| Popis: |
The aim of this chapter is to provide a summary of the theory of linear rewriting and the application of this theory to the construction of free resolutions for associative algebras. In Sect. 2, we present linear polygraphs as an algebraic setting for linear rewriting without a monomial order, and we review the fundamental notion of linear polygraphs. In Sect. 3, we recall several historical constructions on linear rewriting systems for associative algebras, and we show how the confluence properties are studied in these different approaches. We relate the notion of convergent linear polygraph with the notion of noncommutative Grobner basis. In Sect. 4, we describe an algorithmic way to compute free resolutions for algebras using a method introduced by Anick. Section 5 deals with extension of linear polygraphs, seen as higher dimensional linear rewriting systems, into polygraphic resolutions for algebras. We show how to construct such a resolution starting from a convergent presentation. In the last section, we show how to relate Koszulness for algebras with the property of confluence. |
