe →^{ρ} eAe/eIe → 0

is a trivial extension of eAe/eIe by eIe, that is, there exists a morphism of algebras ι : eAe/eIe → eAe such that ρι = id (see [22]). Since eAe/eIe and B are isomorphic A–algebras, eAe is isomorphic to the algebra B ⊕ eIe with product (b, eye)(b′, ey′e) = (bb′, bey′e + eyeb′), and from now on we shall identify them. Hence any element in A can be written as

a = eae + ea(1 − e) + (1 − e)a = b + eye + ea(1 − e) + (1 − e)a

with b ∈ B and y ∈ I.

THEOREM 4.1 If A is a repetitive covering selfinjective algebra then H^{1}(A) ≠ 0.

Proof. Consider the k–linear map δ : A → A defined by

δ(a) = δ(b + eye + ea(1 − e) + (1 − e)a) = eye + ea(1 − e)

Recall that IeI = 0 and 1 − e ∈ I, and observe that δ((1 − e)A) = 0 and δ(x) = x for any x ∈ eA(1 − e) or x ∈ eIe by the definition of δ. A direct computation shows that δ is a derivation. So it is enough to show that δ is not inner. Suppose contrarily that it is inner, that is, δ = δ_{x} for some x ∈ A. Then, since δ(e_{i}) = 0 for all i ∈ (Q_{A})_{0}, 0 = δ(e_{i}) = δ_{x}(e_{i}) = e_{i}x − xe_{i} and so x = ∑_{i} e_{i}x = ∑_{i} e_{i}xe_{i} = ∑_{i} λ_{i}e_{i} + y with y ∈ ⊕_{i}(e_{i} rad A e_{i}). For any arrow α in Q_{B}, we have

0 = δ_{x}(α) = αx − xα = (λ_{t(α)} − λ_{s(α)})α + αy − yα.

Since αy − yα ∈ rad^{2} A and α ∉ rad^{2} A, we get λ_{t(α)} = λ_{s(α)}. Since this holds for any arrow α in Q_{B} and Q_{B} is connected, we have λ_{i} = λ_{j} for all i, j ∈ (Q_{B})_{0}. Let i be a fixed source in Q_{B}, and let p ∈ P be a maximal path in kQ_{B} such that s(p) = i. For γ := pβ_{p} and 0 ≤ m < η(i) we have γ(m) = p(m)β_{p}(m) : e_{ν^{m}(i)} → e_{ν^{m+1}(i)} are maximal paths in kQ_{A} (see Proposition 3.5 i), ii)). By definition, δ(\overline{γ}(0)) = \overline{γ}(0) since \overline{γ}(0) belongs to eIe or eA(1 − e), and δ(\overline{γ}(m)) = 0 for 0 < m < η(i) since δ((1 − e)A) = 0. But

δ(\overline{γ}(m)) = δ_{x}(\overline{γ}(m)) = (λ_{ν^{m+1}(i)} − λ_{ν^{m}(i)})\overline{γ}(m) + \overline{γ}(m)y − y\overline{γ}(m) = (λ_{ν^{m+1}(i)} − λ_{ν^{m}(i)})\overline{γ}(m)

since γ(m) is a maximal path. Then λ_{ν^{m}(i)} = λ_{ν^{m+1}(i)} for any m with 0 < m < η(i). But i, ν^{η(i)}(i) are both vertices in Q_{B}, hence λ_{ν^{η(i)}(i)} = λ_{i}. So λ_{ν^{m}(i)} = λ_{ν^{m+1}(i)} for any m with 0 ≤ m < η(i) and δ(\overline{γ}(0)) = δ_{x}(\overline{γ}(0)) = (λ_{ν(i)} − λ_{i})\overline{γ}(0) = 0, a contradiction. □

REMARK 4.2 If A is a repetitive covering selfinjective algebra, A ≃ B̂/G and B = k, then A = kQ/J^{2} with Q the oriented cycle

1 → ν(1) → ν^{2}(1) → … → ν^{η(1)−1}(1) → 1

and J the ideal generated by the arrows. In this case H^{1}(A) is well known: H^{1}(A) = k^{2} if ν(1) = 1 and k is a field of characteristic 2, and H^{1}(A) = k otherwise [6, Proposition 2.3,2.4].

From now on we assume that B is not the trivial k–algebra k. This means that the set of vertices Q_{0} has at least two elements, since Q has no oriented cycles.

LEMMA 4.3 [14] Let A be a basic, connected finite dimensional algebra, and let {e_{1}, …, e_{n}} be a complete set of primitive orthogonal idempotents, n > 1. For any δ ∈ Der(A) there exists an inner derivation δ_{x} such that

(δ − δ_{x})(e_{i}) = 0 for all i = 1, …, n.

Proof. Let δ : A → A be a derivation. Then

δ(e_{i}) = δ(e_{i}e_{i}) = e_{i}δ(e_{i}) + δ(e_{i})e_{i}
= e_{i} (e_{i}δ(e_{i}) + δ(e_{i})e_{i}) + (e_{i}δ(e_{i}) + δ(e_{i})e_{i}) e_{i}  (1)
= e_{i}δ(e_{i}) + δ(e_{i})e_{i} + 2e_{i}δ(e_{i})e_{i}

For any pair (i, j) with i ≠ j, we have

0 = δ(e_{i}e_{j}) = e_{i}δ(e_{j}) + δ(e_{i})e_{j}. (2)

Let x = ∑_{j=1}^{n} e_{j}δ(e_{j}). Then δ_{x}(e_{i}) = e_{i}x − xe_{i} = e_{i}δ(e_{i}) − ∑_{j=1,j≠i}^{n} e_{j}δ(e_{j})e_{i} − e_{i}δ(e_{i})e_{i}. By (2), we have δ_{x}(e_{i}) = e_{i}δ(e_{i}) + ∑_{j=1,j≠i}^{n} e_{j}e_{j}δ(e_{i}) − e_{i}δ(e_{i})e_{i} = δ(e_{i}) − e_{i}δ(e_{i})e_{i}. Hence it suffices to show that e_{i}δ(e_{i})e_{i} = 0, which follows from (1). In fact, (1) implies 2e_{i}δ(e_{i})e_{i} = 0 and so e_{i}δ(e_{i})e_{i} = 0 if char k ≠ 2. In case char k = 2, (1) shows that δ(e_{i}) = e_{i}δ(e_{i}) + δ(e_{i})e_{i}, hence e_{i}δ(e_{i})e_{i} = 0 by multiplying e_{i} from the left. □

We denote Der^{n}(A) = {δ ∈ Der(A) : δ(e_{i}) = 0 for all i} the subspace of normalized derivations and Inn^{n}(A) = Der^{n}(A) ∩ Inn(A).

COROLLARY 4.4

H^{1}(A) ≃ Der^{n}(A)/ Inn^{n}(A)

and Inn^{n}(A) = { δ_{x} : x = ∑_{i=1}^{n} λ_{i}e_{i} + y, y ∈ ⊕_{i}e_{i} rad A e_{i} }.

Proof. By the previous lemma we have H^{1}(A) ≃ Der^{n}(A)/ Inn^{n}(A). Let δ_{x} ∈ Inn^{n}(A) for x ∈ A. Then 0 = δ_{x}(e_{i}) = e_{i}x − xe_{i} for all i. So x = ∑_{i=1}^{n} e_{i}x = ∑_{i=1}^{n} e_{i}xe_{i} = ∑_{i=1}^{n} λ_{i}e_{i} + y for some y ∈ ⊕_{i}e_{i} rad A e_{i}. □

The proof of the following proposition is a special case of [21, Proposition 1.7].

PROPOSITION 4.5 Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G = (νc ; B̂), c ≠ 0, as in Theorem 2.4, and assume that B is schurian. Then, for any arrow α : i → j in Q_{A}, dim_{k} e_{i}Ae_{j} = 1.

COROLLARY 4.6 Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G = (νc ; B̂), c ≠ 0, as in Theorem 2.4, and B schurian. If δ ∈ Der^{n}(A) then δ(α) = λ_{α}α for any arrow α ∈ (Q_{A})_{1} and some λ_{α} ∈ k.

Proof. Let δ ∈ Der^{n}(A). Since δ is a derivation, for any arrow α ∈ (Q_{A})_{1} we have δ(α) = δ(e_{s(α)}αe_{t(α)}) = e_{s(α)}δ(α)e_{t(α)}. By the previous proposition dim_{k} e_{s(α)}Ae_{t(α)} = 1, so δ(α) = λ_{α}α. □

REMARK 4.7 Given δ ∈ Der^{n}(A) and a path γ = α_{1} … α_{r} with α_{i} arrows in Q_{A},

δ(\overline{γ}) = ∑_{i=1}^{r} \overline{α_{1} … α_{i−1}}δ(α_{i})\overline{α_{i+1} … α_{r}} = (∑_{i=1}^{r} λ_{α_{i}})\overline{γ}.

Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G = (νc ; B̂), c ≠ 0, as in Theorem 2.4, and assume that B is schurian. Let p_{0} be a fixed maximal path in kQ_{B} from i to j. We define a k–linear map L : Der^{n}(A) → k in the following way:

L(δ) = ∑_{m=0}^{c−1} λ_{m}

where δ(\overline{p_{0}β_{p_{0}}}(m)) = λ_{m}\overline{p_{0}β_{p_{0}}}(m).

THEOREM 4.8 The map L induces a surjective group morphism

H^{1}(A) →^{L̂} k → 0.

Proof. First we will show that L(Inn(A)) = 0. Let δ_{x} be an inner derivation with x = ∑ μ_{i}e_{i} + y, y ∈ rad A. Since \overline{p_{0}β_{p_{0}}}(m) are maximal paths, then

δ_{x}(\overline{p_{0}β_{p_{0}}}(m)) = \overline{p_{0}β_{p_{0}}}(m)x − x\overline{p_{0}β_{p_{0}}}(m) = (μ_{ν^{m+1}(i)} − μ_{ν^{m}(i)})\overline{p_{0}β_{p_{0}}}(m).

So λ_{m} = μ_{ν^{m+1}(i)} − μ_{ν^{m}(i)}, and L(δ_{x}) = ∑_{m=0}^{c−1} λ_{m} = μ_{ν^{c}(i)} − μ_{i} = 0 since ν^{c}(i) = i. Hence L induces a morphism L̂ : H^{1}(A) → k. Let δ be the derivation defined in Theorem 4.1. Then δ(\overline{p_{0}β_{p_{0}}}(m)) = 0 for any m with 0 < m < c, because δ((1 − e)A) = 0. Moreover, δ(\overline{p_{0}β_{p_{0}}}(0)) = \overline{p_{0}β_{p_{0}}}(0), because, if c > 1 or c = 1, \overline{p_{0}β_{p_{0}}}(0) ∈ eA(1 − e) or \overline{p_{0}β_{p_{0}}}(0) ∈ eIe, respectively. Then L̂([δ]) = 1. So L̂ is surjective. □

Now we will prove several lemmas that will be used in the proof of the main theorem of this section.

Recall that if p is a maximal path in kQ_{B} that goes through the vertex i in (Q_{B})_{0}, we write p = p_{−}p_{+} for some subpaths p_{−}, p_{+} with t(p_{−}) = i = s(p_{+}).

Let δ ∈ Der^{n}(A), i ∈ (Q_{B})_{0}, and let p be a maximal path going through i. Denote by λ_{p}^{(i,m)} the element in k such that

δ(\overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)) = λ_{p}^{(i,m)} \overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1).

LEMMA 4.9 If p, q are maximal paths going through i, then λ_{p}^{(i,m)} = λ_{q}^{(i,m)} for any m with 0 ≤ m < c.

Proof. By Proposition 3.5 iv) we have

\overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1) = μ\overline{q_{+}β_{q}}(m)\overline{q_{−}}(m + 1)

for some 0 ≠ μ ∈ k. So

δ(\overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)) = μδ(\overline{q_{+}β_{q}}(m)\overline{q_{−}}(m + 1)),

that is,

λ_{p}^{(i,m)} \overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1) = λ_{q}^{(i,m)} μ \overline{q_{+}β_{q}}(m)\overline{q_{−}}(m + 1).

Since \overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1) ≠ 0 (see Remark 2.16) we have λ_{p}^{(i,m)} = λ_{q}^{(i,m)}. □

So we may denote λ^{(i,m)} = λ_{p}^{(i,m)} for any maximal path p going through i.

REMARK 4.10 If L is the map defined in Theorem 4.8, then

L̂([δ]) = ∑_{m=0}^{c−1} λ_{m} = ∑_{m=0}^{c−1} λ_{p_{0}}^{(s(p_{0}),m)} = ∑_{m=0}^{c−1} λ^{(s(p_{0}),m)}.

LEMMA 4.11 For any i, j ∈ (Q_{B})_{0}, ∑_{m=0}^{c−1} λ^{(i,m)} = ∑_{m=0}^{c−1} λ^{(j,m)}.

Proof. Since Q_{B} is connected it is enough to show that the assertion holds whenever there exists an arrow α : i → j. Let p be a maximal path containing the arrow α, that is, p = p_{1}αp_{2}. Now we claim that λ_{α(0)} = λ_{α(c)}. In fact, α(c) ∈ e_{ν^{c}(i)}Ae_{ν^{c}(j)} = e_{i}Ae_{j} and dim_{k} e_{i}Ae_{j} = dim_{k} e_{i}Be_{j} = 1 by Proposition 4.5. Then α(c) = μα(0) for some 0 ≠ μ ∈ k, and δ(α(c)) = μδ(α(0)) = λ_{α(0)}α(c), while δ(α(c)) = λ_{α(c)}α(c). Therefore it follows that

∑_{m=0}^{c−1} λ^{(i,m)} = ∑_{m=0}^{c−1} λ_{p}^{(i,m)} = ∑_{m=0}^{c−1}(λ_{α(m)} + λ_{p_{2}(m)} + λ_{p_{1}(m+1)})

= λ_{α(0)} + ∑_{m=0}^{c−1}(λ_{p_{2}(m)} + λ_{p_{1}(m+1)} + λ_{α(m+1)}) − λ_{α(c)}

= ∑_{m=0}^{c−1}(λ_{p_{2}(m)} + λ_{p_{1}(m+1)} + λ_{α(m+1)})

= ∑_{m=0}^{c−1} λ_{p}^{(j,m)} = ∑_{m=0}^{c−1} λ^{(j,m)}.

□

Let (Q_{B}, I) be a presentation of B and recall the following construction from [18, 1.2]. Let k^{+} be the underlying additive group of the field k, and C^{0}(B, I, k^{+}) the set of all k^{+}– valued functions on (Q_{B})_{0}. Let Z^{1}(B, I, k^{+}) be the set of all k^{+}–valued functions f on (Q_{B})_{1} such that ∑_{i=1}^{s} f(α_{i}) = ∑_{j=1}^{t} f(β_{j}) whenever there exists a minimal relation ρ = ∑_{i=1}^{m} λ_{i}w_{i} such that w_{1} = α_{1}α_{2} … α_{s} and w_{2} = β_{1}β_{2} … β_{t}. We have an exact sequence of abelian groups

0 → k^{+} →^{d_{0}} C^{0}(B, I, k^{+}) →^{d_{1}} Z^{1}(B, I, k^{+}) →^{p} Hom(π_{1}(Q_{B}, I), k^{+}) → 0,

where d_{0}(m) : (Q_{B})_{0} → k^{+} is the constant function with value m; the map d_{1}(g) : (Q_{B})_{1} → k^{+} is defined by d_{1}(g)(α) = g(t(α)) − g(s(α)) and p(f) ∈ Hom(π_{1}(Q_{B}, I), k^{+}) is defined by p(f)([α_{1}^{ϵ_{1}} … α_{t}^{ϵ_{t}}]) = ∑_{i=1}^{t} ϵ_{i}f(α_{i}).

THEOREM 4.12 Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G = (νc ; B̂), c ≠ 0, as in Theorem 2.4, and assume that B is schurian. Then we have the following exact sequence of abelian groups

0 → k^{+} →^{d_{0}} C^{0}(B, I, k^{+}) →^{d_{1}} Z^{1}(B, I, k^{+}) →^{h} H^{1}(A) →^{L̂} k → 0.

Proof. We first define the group morphism h : Z^{1}(B, I, k^{+}) → H^{1}(A). Let Q_{A} be the ordinary quiver of A whose description was given in Proposition 3.4. Let f ∈ Z^{1}(B, I, k^{+}), and extend f to paths by writing f(α_{1} … α_{m}) := ∑_{i=1}^{m} f(α_{i}) for α_{i} ∈ (Q_{B})_{1}. We define a k–linear map δ_{f} : A → A in the following way:

δ_{f}(e_{i}) =  0  for i ∈ (Q_{A})_{0},
δ_{f}(α(m)) = f(α)α(m)  for α(m) ∈ (Q_{A})_{1}, 0 ≤ m < c,
δ_{f}(β_{p}(m)) = −f(p)β_{p}(m) for β_{p}(m) ∈ (Q_{A})_{1}, 0 ≤ m < c,

and, for any path γ = γ_{1} … γ_{s}, γ_{i} ∈ (Q_{A})_{1},

δ_{f}(\overline{γ}) = ∑_{i=1}^{s} \overline{γ_{1} … γ_{i−1}}δ_{f}(γ_{i})\overline{γ_{i+1} … γ_{s}}.

To see that δ_{f} is well defined we have to check the minimal relations described in Proposition 3.5. If \overline{γ}_{1} = μ\overline{γ_{2}} in B with 0 ≠ μ ∈ k, then f(γ_{1}) = f(γ_{2}) by the definition of Z^{1}(B, I, k^{+}). So

δ_{f}(\overline{γ_{1}}(m)) = f(γ_{1})\overline{γ_{1}}(m) = f(γ_{2})\overline{γ_{1}}(m) = f(γ_{2})μ\overline{γ_{2}}(m) = μδ_{f}(\overline{γ_{2}}(m)).

If q_{1} and q_{2} are paths in kQB̂ having the same supplement γ(m + 1), say

q_{1}γ(m + 1) = (α_{s} … α_{t}β_{p_{1}})(m)(α_{1} … α_{s′}γ)(m + 1),
q_{2}γ(m + 1) = (δ_{r} … δ_{l}β_{p_{2}})(m)(δ_{1} … δ_{r′}γ)(m + 1),

where p_{1}, p_{2} ∈ P, γ is a path in kQ_{B}, s′ < s, r′ < r and 0 ≤ m < c − 1, then

δ_{f}(\overline{q_{1}}) = (∑_{i=s}^{t} f(α_{i}) − f(p_{1}) + ∑_{i=1}^{s′} f(α_{i})) \overline{q_{1}},

δ_{f}(\overline{q_{2}}) = (∑_{i=r}^{l} f(δ_{i}) − f(p_{2}) + ∑_{i=1}^{r′} f(δ_{i})) \overline{q_{2}}.

Observe that the last equalities are also true for m+1 = c. In fact, given α : i → j in (Q_{B})_{1}, α(c) ∈ e_{ν^{c}(i)}Ae_{ν^{c}(j)} = e_{i}Ae_{j} and dim_{k} e_{i}Ae_{j} = dim_{k} e_{i}Be_{j} = 1, by Proposition 4.5. Then α(c) = μα(0), 0 ≠ μ ∈ k and δ_{f}(α(c)) = μ δ_{f}(α(0)) = μf(α) α(0) = f(α) α(c). Now,

\overline{α_{1} … α_{s′}γα_{s} … α_{t}} = λ_{1}\overline{p_{1}}, \overline{δ_{1} … δ_{r′}γδ_{r} … δ_{l}} = λ_{2}\overline{p_{2}}

imply that

∑_{i=1}^{s′} f(α_{i}) + ∑_{i=s}^{t} f(α_{i}) − f(p_{1}) = −f(γ) = ∑_{i=1}^{r′} f(δ_{i}) + ∑_{i=r}^{l} f(δ_{i}) − f(p_{2}).

So \overline{q_{1}} = μ\overline{q_{2}} implies that δ_{f}(\overline{q_{1}}) = −f(γ) \overline{q_{1}} = −f(γ)μ \overline{q_{2}} = μ δ_{f}(\overline{q_{2}}). Clearly δ_{f} is a derivation. Now we define h : Z^{1}(B, I, k^{+}) → H^{1}(A) as follows: h(f) = [δ_{f}] ∈ H^{1}(A). To finish the proof we only have to show that the following sequence is exact

C^{0}(B, I, k^{+}) →^{d_{1}} Z^{1}(B, I, k^{+}) →^{h} H^{1}(A) →^{L̂} k,

because the exactness in the other places follows from [18, 1.2] and Theorem 4.8.

i) h.d_{1} = 0 since h(d_{1}(g)) = [δ_{d_{1}(g)}] for g ∈ C^{0}(B, I, k^{+}), and we will show that δ_{d_{1}(g)} = δ_{x} with x = ∑_{i∈(Q_{B})_{0}} ∑_{m=0}^{c−1} g(i)e_{ν^{m}(i)}. Since we are working with derivations it is enough to show that δ_{d_{1}(g)}(α) = δ_{x}(α) for any α ∈ (Q_{A})_{1}. Now, for any m with 0 ≤ m < c,

δ_{d_{1}(g)}(α(m)) = d_{1}(g)(α) α(m)
= (g(t(α)) − g(s(α)) α(m)
= α(m) x − x α(m) = δ_{x}(α(m))

since s(α(m)) = ν^{m}(s(α)) and t(α(m)) = ν^{m}(t(α)). For any m with 0 ≤ m < c,

δ_{d_{1}(g)}(β_{p}(m)) = −d_{1}(g)(p) β_{p}(m)
= −(g(t(p)) − g(s(p))) β_{p}(m)
= β_{p}(m) x − x β_{p}(m) = δ_{x}(β_{p}(m))

since s(β_{p}(m)) = ν^{m}(t(p)), t(β_{p}(m)) = ν^{m+1}(s(p)) and t(β_{p}(c − 1)) = ν^{c}(s(p)) = s(p). Hence δ_{d_{1}(g)} ∈ Inn(A).

ii) Let f ∈ Z^{1}(B, I, k^{+}) with h(f) = 0. Then [δ_{f}] = 0, so δ_{f} = δ_{x} for some x = ∑_{i=1}^{n} λ_{i}e_{i} + y ∈ A with y ∈ ⊕_{i}(e_{i} rad A e_{i}) (see Corollary 4.4). Define g ∈ C^{0}(B, I, k^{+}) by g(e_{i}) = λ_{i} for all i ∈ (Q_{B})_{0}. Then, for any arrow α in Q_{B},

δ_{f}(α(0)) = f(α) α(0) = δ_{x}(α(0)) = (λ_{t(α)} − λ_{s(α)}) α(0) + α(0) y − y α(0).

Since α(0) y − y α(0) ∈ rad^{2} A and α(0) ∉ rad^{2} A, we have f(α) = λ_{t(α)} − λ_{s(α)} = d_{1}(g)(α). So f = d_{1}(g).

iii) L̂.h = 0 since

L̂.h(f) = L̂([δ_{f}]) = ∑_{m=0}^{c−1} λ_{m},

where for any m with 0 ≤ m < c

λ_{m} \overline{p_{0}β_{p_{0}}}(m) = δ_{f}(\overline{p_{0}β_{p_{0}}}(m)) = (f(p_{0}) − f(p_{0})) \overline{p_{0}β_{p_{0}}}(m) = 0.

iv) Let [δ] ∈ H^{1}(A) be such that L̂([δ]) = 0. Then ∑_{m=0}^{c−1} λ_{m} = 0, where δ(\overline{p_{0}β_{p_{0}}}(m)) = λ_{m} \overline{p_{0}β_{p_{0}}}(m). Hence, it follows from Remark 4.10 and Lemma 4.11 that ∑_{m=0}^{c−1} λ^{(i,m)} = 0 for any i ∈ (Q_{B})_{0}. Let

x = ∑_{i∈(Q_{B})_{0}} ∑_{m=1}^{c−1}(∑_{r=0}^{m−1} λ^{(i,r)})e_{ν^{m}(i)},

and let p be any maximal path in kQ_{B} such that p = p_{−}p_{+} with t(p_{−}) = i = s(p_{+}). For any m with 0 ≤ m < c − 1, we have

δ_{x}(\overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)) = \overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1) x − x \overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)
= (∑_{r=0}^{m} λ^{(i,r)} − ∑_{r=0}^{m−1} λ^{(i,r)}) \overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)
= λ^{(i,m)} \overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1).

Observe that if m + 1 = c, then

δ_{x}(\overline{p_{+}β_{p}}(c − 1)\overline{p_{−}}(c)) = \overline{p_{+}β_{p}}(c − 1)\overline{p_{−}}(c) x − x \overline{p_{+}β_{p}}(c − 1)\overline{p_{−}}(c)
= (− ∑_{r=0}^{c−2} λ^{(i,r)}) \overline{p_{+}β_{p}}(c − 1)\overline{p_{−}}(c)
= λ^{(i,c−1)}\overline{p_{+}β_{p}}(c − 1)\overline{p_{−}}(c),

since ν^{c}(i) = i and ∑_{r=0}^{c−1} λ^{(i,r)} = 0. So

δ(\overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)) = δ_{x}(\overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)).

Since [δ] = [δ − δ_{x}], we may assume that δ(\overline{p_{+}β_{p}}(m)\overline{p_{−}}(m + 1)) = 0 for any maximal path p and any m with 0 ≤ m < c. Given any arrow α in Q_{B}, let p = p_{1}αp_{2} be a maximal path containing α. Then

δ(\overline{αp_{2}β_{p}}(m)\overline{p_{1}}(m + 1)) = 0 = δ(\overline{p_{2}β_{p}}(m)\overline{p_{1}α}(m + 1))

and \overline{p_{2}β_{p}}(m)\overline{p_{1}α}(m + 1) ≠ 0 ≠ \overline{αp_{2}β_{p}}(m)\overline{p_{1}}(m + 1) implies that

λ_{α(m)} + λ_{p_{2}(m)} + λ_{β_{p}(m)} + λ_{p_{1}(m+1)} = 0 = λ_{p_{2}(m)} + λ_{β_{p}(m)} + λ_{p_{1}(m+1)} + λ_{α(m+1)},

where δ(\overline{p_{2}}(m)) = λ_{p_{2}(m)}\overline{p_{2}}(m) and δ(\overline{p_{1}}(m + 1)) = λ_{p_{1}(m+1)}\overline{p_{1}}(m + 1) (see Remark 4.7). So λ_{α(m)} = λ_{α(m+1)}, and therefore λ_{α(m)} = λ_{α(0)} for any m with 0 ≤ m < c. Since this holds for any arrow α in Q_{B} we get λ_{β_{p}(m)} = −λ_{p(0)} for any p ∈ P and m with 0 ≤ m < c. Let f : (Q_{B})_{1} → k^{+} be the function defined by f(α) = λ_{α(0)}. Hence δ = δ_{f} and we claim that f ∈ Z^{1}(B, I, k^{+}). In fact, given a minimal relation ρ = ∑_{i=1}^{m} λ_{i}w_{i}, w_{1} and w_{2} are two paths of positive length in kQ_{B} sharing starting and terminal points. Then \overline{w_{1}} = μ\overline{w_{2}} for some 0 ≠ μ ∈ k because B is schurian. Suppose w_{1} = α_{1} … α_{s} and w_{2} = β_{1} … β_{r}. Then δ(\overline{w_{1}}) = μδ(\overline{w_{2}}), that is,

∑_{i=1}^{r} λ_{α_{i}(0)}\overline{w_{1}} = μ ∑_{i=1}^{s} λ_{β_{i}(0)}\overline{w_{2}}.

But \overline{w_{1}} ≠ 0 implies that ∑_{i=1}^{r} λ_{α_{i}(0)} = ∑_{i=1}^{s} λ_{β_{i}(0)}. So, by the definition of f, ∑_{i=1}^{r} f(α_{i}) = ∑_{i=1}^{s} f(β_{i}). □

COROLLARY 4.13 Let A be a repetitive covering selfinjective algebra, F : B̂ → A a Galois covering with the Galois group G = (νc ; B̂), c ≠ 0, and assume that B is schurian. Then

0 → Hom(π_{1}(Q_{B}, I), k^{+}) → H^{1}(A) → k^{+} → 0

is an exact sequence of abelian groups.

COROLLARY 4.14 If B is simply connected, then H^{1}(A) = k.

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Instituto de Matemática Universidad Nacional del Sur Av. Alem 1253 (8000) Bahía Blanca, Argentina e-mail: mredondo@criba.edu.ar