I have a couple of questions about existence of certain inverse limits in the category of schemes (I am also happy about links to relevant literature... in the stacksproject I only found the affine case). Let $R$ be a noetherian ring.
Given an inverse system of (graded) morphisms of graded $R$-algebras $(S_{j+1} \rightarrow S_{j})_{j \in \mathbb{N}}$, $S_j = \bigoplus_{i \in \mathbb{N}} S_j^{(i)}$ generated in degree $1$ and with $S_j^{(0)}=R$ for all $j$,
- Does the inverse limit $\lim\mathrm{Proj}(S_j)$ exist (as a scheme or some other object)?
- Do we have $\lim S_j=\mathrm{Proj}(\mathrm{colim}(S_j))$?
- If not in this generality, is at least true that $\lim \mathbb{P}^j_R=\mathrm{Proj}(R[x_0,x_1,\ldots])$?
- If $I \subseteq R[x_0,x_1,\ldots]$ is a homogeneous ideal, and $I_1 \subseteq I_2 \subseteq \cdots$ a chain of homogeneous ideals with $I_j \subset R[x_0,\ldots,x_j]$ and such that $\bigcup I_i = I$, and such that $I_{j+1} \subseteq I_j+(x_{j+1})$, write $V_{+}(I_j) \subseteq \mathbb{P}^j_R$ for the associated closed subset. Does the inverse limit $\lim V_{+}(I_j)$ exist? Is it $\mathrm{Proj}(R[x_0, \ldots]/I)$?