Exercise 16, § 3

Let $(A_\lambda, f_{\mu\lambda})$ be a filtered inductive system of local rings, such that the $f_{\mu\lambda}$ are local homomorphisms; let $m_\lambda$ be the maximal ideal of $A_\lambda$, and let $K_\lambda = A_\lambda / m_\lambda$. Then $A = \lim \leftarrow A_\lambda$ is a local ring whose $m = \lim \leftarrow m_\lambda$ is the maximal ideal and $K = \lim \rightarrow K_\lambda$ the residue field. Moreover, if one has $m_\mu = A_\mu m_\lambda$ for $\lambda < \mu$, one has $m = Am_\lambda$ for every $\lambda$.