Complete rewriting systems and homology of monoid algebras.

*(English)*Zbl 0711.20035Let \(M\) be a monoid presented by a complete rewriting system \(R\) over an alphabet \(\Sigma\). In other words, \(M\) is the image of the free monoid \(\Sigma^*\) determined by the congruence on \(\Sigma^*\) that is generated by a relation \(@>R>>\) which is Noetherian (that is, there are no infinite chains \(x_ 1\to^{R}x_ 2\to^{R}...)\) and confluent (that is, for any words \(x,y,w\in \Sigma^*\) with \(w@>R^*>>x\), \(w@>R^*>>y\) there exists \(z\in\Sigma^*\) such that \(x@>R^*>>z\), \(y@>R^*>>z\), where \(@>R^*>>\) denotes the transitive reflexive closure of \(@>R>>\)). Let \(A\) be the monoid algebra of \(M\) over a commutative ring \(K\). A method of constructing a free resolution of the trivial \(A\)-module is given. (More generally, free resolutions of the right \(A\)-modules associated with certain right rewriting systems over \(\Sigma\) that are complete over \(R\) are constructed.) This is applied to give an upper bound for the global dimension of \(A\) in terms of the lengths of chains in the graph associated to \(R\). The paper is based on the ideas of C. C. Squier [J. Pure Appl. Algebra 49, 201-217 (1987; Zbl 0648.20045)] and D. J. Anick [Trans. Am. Math. Soc. 296, 641-659 (1986; Zbl 0598.16028)].

Reviewer: J.Okniński

##### MSC:

20M50 | Connections of semigroups with homological algebra and category theory |

20M25 | Semigroup rings, multiplicative semigroups of rings |

16E10 | Homological dimension in associative algebras |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

##### Keywords:

complete rewriting system; free monoid; congruence; monoid algebra; free resolutions; global dimension
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\textit{Y. Kobayashi}, J. Pure Appl. Algebra 65, No. 3, 263--275 (1990; Zbl 0711.20035)

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##### References:

[1] | Anick, D.J., On the homology of associative algebras, Trans. amer. math. soc., 296, 641-659, (1986) · Zbl 0598.16028 |

[2] | Cartan, H.; Eilenberg, S., Homological algebra, (1956), Princeton Univ. Press Princeton, NJ · Zbl 0075.24305 |

[3] | Kapur, D.; Narendran, P., The Knuth-bendix completion procedure and thue systems, SIAM J. comput., 14, 1052-1072, (1985) · Zbl 0576.68010 |

[4] | Kapur, D.; Narendran, P., A finite thue system with decidable word problem and without equivalent finite canonical system, Theoret. comput. sci., 35, 337-344, (1985) · Zbl 0588.03023 |

[5] | Squier, C.C., Word problems and a homological finiteness condition for monoids, J. pure appl. algebra, 49, 201-217, (1987) · Zbl 0648.20045 |

[6] | Squier, C.; Otto, F., The word problem for finitely presented monoids and finite canonical rewriting systems, Rewriting techniques and applications, Lecture notes in computer science, 256, 74-82, (1987) |

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