### Abstract

Let Lambda be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k, satisfying J(3)(Lambda)=- 0. Let S-1 ,..., S-r be representatives of the isomorphism classes of simple A-modules, and let E be the r x r matrix whose (i, j) entry is dim(k) Ext(Lambda)(1)(S-i, S-j). If there exists an eigenvalue lambda of E satisfying vertical bar lambda vertical bar > 2 then the minimal resolution of each non-projective finitely generated A-module has exponential growth, with radius of convergence 1/2 (lambda - root lambda(2) - 4). On the other hand, if all eigenvalues lambda of E satisfy vertical bar lambda vertical bar <= 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated A-module are either bounded or grow linearly. In this case, we classify the possibilities for the matrix E. The proof is an application of the Perron-Frobenius theorem. (C) 2008 Elsevier Inc. All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 48-56 |

Number of pages | 9 |

Journal | Journal of Algebra |

Volume | 320 |

Issue number | 1 |

Early online date | 24 Mar 2008 |

DOIs | |

Publication status | Published - 1 Jul 2008 |

### Keywords

- representations of algebras
- projective resolutions

### Cite this

**Resolutions over symmetric algebras with radical cube zero.** / Benson, David J.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 320, no. 1, pp. 48-56. https://doi.org/10.1016/j.jalgebra.2008.02.033

}

TY - JOUR

T1 - Resolutions over symmetric algebras with radical cube zero

AU - Benson, David J.

PY - 2008/7/1

Y1 - 2008/7/1

N2 - Let Lambda be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k, satisfying J(3)(Lambda)=- 0. Let S-1 ,..., S-r be representatives of the isomorphism classes of simple A-modules, and let E be the r x r matrix whose (i, j) entry is dim(k) Ext(Lambda)(1)(S-i, S-j). If there exists an eigenvalue lambda of E satisfying vertical bar lambda vertical bar > 2 then the minimal resolution of each non-projective finitely generated A-module has exponential growth, with radius of convergence 1/2 (lambda - root lambda(2) - 4). On the other hand, if all eigenvalues lambda of E satisfy vertical bar lambda vertical bar <= 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated A-module are either bounded or grow linearly. In this case, we classify the possibilities for the matrix E. The proof is an application of the Perron-Frobenius theorem. (C) 2008 Elsevier Inc. All rights reserved.

AB - Let Lambda be a finite dimensional indecomposable weakly symmetric algebra over an algebraically closed field k, satisfying J(3)(Lambda)=- 0. Let S-1 ,..., S-r be representatives of the isomorphism classes of simple A-modules, and let E be the r x r matrix whose (i, j) entry is dim(k) Ext(Lambda)(1)(S-i, S-j). If there exists an eigenvalue lambda of E satisfying vertical bar lambda vertical bar > 2 then the minimal resolution of each non-projective finitely generated A-module has exponential growth, with radius of convergence 1/2 (lambda - root lambda(2) - 4). On the other hand, if all eigenvalues lambda of E satisfy vertical bar lambda vertical bar <= 2 then the dimensions of the modules in the minimal projective resolution of each finitely generated A-module are either bounded or grow linearly. In this case, we classify the possibilities for the matrix E. The proof is an application of the Perron-Frobenius theorem. (C) 2008 Elsevier Inc. All rights reserved.

KW - representations of algebras

KW - projective resolutions

U2 - 10.1016/j.jalgebra.2008.02.033

DO - 10.1016/j.jalgebra.2008.02.033

M3 - Article

VL - 320

SP - 48

EP - 56

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -