# Regular morphisms and formal power series

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?

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 local domains $$B$$ of dimension two and 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, and I believe Sharp's construction gives a ring $$B$$ that also has non-Cohen–Macaulay formal fibers.
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.