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Complete rewriting systems and homology of monoid algebras. (English) Zbl 0711.20035
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

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
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[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
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