Commutative Algebra

Let $(A,\mathfrak{m})$ and $(B, \mathfrak{n})$ be local rings, and $\phi\colon A\to B$ a ring homomorphism.

For convenience in this problem only, call a data tuple $(\psi,\mathfrak{p}, \mathfrak{q})$ good if $\psi\colon R\to S$ is a ring homomorphism and $\mathfrak{p}$ and $\mathfrak{q}$ are prime ideals of $R$ and $S$, resp., such that $\psi^{-1} (\mathfrak{q}) = \mathfrak{p}$.

Now consider the following two conditions on $\phi$:

• $\phi$ is a local homomorphism.
• There exists a good tuple $(\psi, \mathfrak{p}, \mathfrak{q})$ and a commutative diagram down from $\psi$ to $\phi$ such that the vertical maps $R\to A$ and $S\to B$ induce isomorphisms $R_{ \mathfrak{p}} \to A$ and $S_{ \mathfrak{q}} \to B$, meaning, precisely, that $R\to A$ factors through the localization map $R\to R_{ \mathfrak{p}}$ with $R_{ \mathfrak{p}} \to A$ an isomorphism, and similarly for $S\to B$.