Kobayashi, Yuji Complete rewriting systems and homology of monoid algebras. (English) Zbl 0711.20035 J. Pure Appl. Algebra 65, No. 3, 263-275 (1990). Let \(M\) 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 Cited in 31 Documents 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 Citations:Zbl 0648.20045; Zbl 0598.16028 PDF BibTeX XML Cite \textit{Y. Kobayashi}, J. Pure Appl. Algebra 65, No. 3, 263--275 (1990; Zbl 0711.20035) Full Text: DOI OpenURL 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) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.