Hi, everyone, I want to ask following problem:


Is strict Henselian ring a excellent ring?


If not, could you give me a example?



Is strict Henselian ring a excellent ring?
Let $k$ be a field of characteristic $p>0$. Consider the field $k((t))$ of Laurent series, endowed with its $t$-adic valuation. Let $L$ be a subextension of $k((t))/k$ which is of finite type over $k$ and of positive transcendence degree over $k$. Consider the subring $A$ of $k[[t]]$ corresponding to $L$. Then the discrete valuation ring $A$ is not excellent (in fact, not even quasi-excellent); see (the proof of) Proposition 11.6 in Exposé I, by M. Raynaud, in [Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’École polytechnique 2006–2008](http://www.math.polytechnique.fr/~orgogozo/travaux_de_Gabber/GTG/GTG.pdf). By Theorem 8.1 of loc. cit., the strict henselisation of $A$ is not quasi-excellent.


