An infinite dimensional local ring with finite residue field It is well-known that there is examples of infinite (Krull) dimensional ring. For example, Kang and Park, in [Example, pages 111 and 112, A localization of a power series ring over a valuation domain , JPAA 140 (1999) 107-124], constructs an infinite-dimensional discrete valuation ring.
So, is there any example of a local ring $(R,M)$ such that $\dim(R)=\infty$ and $R/M$ is finite?
 A: Let $F$ be any finite field, and let $R$ be the localization of the polynomial ring $F[x_n:n\in\mathbb{N}]$ at the maximal ideal $\langle x_n:n\in\mathbb{N}\rangle$. $R$ is local, with unique maximal ideal $M=\langle x_n/1:n\in\mathbb{N}\rangle$, and $R\big/M\cong F$ is finite. But $$0<\langle x_1/1\rangle<\langle x_1/1,x_2/1\rangle<\dots$$ is an infinite strictly ascending chain of prime ideals of $R$, so that $\dim R=\infty$.

As an aside, I am not sure about your claim that the paper you cite constructs an "infinite-dimensional discrete valuation ring". By definition, a discrete valuation ring has Krull dimension $1$, at least for the usual definitions of "discrete valuation ring" and "Krull dimension"; see for example characterization 5 here. Are the authors of the paper you cite using different definitions?
A: According to Example 175 of [H.C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, (1981)], if we take $K_0$ a finite field then $R:=K_0+P$ is a local infinite-dimensional ring with maximal ideal $P$ and hence $R/P$ is finite.
