Let $f: Y \to X$ be an etale morphism of schemes.



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> If $X$ has pseudo-rational singularities then does $Y$ also have pseudo-rational singularities?
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For the definition of pseudo-rational see, for example, Definition 9.4 [here](https://arxiv.org/pdf/1703.02269.pdf).


Note that if $X$ has a resolution of singularities, then it follows from Lemma 9.3 of the paper of Kovacs linked above (and flat base change) that $Y$ has pseudo-rational singularities if $X$ does, but I would like to know whether there is an unconditional proof.


The question above is related to the following (to which I would be happy to get an answer as well):



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> Let $R$ be an excellent noetherian local ring. If $R$ is pseudo-rational then is its completion also pseudo-rational?
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Is pseudo-rationality preserved by etale morphisms?
I believe the answer is yes, although I do not know of a reference.


We will use the original definition of pseudo-rationality due to Lipman and Teissier [[Lipman–Teissier 1981](https://doi.org/10.1307/mmj/1029002461), p. 102], and the following characterization:


**Lemma** [[Lipman–Teissier 1981](https://doi.org/10.1307/mmj/1029002461), Remark (a) on p. 102]**.** *Let $(R,\mathfrak{m})$ be an $n$-dimensional normal Cohen–Macaulay local ring. Then, $R$ is pseudo-rational if and only if for every projective birational morphism $f\colon W \to \operatorname{Spec}(R)$ where $W$ is integral, there exists a proper birational morphism $g\colon W' \to W$ such that $W'$ is normal and integral and such that*
$$\delta\_{fg}\colon H^n\_{\mathfrak{m}}(R) \longrightarrow H^n\_{(fg)^{-1}(\{\mathfrak{m}\})}(\mathcal{O}\_{W'})$$
*is injective.*


**Proposition.** *Let $(R,\mathfrak{m})$ be an $n$-dimensional noetherian local ring that is pseudo-rational, and let $\varphi\colon R \to S$ be an étale map. Then, for every prime ideal $\mathfrak{n} \subseteq S$ lying over $\mathfrak{m}$, the ring $S\_{\mathfrak{n}}$ is pseudo-rational.*


*Proof.* Since $\varphi$ is étale, $S$ is normal and Cohen–Macaulay [[Matsumura 1989](https://doi.org/10.1017/CBO9781139171762), Corollary to Theorem 23.9 and Corollary to Theorem 23.3], and $S\_\mathfrak{n}$ is of dimension $n$ [Stacks, Tag [04N4](https://stacks.math.columbia.edu/tag/04N4)].


Now let $f\_\mathfrak{n}\colon W\_\mathfrak{n} \to \operatorname{Spec}(S\_\mathfrak{n})$ be a projective birational morphism, where $W\_\mathfrak{n}$ is integral. Since $S\_{\mathfrak{n}}$ is integral and noetherian, there exists an ideal $I\_{\mathfrak{n}} \subseteq S\_{\mathfrak{n}}$ such that $W\_\mathfrak{n}$ is the blowup along $I\_{\mathfrak{n}}$ [[EGAIII$\_1$](http://www.numdam.org/item?id=PMIHES_1961__11__5_0), Corollaire 2.3.6]. Clearing denominators, there exists an ideal $I \subseteq S$ localizing to $I\_{\mathfrak{n}}$ such that the blowup $f\colon W \to \operatorname{Spec}(S)$ along $I$ localizes to $f\_\mathfrak{n}$.


By [Stacks, Tag [087B](https://stacks.math.columbia.edu/tag/087B)] or [[Rydh](https://people.kth.se/%7Edary/tamecompactification20110517.pdf), Proposition 4.14], there exists an ideal $J \subseteq R$ such that we have the commutative diagram
$$\require{AMScd}\begin{CD}
 W' @>g>> W @>f>> \operatorname{Spec}(S)\\
 @VVV @. @VVV\\
 \operatorname{Bl}\_J\bigl(\operatorname{Spec}(R)\bigr) @= \operatorname{Bl}\_J\bigl(\operatorname{Spec}(R)\bigr) @>h>> \operatorname{Spec}(R)
\end{CD}$$
where $W' \to \operatorname{Spec}(S)$ is the blowup along $JS$, and the outer rectangle is cartesian by flat base change [Stacks, Tag [0805](https://stacks.math.columbia.edu/tag/0805)]. Replacing $\operatorname{Bl}\_J(\operatorname{Spec}(R))$ by its normalization, we may assume that $\operatorname{Bl}\_J(\operatorname{Spec}(R))$ is normal.


Now by flat base change for local cohomology [[Hashimoto–Ohtani 2008](https://doi.org/10.1307/mmj/1220879415), Theorem 6.10], the homomorphism
$$\delta\_{f\_\mathfrak{n}g\_\mathfrak{n}}\colon H^n\_{\mathfrak{n}}(S\_{\mathfrak{n}}) \longrightarrow H^n\_{(f\_\mathfrak{n}g\_\mathfrak{n})^{-1}(\{\mathfrak{n}\})}(\mathcal{O}\_{W'\_{\mathfrak{n}}})$$
is obtained from $\delta\_h$ by tensoring with $S\_{\mathfrak{n}}$, and hence is injective. Finally, since $fg$ is a blowup and $W'$ is normal by [[Matsumura 1989](https://doi.org/10.1017/CBO9781139171762), Corollary to Theorem 23.9], we see that $S\_\mathfrak{n}$ is pseudo-rational by the Lemma above. $\blacksquare$


It should be possible to prove that completions of pseudo-rational G-rings are pseudo-rational, using Néron–Popescu desingularization [[Popescu 1986](https://doi.org/10.1017/S0027763000022698), Theorem 2.4; [Popescu 1990](https://doi.org/10.1017/S0027763000002981), p. 45; [Swan 1998](https://mathscinet.ams.org/mathscinet-getitem?mr=1697953), Theorem 1.1] to reduce to showing that pseudo-rationality is preserved under étale extensions (shown above) and polynomial extensions.


