Regular morphisms and formal power series

Question

Let $A$ be a local noetherian ring. When (besides when $A$ is excellent) do we have that $\operatorname{Spec}(A[[t]])\rightarrow \operatorname{Spec}(A[t])$ is regular?

Answer 1

As you probably are aware, a sufficient condition is that $A$ is a $G$-ring [Matsumura, Thm. 79]. On the other hand, the following result says that one must put some strong assumptions on $A$ to have $A[t] \to A[[t]]$ be regular.

Theorem [Sharp, Thm. 2.9]. There exist a local domain $B$ of dimension either two or three such that setting $L = \operatorname{Frac}(B)$, the generic fiber $B[[t_1,\ldots,t_n]] \otimes_B L$ of the homomorphism $B \to B[[t_1,\ldots,t_n]]$ is not Cohen–Macaulay for every integer $n > 0$.

The example is based on Ferrand and Raynaud's construction of a local domain $A$ such that the zero ideal in the completion $\hat{A}$ has an embedded prime [Ferrand–Raynaud, Prop. 3.3]. This ring $A$ therefore has non-Cohen–Macaulay formal fibers. Sharp's construction also gives a ring $B$ that also has non-Cohen–Macaulay formal fibers by [Ooishi, Cor. 36].

We can now consider the composition $$B \longrightarrow B[t] \longrightarrow B[[t]],$$ where $B$ is as in Sharp's theorem. The first map is a regular homomorphism. If the second map were also regular, then the composition would be regular. But Sharp's theorem says that this cannot be the case.