Flat, unramified base change preserves regularity I am reading Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves," and on page 337, he claims that if $C$ is a regular $R$-scheme, where $R$ is a DVR, then if $R^{sh}$ denotes the strict henselization of $R$, then $C \times_R R^{sh}$ is regular, because $C$ is regular and $R^{sh}/R$ is flat and unramified. I understand all the conditions being claimed are true, but I do not see why flat, unramified base change preserves regularity. Can someone either provide a reference or a detailed proof?
 A: Just so this is off the unanswered list.
Note that $\mathrm{Spec}(R^\mathrm{sh})\to\mathrm{Spec}(R)$ is faithfully flat (see Tag 07QM) so that $C_{R^\mathrm{sh}}\to C$ is faithfully flat. The claim then follows from Tag 07NG which shows that if $Y\to X$ is faithfully flat, $X$ is regular, then so is $Y$.
EDIT EDIT: See the comments below—the EDIT is correct, I believe.
EDIT: After a more careful rereading of the question, I’d like to add that while the answer I supplied is ‘better’ in my opinion, it’s likely that Silverman meant something else. Namely, Silverman seems to be emphasizing the ‘ind-etale’ nature of $R\to R^\mathrm{sh}$. So, with this in mind here is an alternative proof.
Note that, by definition, we may write $R^\mathrm{sh}=\varinjlim R_i$ where each $R\to R_i$ is an etale map. So then, note that if $\mathrm{Spec}(A)\subseteq C$ is any affine open then
$$\mathrm{Spec}(A)_{R^\mathrm{sh}}=\mathrm{Spec}(\varinjlim (A\otimes_R R_i))$$,
and such affine schemes form an affine open cover of $C_{R^\mathrm{sh}}$. Thus, it suffices to show that each $\varinjlim(A\otimes_R R_i)$ is regular. But, the local rings of this colimit are of the form $\varinjlim (A\otimes_R R_i)_{\mathfrak{p}_i}$ for $\mathfrak{p}_i$ compatible choices of primes of $A\otimes_R R_i$. But, such a ring is Noetherian (since $A\otimes_R R^\mathrm{sh}$ is finite type over $R^\mathrm{sh}$ which is Noetherian by Tag 06LJ), and so by Tag 07DX it suffices to show that each $(A\otimes_R R_i)_{\mathfrak{p}_i}$ is regular or, in fact, just that $A\otimes_R R_i$ is regular. But, since $R\to R_i$ is etale, so is $A\to A\otimes_R R_i$, and so the claim follows from Tag 025N.
Again, I actually think this is an overly-complicated proof, but I wanted to address the direct implication that Silverman is making. I hope it’s helpful!
