The Stacks project

Lemma 15.33.8. Let $A \to B$ be a local homomorphism of local rings. Let $A^ h \to B^ h$, resp. $A^{sh} \to B^{sh}$ be the induced map on henselizations, resp. strict henselizations (Algebra, Lemma 10.155.6, resp. Lemma 10.155.10). Then $\mathop{N\! L}\nolimits _{B/A} \otimes _ B B^ h \to \mathop{N\! L}\nolimits _{B^ h/A^ h}$ and $\mathop{N\! L}\nolimits _{B/A} \otimes _ B B^{sh} \to \mathop{N\! L}\nolimits _{B^{sh}/A^{sh}}$ induce isomorphisms on cohomology groups.

Proof. Since $A^ h$ is a filtered colimit of ├ętale algebras over $A$ we see that $\mathop{N\! L}\nolimits _{A^ h/A}$ is an acyclic complex by Algebra, Lemma 10.134.9 and Algebra, Definition 10.143.1. The same is true for $B^ h/B$. Using the Jacobi-Zariski sequence (Algebra, Lemma 10.134.4) for $A \to A^ h \to B^ h$ we find that $\mathop{N\! L}\nolimits _{B^ h/A} \to \mathop{N\! L}\nolimits _{B^ h/A^ h}$ induces isomorphisms on cohomology groups. Moreover, an ├ętale ring map is a local complete intersection as it is even a global complete intersection, see Algebra, Lemma 10.143.2. By Lemma 15.33.7 we get a six term exact Jacobi-Zariski sequence associated to $A \to B \to B^ h$ which proves that $\mathop{N\! L}\nolimits _{B/A} \otimes _ B B^ h \to \mathop{N\! L}\nolimits _{B^ h/A}$ induces isomorphisms on cohomology groups. This finishes the proof in the case of the map on henselizations. The case of strict henselization is proved in exactly the same manner. $\square$


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