Dimension from Hilbert series with variable-weighted grading?

Question

Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{e_n}$ to be the weighted sum $\sum_i e_i d(i)$ makes $R$ into a graded ring (equivalently, makes $I$ a homogeneous ideal). Under this grading, consider the Hilbert series of $R$.

In general, can we easily read off the Krull dimension of $R$ from the Hilbert series?

In particular, if $d(i)=1$ for all $i$, we know that if you write the Hilbert series as a rational function in reduced terms, then the Krull dimension is the number of factors of the form $(1-t)$ in the denominator. It seems like, in the above more general situation, we should be able to say the same, but where we count the number of factors of the form $(1-t^a)$ for any $a$. This works for $I=0$ rather trivially even for arbitrary function $d$, but:

First, is such a result true in general? And second, is there a good reference for this?

Related sources I know of that don't seem to answer this question:

• Graded Grothendieck Group and Hilbert Polynomial
• Exercises in Eisenbud on pp. 245-246, 280
• Kreuzer & Robbiano Sections 5.6 and 5.8 (unless I missed something...)

Answer 1

What you have to do is look at the pole at $t=1$, and its order gives you the Krull dimension. This is discussed in detail in Proposition 5.3.2 in my book, "Representations and Cohomology II" (combined with Theorem 5.4.6).

The problem with the way you describe it is that it isn't quite well defined, because in reduced form the bottom may not necessarily be expressible as a product of terms of the form $(1-t^a)$.