 
Summary: NONCOMMUTATIVE GR ˜
OBNER BASES
FOR THE COMMUTATOR IDEAL
SUSAN HERMILLER 1 AND JON MCCAMMOND 2
Abstract. 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 Gr˜obner bases for I, and uncountably many
infinite Gr˜obner bases for I with respect to total division orderings.
In addition, for m = 3 we give a complete description of its universal
Gr˜obner 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 poly
nomial ring over K with commuting variables in A by K[A]. The kernel
of the natural map # : K#A# # K[A] is called the commutator ideal. The
commutator ideal in K#A#, and in particular its noncommutative Gr˜obner
bases, have been used to investigate properties of finite 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 Gr˜obner bases for the commutator ideal
