Ibrahim Assem
email: ibrahim.assem@usherbrooke.ca

AND Juan Carlos Bustamante
email: juan.carlos.bustamante@usherbrooke.ca

Patrick Le Meur
email: patrick.lemeur@math.univ-bpclermont.fr

Special biserial algebras with no outer derivations

Keywords: . 2000 Mathematics Subject Classification. Primary 16E40, 16G60 Special biserial algebras, Hochschild cohomology, simple connectedness, fundamental group

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Let k be an algebraically closed field and A a finite dimensional k-algebra. It is a reasonable question to try to relate the Hochschild cohomology groups of A with the representation theory of A, that is, with properties of the category mod A of finitely generated right A-modules.

We are here interested in the vanishing of the first Hochschild cohomology group HH 1 (A) of A with coefficients in the bimodule A A A . In [START_REF] Skowroński | Simply connected algebras and Hochschild cohomologies[END_REF], Skowroński asked for which triangular algebras A do we have HH 1 (A) = 0 if and only if A has no proper Galois covering.

Since then, this problem has been investigated by several authors due to its connection with the notion of simple connectedness. In [START_REF] Bongartz | Covering spaces in representation-theory[END_REF], Bongartz and Gabriel define a representation finite algebra to be simply connected if its Auslander-Reiten quiver is simply connected as a two-dimensional simplicial complex. For a not necessarily representation finite algebra, it is easily seen that A has no proper Galois covering if and only if for every bound quiver presentation A ≃ kQ/I, the fundamental group of (Q, I) is trivial, see [START_REF] Martínez-Villa | The universal cover of a quiver with relations[END_REF]. In [START_REF] Assem | On some classes of simply connected algebras[END_REF], the first author and Skowroński call an algebra simply connected if it is triangular and has no proper Galois covering. This definition restricts to that of Bongartz and Gabriel in the representation finite case. In this terminology, Skowroński's question can be reformulated to ask which algebras A satisfy HH 1 (A) = 0 if and only if A is simply connected. This statement was shown to hold true for several classes of algebras, in particular for representation finite algebras [START_REF] Buchweitz | Hochschild cohomology and representation-finite algebras[END_REF]. Note also that another definition of simple connectedness, which does not assume triangularity is used in [START_REF] Assem | Coverings of laura algebras: the standard case[END_REF][START_REF] Meur | Topological invariants of piecewise hereditary algebras[END_REF]. Our objective in this paper is to study this problem in case A is special biserial (not necessarily triangular). Throughout, we use simply connected only for representation finite algebras, that is in the sense of Bongartz and Gabriel.

The class of special biserial algebras was introduced by Skowroński and Waschbüsch in [START_REF] Skowroński | Representation-finite biserial algebras[END_REF]. It has played an important rôle in the study of self-injective algebras. Special biserial algebras have a well-understood representation theory. In fact, if A is special biserial, then it has a two-sided ideal S such that the quotient A/S is a monomial algebra, and actually a string algebra in the sense of Butler and Ringel [START_REF] Butler | Auslander-Reiten sequences with few middle terms and applications to string algebras[END_REF]. In this paper, we prove the following theorem.

Theorem. Let A ≃ kQ/I be a special biserial algebra. The following conditions are equivalent: (a) The fundamental group of the bound quiver (Q, I) is trivial, (b) A is representation finite and simply connected, (c) HH 1 (A) = 0, (d) HH j (A) = 0, for every j 1, (e) χ(Q) = dim k S.

Thus, in particular, if A satisfies the equivalent conditions of the theorem then it is necessarily triangular. Moreover, it is constrained in the sense of Bardzell and Marcos [START_REF] Bardzell | H 1 and presentations of finite dimensional algebras[END_REF], therefore, the fundamental group of any bound quiver presentation of A is trivial, or, equivalently, A has no proper Galois covering.

The paper is organised as follows. In section 1, we briefly recall the necessary definitions. Sections 2 and 3 are technical: in section 2 we study the cycles involved in binomial relations for special biserial bound quivers (Q, I), and in section 3 we relate the number of projective -injective indecomposable non uniserial kQ/I modules to the Euler characteristic of Q and we prove the key Proposition 3.3. Section 4 studies the relations between simple connectedness, triangularity and derivations. In it we prove some lemmata used in the proof of the main Theorem, in section 5.

Algebras and quivers

Throughout this paper, k denotes an algebraically closed field and all algebras are finite dimensional associative k-algebras with identity.

Given a finite quiver Q = (Q 0 , Q 1 , s, t), we denote by kQ its path algebra. Two paths sharing source and target are parallel. A bypass is a pair (α, p), where α is an arrow, and p = α is a path parallel to α. Given two points x, y ∈ Q 0 , a relation from x to y is a linear combination r i=1 λ i w i , where λ i ∈ k\{0}, and the w i are distinct paths from x to y. If r = 1, the relation is monomial, whereas if r = 2 the relation is binomial. Let kQ + be the two-sided ideal of kQ generated by Q 1 . An ideal I of kQ is admissible if there exists an integer m 2 such that (kQ + ) m ⊆ I ⊆ (kQ + ) 2 . In this case, the pair (Q, I) is called a bound quiver. The algebra kQ/I is basic, connected if so is Q, and finite-dimensional since Q is finite. Given a path u in Q, its class modulo I is denoted by u.

Conversely, for every finite dimensional, connected and basic algebra A over an algebraically closed field k, there exists a unique connected quiver Q and at least one surjective map ν : kQ → A with I = Ker ν admissible, see [START_REF] Bongartz | Covering spaces in representation-theory[END_REF]. The pair (Q, I) is called a presentation of A by a bound quiver.

For more details concerning the use of bound quivers in the representation theory of algebras we refer to [START_REF] Assem | Elements of the representation theory of associative algebras[END_REF], for instance.

A bound quiver (Q, I) is special biserial [START_REF] Skowroński | Representation-finite biserial algebras[END_REF] if it satisfies the following conditions: (i) Every point in Q is the source of at most two arrows and the target of at most two arrows;

(ii) Given an arrow α : x → y in Q, there is at most one arrow β starting from y such that αβ ∈ I and there is at most one arrow γ arriving at x such that γα ∈ I. If, moreover, the ideal I is monomial, that is, I is generated by monomial relations, then (Q, I) is a string bound quiver [START_REF] Butler | Auslander-Reiten sequences with few middle terms and applications to string algebras[END_REF].

A special biserial algebra (or a string algebra) is an algebra admitting a presentation by a special biserial bound quiver (or a string bound quiver, respectively). If (Q, I) is special biserial, then I is generated by a collection of paths and a collection R of binomial relations (that is relations of the form uλv, with u, v parallel paths not in I, λ ∈ k\{0}) in bijection with the isomorphism classes of non-uniserial indecomposable projective-injective modules. For every such module, we arbitrarily fix a path u in Q such that there is a binomial relation uλv (the relation is fixed once the path u is fixed). It can be seen that a non trivial path cannot be prefix nor suffix of more than one path appearing in a binomial relation in R, and that two binomial relations associated to non-isomorphic projective-injective modules have distinct starting points and distinct ending points. In the sequel, A = kQ/I denotes a special biserial algebra, unless otherwise specified.

We recall some basic facts on special biserial algebras (see [START_REF] Wald | Tame biserial algebras[END_REF][START_REF] Butler | Auslander-Reiten sequences with few middle terms and applications to string algebras[END_REF]).

Let A = kQ/I be special biserial. Let S be the socle of the direct sum of the indecomposable non uniserial projective -injective A modules. Hence, as subspace of A (considered as a vector space) S is generated by the classes u (= λv) of paths appearing in a binomial relation uλv ∈ R. Since (radA) S = S(rad A) = 0, S is a two-sided ideal of A and, in fact, a semisimple subbimodule of A A A . In particular, given points x, y in Q, we have e x Se y = 0 if and only if there is a binomial relation uλv from x to y. Moreover, in this case, we have e x Se y = ku = kv. Note that dim k S equals the cardinality |R| of R.

Given a special biserial algebra A, the quotient algebra A/S is a string algebra. If A = kQ/I with (Q, I) special biserial, then A/S ≃ kQ/J where J is the two-sided monomial ideal of kQ generated by I ∪ {u, v | uλv a binomial relation}.

Cycles in binomial relations

A cycle in Q is a path a from a point x to itself, and it is a simple cycle if x occurs only at the beginning and at the end of a. Given an arrow α : x → y, its formal inverse is the arrow α -1 : y → x. A walk in Q is a composition of arrows and formal inverses of arrows of

Q, w = α ε1 1 α ε2 2 • • • α εr r , ε i ∈ {±1}, such that s(α εi i ) = t(a εi-1 i-1
). The walk w is reduced if contains no subwalk of the form αα -1 or α -1 α, with α ∈ Q 1 and it is closed if the source of the first arrow coincides with the target of the last one. A closed walk is simple if each point appears at most once in it, except of course its starting and ending point. In the sequel by closed walk we mean closed reduced walk.

The conditions in the definition of a special biserial bound quiver impose that binomial relations involving oriented cycles are very particular, and can be described precisely, as follows: (3) a nλb m , α r β 1 , β s α 1 , for some natural numbers n, m, and a scalar λ ∈ k\{0}, (4) (ab) mλ(ba) m , α r α 1 , β s β 1 for some natural number m, and a scalar λ ∈ k\{0}, (5) (ab) m aλ(ba) m b, α r α 1 , β s β 1 for some natural number m, and a scalar λ ∈ k\{0}.

Proof. Let u = ap, with a :

x α1 / / x 1 α2 / / • • • αr / /
x a simple cycle at x, and p be a path such that u = ap. (a) Assume first that x = y, and that u and v share an arrow. Since (Q, I) is special biserial, no path can be a common prefix (nor suffix) of two paths appearing in a binomial relation. Thus, v does not start with α 1 . If p starts with α 1 , then since ap ∈ I, the point y must lie in the support of the cycle a, so that p = a n a 1 for some n and some suffix a 1 of a. But then, since v does not start with α 1 , it cannot have a common arrow with u. Thus we must have that p starts with the first arrow of v (and α r α 1 ∈ I), so that v = pb with b a cycle at y. If the cycle b were stationary, we would have a relation of the form apλp, but this would contradict the fact that a is nilpotent. Now, if u and v do not share an arrow, then p must start with α 1 , and, as before, p = a n a 1 for some n and some suffix a 1 of a.

x p / / / o / o / o α1 n n y β1 0 0 αr E E βs Z Z x a1 / / / o / o / o / o / o / o / o @A BC v / o / o / o / o / o / o O O y ED GF a2 o / o / o / o / o / o / case (1) : ap -λpb, α r α 1 , β s β 1 ∈ I case (2) : a n a 1 -λv ∈ I (b) Assume that a = α 1 • • • α r and b = β 1 • • • β s are simple cycles at
x, such that u begins with a, and v begins with b. Note that the cycles a and b have no arrow in common. We distinguish two cases, according to whether α r α 1 belongs to I or not.

x α1 p p β1 . . αr A A βn ] ] cases (3) -(4) - (5

Given a cycle a

= α 1 • • • α r , define σ(a) to be the cycle α 2 α 3 • • • α r α 1 .
A cyclic permutation of a is a cycle of the form σ j (a), and ã will denote the set of all cyclic permutations of a. Note that two simple cycles not in I have a common arrow if and only if they are equal up to cyclic permutation. The following remarks will be useful.

Remarks.

(a) Note that in case 2.1, (1) above, there can be a path q from y to a point z (which might be x) at which there is a cycle c (which must be a in case z = x), and a minimal relation of the form bqλqc. However, since any cyclic permutation of If follows from the preceding lemma that the set R can be partitioned as R = R 1 R 2 , where R 1 is the set of binomial relations uλv such that one of the paths starts or ends with a cycle, and R 2 = R\R 1 .

The following section is technical. We establish the key result 3.3.

3.

The cycle graph of (Q, I)

Following [START_REF] Happel | Hochschild cohomology of finite-dimensional algebras[END_REF], given a quiver Q, with N connected components, the Euler characteristic of

Q is χ(Q) = |Q 1 | -|Q 0 | + N .
This number equals the rank of the first homology group H 1 (Q) of the underlying graph of Q, which is free abelian. In order to compare dim k S and χ(Q) we introduce an auxiliary graph Γ, defined as follows:

-Γ 0 is the set of of simple oriented cycles in Q, considered up to cyclic permutation, which are prefix or suffix of a path appearing in a binomial relation in R 1 . -Given two points ã and b in Γ, we attach edges between them according to the following rules (see 2. (a) Note that we may have Γ 0 = ∅, but still have oriented cycles in Q. However, this would mean that R = R 2 .

(b) By construction |Γ 1 | = |R 1 |, this will be very useful later. Also, note that Γ can have multiple edges, but no loops: in case there are two (classes of) cycles a and b in Q having n common points with relations of the type (3) there are exactly n arrows between them. For instance, the quiver Γ corresponding to the bound quiver given in 2.3 is:

ã b (c)
The set of simple oriented cycles of Q (up to cyclic permutation) which are prefixes or suffixes of a path appearing in a binomial relation gives rise to a linearly independent set in H 1 (Q). Thus, we have a map ϕ 0 : Γ 0 → H 1 (Q) whose image is linearly independent. Note however that this does not mean that we have a natural map R 1 → H 1 (Q) with the same property. Indeed we may have several binomial relations involving different cycles of a class ã (see example 2.

Lemma. There exists an injective homomorphism of groups ϕ

1 : H 1 (Γ) → H 1 (Q) whose image C 1 satisfies C 1 ∩ C 0 = 0.
Proof. First of all, note that each edge p of Γ determines a point of Q, that is the starting point of the binomial relation in Q giving rise to the edge p. We now proceed by induction on χ(Γ).

Assume χ(Γ) = 1 so that there is a simple closed walk

w = c 1 p1 c 2 p2 • • • pr c 1 in Γ.
-Assume first that this cycle lies in a component of type [START_REF] Assem | The fundamental groups of a triangular algebra[END_REF]. That means that there are relations of type (1) in Q: a 1 p 1p 1 a 2 , a 2 p 2p 2 a 3 , . . . , a r p rp r a 1 . Set ϕ 1 (w) = r j=1 p j , as element of H 1 (Q).

-In light of 3.1, (d) we can assume that w lies in a component given by relations of type [START_REF] Assem | On some classes of simply connected algebras[END_REF] or else by relations of type (3). Let x i be the point determined by the edge ci c i+1 of w, see 2.2, (c) and (d). We build a simple cycle at x 1 as follows. Start at x 1 , then:

. If the edge in Γ comes from a relation of type (3), go to x 2 following c 2 , . If the edge in Γ comes from a relation (2), go from x to the ending point of the binomial relation following c 1 , then go to to x 2 following c 2 . and continue in that way. The obtained cycle is not necessarily simple, but by eliminating each proper subpath which is a nontrivial cycle, we obtain a simple cycle. The constructed cycle does not lie in C 0 because it does not contain all the arrows of any cycle of Q it passes through.

Assume now that the statement holds true for bound quivers such that χ(Γ) = k -1, and let (Q, I) be such that χ(Γ) = k 2. Since χ(Γ) > 0, there exists an edge p in Γ such that the graph Γ ′ obtained from G by deleting p has the same number of connected components of Γ, and hence χ(Γ ′ ) = χ(Γ)-1. The graph Γ ′ corresponds to the bound quiver (Q, I ′ ) where I ′ is generated by one less binomial relation than I. Note that the vertices of Γ coincide with those of Γ ′ , so that ϕ 0 = ϕ ′ 0 . In addition, H 1 (Γ ′ ) can be regarded as a subgroup of H 1 (Γ), and by the induction hypothesis there exists an injective map ϕ ′ 1 :

H 1 (Γ ′ ) → H 1 (Q) whose image C ′ 1 does not intersect C 0 . We now extend ϕ ′ 1 to H 1 (Γ).
There exists a cycle in Γ involving the edge p, thus, not belonging to H 1 (Γ ′ ). We construct a cycle in Q in the same way as we did, and it only remains to show that no integer multiple of this cycle belongs to C 0 + C ′ 1 . Using the edge p in Γ corresponds to changing from one cycle to another in Q, say from a to b, at a point x which is uniquely determined by p. Let α 1 and α 2 , (respectively β 1 , and β 2 ) be the arrows of a (respectively of b) entering and leaving x.

x α2 p p β2 . . α1 A A β1 ] ]
These arrows may have appeared in a previously constructed cycle of Q. However, since we have never before used the point x to change from a to b, each time α 1 appeared in the cycles of C 0 + C ′ 1 , the arrow α 2 also appeared. Thus, in every element of C 0 + C ′ 1 , the coefficient of α 1 is the same as that of α 2 , and this is not the case for the cycle we have just constructed.

3.3. Proposition. Let (Q, I) be a special biserial bound quiver. Then dim k S χ(Q). Moreover, if equality holds, then Q is acyclic.

Proof. Let uλv ∈ R 2 be a relation from x to y. Note that neither u nor v starts with a cycle, and they have at least the point y in common. Let z be the first point which is common to u and v and different from x, thus u = u ′ u ′′ and v = v ′ v ′′ , where u ′ and v ′ go from x to z. Moreover, let u 1 be the path constructed from u ′ by deleting every occurrence of a cycle, if there is one, and v 1 constructed in an analogous way. Since uλv ∈ R 2 we have that

u 1 v -1 1 is a reduced non-oriented cycle of Q. Define ϕ 2 : R 2 → H 1 (Q) by ϕ 2 (u -λv) = u 1 -v 1 .
The paths u and u 1 have their first arrow in common, and since no path in Q can be a prefix (nor suffix) of more than one path appearing in some binomial relation, the image of this map is a linearly independent set. Let C 2 be subgroup of H 1 (Q) generated by Im ϕ 2 .

In 3.2 we constructed an injection ϕ 1 :

H 1 (Γ) → H 1 (Q), whose image C 1 satisfies C 1 ∩ C 0 = 0.
By construction, any cycle C 0 + C 1 has only arrows belonging to cycles which are prefixes or sufixes of paths appearing in binomial relations. Since the first arrow of a path of a relation from R 2 cannot appear in any such cycle, we have

C 2 ∩ (C 0 + C 1 ) = 0. Thus C 0 + C 1 + C 2 = C 0 ⊕ C 1 ⊕ C 2 is a subgroup of H 1 (Q), and thus dim k S = |R| = |R 1 | + |R 2 | = |Γ 1 | + rk C 2 N + |Γ 1 | + rk C 2 = χ(Γ) + |Γ 0 | + rk C 2 = rk C 0 + rk C 1 + rk C 2 rk H 1 (Q) = χ(Q).
Assume now that χ(Q) = dim k S. Then all the inequalities must be equalities, in particular N = 0. Therefore Γ is empty, and so is R 1 . Moreover, we must have that C 2 = H 1 (Q), which is only formed by non oriented cycles.

3.4. Corollary. Let (Q, I) be a special biserial bound quiver such that dim k S = χ(Q). If u, v are distinct parallel paths, then u and v are proportional. In particular, Q has no bypasses.

Proof. The hypothesis dim k S = χ(Q) implies that Q has no oriented cycles and R = R 2 .

Let u, v be parallel paths. Consider the shortest non-trivial prefixes u ′ v ′ of u and v, respectively, such that u ′ and v ′ are parallel. Then u ′ v ′-1 is a simple closed walk in Q. Therefore, u ′ and v ′ are bound by a binomial relation. Moreover, if we write u = u ′ u ′′ and v = v ′ v ′′ , then u ′′ is trivial if and only if v ′′ is trivial, because Q has no oriented cycles. Thus, u and v are bound by a binomial relation or both lie in I according to whether u ′′ and v ′′ are both trivial or both non trivial, respectively.

The second statement follows directly, since I is admissible.

We refer the reader to [START_REF] Martínez-Villa | The universal cover of a quiver with relations[END_REF] for the definition of the fundamental group of a bound quiver. See also [START_REF] Skowroński | Simply connected algebras and Hochschild cohomologies[END_REF][START_REF] Assem | The fundamental groups of a triangular algebra[END_REF][START_REF] De La Peña | On the first Hochschild cohomology group of an algebra[END_REF] for relations with Hochschild cohomology. Recall from [START_REF] Bardzell | H 1 and presentations of finite dimensional algebras[END_REF] that an algebra A ≃ kQ/I is called constrained if for each arrow α : x → y in Q we have dim k e x Ae y 1. It is shown in [START_REF] Bardzell | H 1 and presentations of finite dimensional algebras[END_REF] that if A is constrained, then the fundamental groups of any two presentations of A are isomorphic.

3.5. Corollary. Let (Q, I) be a special biserial bound quiver with

χ(Q) = dim k S. If kQ/I ≃ kQ/I ′ , then π 1 (Q, I) ≃ π 1 (Q, I ′ ),
Proof. This follows from [START_REF] Bardzell | H 1 and presentations of finite dimensional algebras[END_REF], and the absence of bypasses in Q.

Derivations, triangularity and fundamental groups

4.1. Lemma. Let (Q, I) be a special biserial bound quiver. If Hom(π 1 (Q, I), k + ) = 0, then dim k S = χ(Q). In particular, Q is acyclic.
Proof. Recall from the description of π 1 (Q, I) given in [START_REF] Martínez-Villa | The universal cover of a quiver with relations[END_REF] for instance, that this group is generated by χ(Q) elements satisfying dim k S relations. Therefore, Hom(π 1 (Q, I), k + ) is isomorphic, as a kvector space, to a subspace of k χ(Q) given by dim k S relations. Therefore, Hom(π 1 (Q, I), k + ) = 0 implies that dim k S χ(Q). The conclusion then follows from 3.3.

Proposition.

Let A be a special biserial algebra such that HH 1 (A) = 0 or else such that there exists a special biserial presentation A ≃ kQ/I such that π

1 (Q, I) = 1, then dim k S = χ(Q). In particular, Q is acyclic. Proof. If HH 1 (A) = 0, then it follows from [1, 3.2] that Hom(π 1 (Q, I), k + ) = 0. Similarly, if π 1 (Q, I) = 1, then Hom(π 1 (Q, I), k + ) = 0. We conclude using 4.1.
Recall that string algebras are tame, and have two kinds of indecomposable modules, the socalled band modules and the string modules. Moreover, with our notations every A-module is either projective-injective or an A/S-module. See [START_REF] Wald | Tame biserial algebras[END_REF][START_REF] Butler | Auslander-Reiten sequences with few middle terms and applications to string algebras[END_REF], for more details. Proof. Let A ≃ kQ/I be a special biserial presentation and let J be the ideal of kQ generated by I and the paths in Q appearing in a binomial relation in I. Hence, A/S ≃ kQ/J. It follows from 3.4 that every simple cycle in Q contains a path or a formal inverse of a path in Q that lies in J. Thus, there is no band in (Q, J) and, therefore, no band module over A/S. This shows that A/S and, therefore, A are both representation finite.

We now need to give a precise description of the fundamental group π 1 (Q, I), in case (Q, I) is a triangular special biserial bound quiver. In order to do so, we recall some terminology and results from [START_REF] Assem | The fundamental groups of a triangular algebra[END_REF][START_REF] Castonguay | On the inductive construction of Galois coverings of algebras[END_REF].

Following [START_REF] Castonguay | On the inductive construction of Galois coverings of algebras[END_REF], given a presentation ν : kQ → A, the algebra A is said to be of the first kind with respect to ν if for every point x and associated indecomposable projective A module P x , every indecomposable summand of radP x is of the first kind with respect to the universal Galois covering associated to ν. The main result of [START_REF] Castonguay | On the inductive construction of Galois coverings of algebras[END_REF] states that if A is a triangular algebra of the first kind with respect to a presentation ν : kQ → A, then the fundamental group π 1 (Q, Ker ν) is free. Now, following [START_REF] Assem | The fundamental groups of a triangular algebra[END_REF], let (Q, I) be a bound quiver with Q acyclic, ν : kQ → A a presentation of A, with kernel I, and x be a source in Q. Let x + be the set of arrows starting at x, and let ≈ be the smallest equivalence relation on x + such that α ≈ β whenever there exist y ∈ Q 0 and a minimal relation r i=1 λ i w i ∈ I, from x to y, such that w 1 = αw ′ 1 and w 2 = βw ′ 2 . Further, denote by t x (ν) = t(ν) the number of equivalence classes of ≈. Let Q ′ be the quiver obtained from Q by deleting x, I ′ = I ∩ kQ ′ , and A ′ = kQ ′ /I ′ , so that A is a one-point extension of A ′ . Then [14, 2.2] asserts that π 1 (Q, I) is the free product of the fundamental groups of the connected components of(Q ′ , I ′ ) and the free group in t(ν) -1 generators. 4.4. Lemma. Let (Q, I) be a triangular special biserial bound quiver. Then

π 1 (Q, I) is free of rank χ(Q) -dim k S.
Proof. First, note that if p : ( Q, Ĩ) → (Q, I) is a Galois covering, then ( Q, Ĩ) is also special biserial. In addition, if w is a string (or a band) in (Q, I), then there exists a string w in ( Q, Ĩ) such that p( w) = w. Furthermore, if M (w) denotes the string module corresponding to w, then M (w) = p λ M ( w), where p λ denotes the push-down functor associated to p (see [START_REF] Bretscher | The standard form of a representation-finite algebra[END_REF]). Because indecomposable projectives are string modules, this implies that special biserial algebras are of the first kind. In light of the main Theorem of [START_REF] Castonguay | On the inductive construction of Galois coverings of algebras[END_REF] cited above, this shows that π 1 (Q, I) is free. Now, if x is a source in Q, then t(ν) is 1 or 2, according to whether the projective P x is also injective or not. The result then follows by induction on the number of points in Q, using [14, 2.2].

The following example shows that the statement does not hold true if one drops the triangularity hypothesis. Then A is a non-triangular special biserial algebra and π 1 (Q, I) ≃ Z × Z.

The main result

We recall that a representation-finite algebra A is called simply connected if its Auslander -Reiten quiver is simply connected as a two-dimensional simplicial complex, see [START_REF] Bongartz | Covering spaces in representation-theory[END_REF]. This is equivalent to saying that A is triangular and has no proper Galois covering, see [START_REF] Assem | On some classes of simply connected algebras[END_REF]. As promised, we now establish a relationship between the simple connectedness of a special biserial algebra, the vanishing of its Hochschild cohomology groups and the dimension of S. Note that, by definition, simple connectedness implies representation -finiteness. Notice that because of 3.5, and the fact that an algebra A has no proper Galois covering if and only if the fundamental group of any presentation is trivial, it follows from 5.1 (a) that the conditions of the theorem are further equivalent to saying that A is representation-finite and has no proper Galois covering.

We now derive some consequences, the first of which deals with the Lie algebra structure of HH 1 (A).

Theorem.

Let A = kQ/I be a special biserial algebra without bypass. Then there is a short exact sequence of k-vector spaces

0 / / HH 1 (A) p * / / HH 1 (A/S) / / k dim k S / / 0
where the map p * is a morphism of Lie algebras.

Proof. Recall that A/S ≃ kQ/J, and that J is a monomial ideal, so that π 1 (Q, J) is the free group on χ(Q) generators. Thus, there is a natural surjective group homomorphism p : π 1 (Q, J) → π 1 (Q, I) obtained by factoring out the binomial relations. Since Q has no bypasses, we have, from [START_REF] De La Peña | On the first Hochschild cohomology group of an algebra[END_REF], that HH 1 (A) ≃ Hom(π 1 (Q, I), k + ) and HH 1 (A/S) ≃ Hom(π 1 (Q, J), k + ). Moreover, the derivations of A and those of A/S are diagonalisable, see [START_REF] Assem | The fundamental groups of a triangular algebra[END_REF][START_REF] De La Peña | On the first Hochschild cohomology group of an algebra[END_REF], that is, for every arrow α, and every derivation ∂ of A or A/S, the image of α under ∂ is a scalar multiple of itself. The morphism p * is obtained from p upon applying the functor Hom(-, k + ). Finally, from 4.4, we obtain dim k HH 1 (A/S) = χ(Q), and dim k HH 1 (A) = χ(Q)dim k S and the dimension of the cokernel of p * follows. Finally, since the derivations are diagonalisable, the Lie algebras HH 1 (A/S) and HH 1 (A) are abelian, and the map p * is thus trivially a Lie algebra homomorphism.

We refer to [START_REF] Skowroński | Simply connected algebras and Hochschild cohomologies[END_REF], for instance, for the definition of separated, coseparated, or strongly simply connected algebras. We now easily deduce conditions equivalent to those of our theorem 5.1 for special biserial representation -finite algebras. (2) Let A be a simply connected triangular special biserial algebra. Then it follows from 5.4, (a), and [START_REF] Bongartz | Quadratic forms and finite representation type[END_REF] that the Auslander-Reiten quiver of A admits both a postprojective and a preinjective component.

(3) Let A be a simply connected special biserial algebra. By [START_REF] Assem | Strongly simply connected algebras[END_REF], there exists a poset Σ and an a ideal J of the incidence algebra kΣ which is generated by classes of paths in the quiver of Σ, such that A ≃ kΣ/J. In particular, A is schurian. (4) In [START_REF] Assem | Quotients of incidence algebras and the Euler characteristic[END_REF] are given criteria for the strong simple connectedness of quotients of incidence algebras.

Recall that for schurian algebras, or, more generally for algebras A having a semi-normed basis, the simplicial homology SH * (A) and cohomology groups SH * (A; k + ) of A (with coefficients in k + ) are defined, see [10, 2.1] and [START_REF] Martins | Comparing the simplicial and the Hochschild cohomologies of a finitedimensional algebra[END_REF]. Moreover, following [START_REF] Bustamante | The classifying space of a bound quiver[END_REF], these groups have a clear interpretation as the homology or cohomology groups of a CW-complex. Proof. Since A is schurian, its quiver has no bypasses. From the previous results, if HH 1 (A) = 0, then π 1 (Q, I) is trivial, and hence so is its abelianisation, SH 1 (A). Then we have, for every presentation (Q, I) of A:

HH 1 (A) ≃ Hom(π 1 (Q, I), k + ) ≃ Hom(SH 1 (A), k + ) ≃ SH 1 (A; k + )
where the first isomorphism comes from [START_REF] De La Peña | On the first Hochschild cohomology group of an algebra[END_REF], the second is the Hurewicz Theorem (see [20, 4.29], and the third is given by the Dual Universal Coefficients Theorem (see [20, 12.11]).

2. 1 . 1 )

 11 Lemma. Let uλv be a binomial relation from x to y. If u starts with a simple cycle a = α 1 • • • α r at x then: (a) If x = y, then (If u and v share at least an arrow, then there exists a path p from x to y and a cycle b at y such that u = ap and v = pb, (2) If u and v have no arrow in common, then there exists a decomposition a = a 1 a 2 with a 1 , a 2 non trivial paths such that u = a n a 1 , for some natural number n 1. (b) If x = y, then there exists a simple cycle b = β 1 • • • β s at x such that the relations involving the simple cycles a and b, and the arrows incident to x are of one of the following forms:

)

  Assume first that α r α 1 ∈ I. Then we must have α r β 1 ∈ I, so that u = a m for some natural number m. Using the same argument, we see that v = b n , and thus the relation is of the form a mλb n . Of course, we must also haveβ s β 1 ∈ I, whereas β s α 1 ∈ I.Assume now that a r α 1 ∈ I. Then α r β 1 ∈ I, β s β 1 ∈ I, and β s α 1 ∈ I. Thus, there are natural numbers n, m such that either u = (ab) n and v = (ba) m , or u = (ab) n b and v = (ba) n b. Assume the first case. If m > n then there exists ǫ 0 such that m -1 = n + ǫ we would have (ba) m = b(ab) m-1 a = b(ab) n+ǫ a = b(ab) n (ab) ǫ a = λb(ba) m (ab) ǫ a ∈ I, a contradiction. Thus m n, and using the same argument we obtain that in fact n = m, and the relation is of the form (ab) mλ(ba) m . The second case is treated in the same way.

  a and b belongs to I, no other cycle of ã nor b can be involved in any other binomial relation. (b) If there is a relation of types 2.1 (4) or (5) involving two cycles a and b, then any cyclic permutation of these cycles belongs to the ideal I. Therefore, there cannot exist another binomial relation involving cycles of ã and b. (c) In case 2.1, (2), the path v may contain a subpath that is a non trivial cycle, say b. There can be relations of type (2) involving other cycles of ã or b. Moreover, if v starts with a cycle, then it also ends with a cycle b at y and there exists a decomposition b = b 1 b 2 such that v = b 1 b m for some natural number m. A cyclic permutation of b may itself be involved in another minimal relation, necessarily of type (2) (see example 2.3, (b)). (d) In case 2.1, (3), cyclic permutations of a or b can be involved in several other minimal relations, either between them, or involving another cycle (see example 2.3, (a)). All such relations must be of type (3). 2.3. Example. Consider the following quiver: Define a = α 1 α 2 α 3 α 4 and b = β 1 β 2 β 3 β 4 . (a) Let I be the ideal generated by σ i (a)σ i (b), α i β i+1 , β i α i+1 , where 1 i 4, and indices are to be read modulo 4. In this case the supports of the cycles a and b are involved in 4 minimal relations of type (3). (b) Let I 1 be the ideal generated by σ i (a)α i+1σ i (b)β i+1 , α i β i+1 , β i α i+1 , where 1 i 4 where, and again indices are to be read modulo 4. In this case, the cycles a and b are involved in 4 relations of type (2).

  1): • For each relation apλpb of type (1), we attach an edge p : ã -b. • For each relation a n a 1λb 1 b m of type (2) we attach an edge p ′ : ã -b. • In case the cycles a and b share a point x and are involved in a relation of one of the forms (3), (4), or (5) we attach an edge x : ã -b. 3.1. Remarks.

  3).Denote by C 0 the the subgroup of H 1 (Q) generated by Im ϕ 0 . (d) In general Γ is disconnected. In light of 2.2 it has 5 types of edges, each given by the type of relation that gives rise to it. Moreover, the type of an edge is an invariant for all the edges in the same connected component of Γ. The latter can be described as follows.-If two cycles a = α 1 • • • α r and b = β 1 • • • β s arelinked by a relation of type 2.1, (4) or (5), then since α r α 1 , β s β 1 ∈ I, no cyclic permutation of a or b can be involved in another binomial relation. Thus, points corresponding to relations of type 2.1 (4) and (5) determine a connected component of the form ã b , -Each point corresponding to binomial relations of type 2.1 (1) gives rise to a connected component of type A or Ã, -Edges corresponding to relations of types 2.1 (2), appear in a separate component with no loops. The same holds for edges coming from relations of type 2.1 (3). See 2.2.

4. 3 .

 3 Lemma. If χ(Q) = dim k S, then the string algebra A/S has no band module. Therefore, A and A/S are representation finite.

  copies of the point 1 are identified, bound by the relations βλ = ξγ = δν = µε = ηζ = ρα = 0, αβγδ = ζξλµ, γδεη = λµνρ, εηαβ = νρζξ .

5. 1 .

 1 Theorem. Let A = kQ/I be a special biserial algebra. Then, the following conditions are equivalent:(a) π 1 (Q, I) = 1, (b) A simply connected, (c) HH 1 (A) = 0, (d) HH j (A) = 0, for every j 1, (e) χ(Q) = dim k S.Proof.(a) implies (b) : If π 1 (Q, I) = 1 then 4.2 implies that Q is acyclic, and that χ(Q) = dim k S. Then 3.5 implies that the fundamental group of every presentation of A is trivial. Finally, 4.3 gives the remaining part.(b) implies (c) : If π 1 (Q, I) = 1 then 4.2 implies that Q is acyclic, and, since A is representation finite, Theorem (4.3) in[START_REF] Buchweitz | Hochschild cohomology and representation-finite algebras[END_REF] gives HH 1 (A) = 0.(c) and (d) are equivalent : If HH 1 (A) = 0, then by 4.3 the algebra A is representation finite. Corollary 4.4 in[START_REF] Buchweitz | Hochschild cohomology and representation-finite algebras[END_REF] then yields that HH i (A) = 0 for i 2. It is trivial that (d) implies (c).(d) implies (e) : Since HH 1 (A) = 0, then 4.2 implies that Q is acyclic and χ(Q) = dim k S. The result follows from 4.4.(d) implies (e) : This follows from 3.3 and 4.4.

5. 3 . 1 )

 31 Corollary. Let A be a special biserial algebra. The following conditions are equivalent: (a) A simply connected (thus, by definition, representation-finite), (b) There exists a string bound quiver presentation (Q,I ′ ) of A/S such that π 1 (Q, I ′ ) is the free group of rank dim k S, (c) Q has no bypass and HH 1 (A/S) ≃ k dim k S .Proof.(a) implies (c) : From 5.1, if A is simply connected, then HH 1 (A) = 0, and then from 3.3 and 3.4, Q has no bypasses. Finally, the exact sequence of 5.2 gives the result.(c) implies (b) : The exact sequence of 5.2 gives HH 1 (A) = 0, and then 5.1 gives that dim k S = χ(Q). On the other hand, since I ′ is monomial, the group π 1 (Q, I ′ ) is free in χ(Q) generators.(b) implies (a) : The hypothesis implies that dim k S = χ(Q), and the result follows from 5.1.5.4. Remarks.(We have further equivalent conditions, namely if A is special biserial and representation -finite, then the following are equivalent:(a) A is simply connected, (b) A is separated, (c) A is co-separated,(d) A is strongly simply connected. Indeed, because of [21, 2.3, 4.1] condition (d) implies (b) and (c), which imply (a). Finally, (a) implies (d) follows from [10].

5. 5 .

 5 Corollary. Let A be a schurian special biserial algebra. The following are equivalent:(a) HH 1 (A) = 0, (b) SH 1 (A) = 0, (c) SH 1 (A; k + ) = 0.

Acknowledgements

The first author gratefully acknowledges partial support from NSERC of Canada and the Université de Sherbrooke. The second author acknowledges partial support from the Universidad San Francisco de Quito and financial support from Université de Sherbrooke, as well as the Sherbrooke group for their hospitality. The third author acknowledges financial support from Université de Sherbrooke and from the CMLA, ENS de Cachan. He also wishes to thank the first two authors for their warm hospitality during visits at Université de Sherbrooke. The authors thank Jean-Philippe Morin for useful discussions.

4.5.

Example. Let A be the algebra given by the quiver


François-Régis Chaumartin

Etienne Coumont

Fabio Mancinelli
email: fabio.mancinelli@xwiki.com

Olivier Grisel
email: olivier.grisel@nuxeo.com

Bridging Mono/.NET and Java in the SCRIBO Project: The Way to UIMA.NET

In this paper we introduce the project SCRIBO (Semi-automatic and Collaborative Retrieval of Information Based on Ontologies) and we describe how we have leveraged the UIMA framework in order to integrate existing tools in a general architecture. The paper focuses on how we have bridged the Java and .Net platforms (using the Mono framework), describing the problems and an effective solution to make UIMA interoperability possible.

Introduction

The SCRIBO project's goal is to develop algorithms and collaborative tools for extracting knowledge from unstructured documents and images. Its distinguishing features rely on the combination of semantic and statistical approaches for natural language processing (NLP) and on its focus on the collaborative dimension of the knowledge extraction process. SCRIBO provides an integrated tool chain for extracting light ontologies from a corpus of documents, for acquiring knowledge by leveraging ontologies, and for extracting structures from digitalized documents. The tool chain will operate in an integrated modular environment built upon standard APIs that facilitates advanced collaboration by providing graphical widgets for annotating semantically evolving texts and images.

One of the main challenges of the SCRIBO project is that of integrating the existing components provided by project's partners and make them interoperate. We chose to leverage the UIMA framework in order to interoperate and to provide components that exposes a standard interface that could be re-used also in context other than SCRIBO itself. -Tool-chains. These are the existing tools that are integrated using the UIMA frameworks. These tools are already existing and developed using different platforms (i.e., Mono (.Net), C++ and a Unix-like integration through Perl and shell scripts), providing the following functionalities:

Architecture

• Semi-Automatic extraction and refinement of ontologies.

• Named entities and relationships extraction.

• Digital document analysis and partitioning and information extraction. These tool-chains are wrapped using the API provided by UIMA framework.

-Document bases and Products. These are existing products developed by SCRIBO partners that provides document management facilities and a document base that is used as input to the analysis tool-chains. These document bases are exposed using UIMA collection readers and documents are feed directly to the tool-chains using the UIMA workflow. -Controller. The controller is used to realize a model-view-controller architecture. The controller mediate the interaction between the integrated environments (i.e., the view), the tool-chains and the knowledge base that will store all the information extracted by the analysis (i.e., the model). -Integrated environments. These components provide the user interface to the SCRIBO framework and allow the user interact with it by changing the parameters, visualize the analysis results and modify them collaboratively in order to detect mismatches and correct errors.

3 Bridging Mono/.NET and Java: the way to UIMA.NET

In this section we describe how we succeeded in integrating MONO and .NETbased tools in the UIMA framework, in the context of the SCRIBO project.

Our company, Proxem, has already under development a linguistic platform, Antelope. It includes several Natural Language Processing components (taggers, parsers, semantic, lexicons, Named Entities recognizers. . . ) written mainly with the C# language, and running under the Microsoft .NET Framework or Mono, its open source cross-platform counterpart. Our problem was to integrate them into the UIMA framework, for which only two default implementations (C++ and Java) exist. Rewriting all our components was not something possible; we decided to find a solution to create a UIMA annotator component, running on a MONO or .NET runtime instead of a JVM, and being able to be called from any UIMA client process. First, we tried to create a Web service based on the SOAP standard protocol. Despite much effort, we didn't succeed in this way; there are incompatibility issues between the Web services exposed in UIMA, and their implementation in .NET. Moreover, UIMA Web services expose some classes using Axis libraries, which cannot be easily translated into the .NET equivalent.

We then explored a lower-level solution, using sockets to communicate between both virtual machines; we decided to use Vinci, which presents two advantages:

-It is provided as a standard protocol of the UIMA framework.

-It uses only standard Java libraries, so is it easily transposable into .NET.

Vinci services

Vinci services can be used directly by the Collection Processing Manager (CPM). Vinci handles service naming and location and data transport. Service naming and location are provided by a Vinci Naming Service (VNS). When a Vinci service is started, it has to register itself to the VNS. Then, when a CPM needs to use the service, it has to know the exact name of the wanted service. A query is made to the VNS in order to discover the machine and port hosting the service, which can then be directly invoked.

Figure 1

 1 Figure1shows the current architecture of the SCRIBO framework:

Fig. 2 .

 2 Fig. 2. Architecture of UIMA.NET showing the communication between components.

Conception of a UIMA annotator for the MONO/.NET Framework

We first had to convert the UIMA libraries (.jar files) into a MONO/.NET assembly. We did it by using the IKVM static compiler, a tool that can translate Java bytecode into MONO/.NET Intermediate Language (the equivalent of Java bytecode). We then created a service launcher, named StartVinciService, a simple MONO/.NET executable which is in charge to call the main function of the Vinci service implementation class, from the library previously translated:

VinciAnalysisEngineService impl.main(args)

We also created a new MONO/.NET class library, named AnnotatorLibrary, that contains our annotators. Each annotator is a class that inherits from the UIMA.NET class CasAnnotator ImplBase, and overrides the process(CAS) method to perform the annotation job, by calling the CAS methods the same way it is done in Java.

This class must be referenced in the annotator XML descriptor file, by specifying its namespace, name, and assembly: <annotatorImplementationName> MyNS.MyDotNetAnnotator, AnnotatorLibrary </annotatorImplementationName> If the assembly is signed, its full name must be specified: <annotatorImplementationName> MyNS.MyDotNetAnnotator, AnnotatorLibrary, Version=1.0, Culture=neutral, PublicKeyToken=0123abcdef </annotatorImplementationName> Now we have to create a Vinci deployment descriptor file that references the previous annotator descriptor file as the "resourceSpecifierPath". We are now ready to launch our MONO/.NET Vinci service. We start the Vinci Naming Service if not already done. Then we launch our StartVinciService executable, with one argument, the path to the annotator descriptor file. We can now use this service from a UIMA client, such as the CAS Visual Debugger, which is one of the tools provided with the UIMA Framework.

Conclusion

Our first intuition that a SOAP-based standard component should be easy to implement was wrong; in fact, the low-level solution was much simpler to deploy. We demonstrated a fast and easy way to build cross-platforms UIMA applications, and we are proud to provide what we expect to be the first MONO/.NET UIMA implementation.