The question concerns a definition of Bruns, Herzog: Cohen-Macaulay Rings (before Prop. 3.6.16):
The Noetherian *local ring $(R,m)$ is said to be *complete if $(R_0,m_0)$ is complete.
(a graded ring $R$ is *local if it has a unique maximal homogeneous ideal $m$).
Given an inverse system of $\mathbb{Z}$-graded rings $(R^i)_{i \in I}$, it's routine to show that the inverse limit exists in the category of graded rings and the components are $$(\varprojlim_i R^i)_n = \varprojlim_i (R^i)_n.$$
Question: Is a Noetherian *complete *local $\mathbb{Z}$-graded commutative ring a complete object in the category of graded rings ?
In the $\mathbb{N}$-graded case, I can answer the question affirmatively, but I don't know if it's also true with $\mathbb{Z}$-gradings.
