Noncommutative Gröbner Bases for the Commutator Ideal

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I , and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis. Let A be a finite set and let K be a field. We denote the free associative algebra over K with noncommuting variables in A by K〈A〉 and the polynomial ring over K with commuting variables in A by K[A]. The kernel of the natural map γ : K〈A〉 → K[A] is called the commutator ideal. The commutator ideal in K〈A〉, and in particular its noncommutative Gröbner bases, have been used to investigate properties of finitely generated ideals in the commutative polynomial ring K[A]. This has occurred, for example, in the study of free resolutions [1] and the homology of coordinate rings of Grassmannians and toric varieties [12]. In this article we establish several new results about noncommutative Gröbner bases for the commutator ideal in K〈A〉. Our main results are as follows: Theorem A. There are exactly m! finite reduced complete rewriting systems for the m-generated free commutative monoid and exactly m! finite reduced Gröbner bases for the commutator ideal of the m-variable free associative algebra. In addition, each such rewriting system and Gröbner basis is induced by a shortlex ordering. Theorem B. There are uncountably many reduced complete rewriting systems for the m-generated free commutative monoid which are compatible with a total division ordering when m ≥ 3. As a consequence, there are uncountably many reduced noncommutative Gröbner bases with respect to total division orderings for the commutator ideal in the m-variable free associative algebra, each with a distinct set of normal forms. Date: June 15, 2007. Supported under NSF grant no. DMS-0071037 Supported under NSF grant no. DMS-0101506