Hilbert series of PI relatively free G-graded algebras are rational functions

Details Hilbert series of PI relatively free G-graded algebras are rational functions Hilbert series of PI relatively free G-graded algebras are rational functions

2010-11-14

arxiv.org/abs/1011.3253v1

Mathematics

Rings and Algebras

14 pages

Scientific paper

Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F/I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F/I is a rational function.

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