Geodesic
Special biserial algebras with no outer derivations
Ibrahim Assem 1   ; Juan Carlos Bustamante 2   ; Patrick Le Meur 3
1 Département de Mathématiques Université de Sherbrooke Sherbrooke, Québec, Canada J1K 2R1
2 Departamento de Matemáticas Universidad San Fransisco de Quito Cumbayá - Quito, Ecuador and Département de Mathématiques Université de Sherbrooke Sherbrooke, Québec, Canada J1K 2R1
3 Laboratoire de Mathématiques Université Blaise Pascal & CNRS Campus Scientifique des Cézeaux BP 80026 63171 Aubière Cedex, France
Colloquium Mathematicum, Tome 125 (2011) no. 1, pp. 83-98 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Let $A$ be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of $A$ with coefficients in the bimodule $A$ vanishes if and only if $A$ is representation-finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of $Q$ equals the number of indecomposable non-uniserial projective-injective $A$-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups of $A$ vanish.
DOI : 10.4064/cm125-1-6
Keywords: special biserial algebra algebraically closed field first hohchshild cohomology group coefficients bimodule vanishes only representation finite simply connected sense bongartz gabriel only euler characteristic equals number indecomposable non uniserial projective injective a modules isomorphism moreover higher hochschild cohomology groups vanish

Ibrahim Assem  1   ; Juan Carlos Bustamante  2   ; Patrick Le Meur  3

1 Département de Mathématiques Université de Sherbrooke Sherbrooke, Québec, Canada J1K 2R1
2 Departamento de Matemáticas Universidad San Fransisco de Quito Cumbayá - Quito, Ecuador and Département de Mathématiques Université de Sherbrooke Sherbrooke, Québec, Canada J1K 2R1
3 Laboratoire de Mathématiques Université Blaise Pascal & CNRS Campus Scientifique des Cézeaux BP 80026 63171 Aubière Cedex, France 
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Ibrahim Assem; Juan Carlos Bustamante; Patrick Le Meur. Special biserial algebras with no outer derivations. Colloquium Mathematicum, Tome 125 (2011) no. 1, pp. 83-98. doi: 10.4064/cm125-1-6

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