# Inverse limit of Noetherian rings

Let $R_i$ be a noetherian regular local ring of Krull dimension being finite. Suppose we are given a surjective homomorphism $\phi_{i,j} \colon R_{j} \twoheadrightarrow R_i$ for each $i,j$ with $j > i$. We consider the ring $R_{\infty}$ defined by

$R_{\infty} \colon= \underset{i \geq 1}\varprojlim \,R_i$.

$R_{\infty}$ might not have finite Krull dimension, nor be noetherian. Choose finitely many non-zero elements $r_1,...,r_{n} \in R_{\infty}$ and consider the ideal ${\frak a} \colon= (r_1,...,r_{n})$ of $R_{\infty}$.

## Q1. Does a prime ideal ${\frak P}$ of $R_{\infty}$ always exist such that ${\mathrm{ht}}({\frak P}) < \infty$ and ${\frak P} \supset {\frak a}$ ?

Next let us localize $R_{\infty}$ at a prime ${\frak Q}$ of $R_{\infty}$ such that ${\mathrm{ht}(\frak Q)} < \infty$ and we denote it by $R_{\infty, {\frak Q}}$.

## Q2. Is $R_{\infty, {\frak Q}}$ a noetherian regular local ring of Krull-dimension ${\mathrm{ht}({\frak Q})}$?

• I'm puzzled how you choose your tags. cv-complex-variables has nothing to do here. neither does galois-representations. You don't specify whether your rings are commutative, is this the case? then ac.commutative-algebra is enough and ra.rings-and-algebras does not fit. Finally, your question being asked in a purely algebraic way, ag.algebraic-geometry is also unclear. – YCor Jun 27 '16 at 7:10
• Just to add to YCor's comment: if there are hidden motivations coming from complex variables or Galois representations, the question would probably be enhanced by mentioning them. The issue of commutativity should be addressed; that sort of thing is usually important. – Todd Trimble May 13 '18 at 15:29