In the theory of Cohen-Macaulay rings, a basic theorem is that if $(R,\mathfrak{m})$ is a Cohen-Macaulay local noetherian ring, and $x_1,\dots,x_r\in\mathfrak{m}$ is a sequence satisfying $\dim R/(x_1,\dots,x_r)R = \dim R - r$, then $x_1,\dots,x_r$ is a regular sequence.
This theorem fails without the local assumption. For example, let $R = k[x,y,z]$ for $k$ a field (clearly a Cohen-Macaulay ring), and consider the sequence $xy,xz,1-x$. (This is a standard example of a sequence that is not regular in this order but it is regular in a different order.) Then the ideal $(xy,xz,1-x) = (y,z,1-x)$ is maximal and we have $R/(y,z,1-x)$ is dimension zero, but the sequence isn't regular since $xz$ is not regular in $R/(xy)$.
But there is often an analogy between the local case and the graded case, so I'm wondering if there is a graded analogy. In particular, I want to know:
If $R$ is a Cohen-Macaulay, nonnegatively graded ring, and $x_1,\dots,x_r$ are homogeneous elements of positive degree such that $\dim R/(x_1,\dots,x_r)R = \dim R - r$, is $x_1,\dots,x_r$ a regular sequence?
Note that I am not assuming that the degree zero component of $R$ is a field. (In the case of interest to me, $R_0=\mathbb{Z}$.) But if that assumption would change the status from false to true, then I'd like to know that.