Let $A$ be a local artinian ring with residue field $k$, $S$ a $k$-algebra. Suppose there is a formally etale deformation $B$ of $S$ over $A$, i.e. a flat $A$-algebra $B$ such that $S\cong B\otimes_Ak$ whose structural morphism $A\to B$ is formally etale.

Question 1: Does it follow that every (flat) deformation $B'/A$ of $S/k$ is isomorphic to $B/A$?

In fact my question originates from the following lemma in a paper of Kato:

Lemma (=Lemma 1 of K. Kato, A generalization of local class field theory by using K-groups III): Let $R=A/I$ be the quotient of a ring $A$ by a nilpotent ideal and assume $R$ has prime characteristic $p$. Let $S$ be an $R$-algebra such that the Frobenius of $S$ ($x\mapsto x^p$) coincides with the base change of the Frobenius of $R$.

Then there exists a formally etale $A$-algebra $B$ such that $S\cong B\otimes_AR$. If $S$ is flat over $R$, then $B$ is characterized by the properties that it is flat over $A$ and $S\cong B\otimes_AR$.

In Kato's paper the algebra $B$ is constructed explicitly and is shown to be flat over $A$ in case $S$ is flat over $R$. But it does not seem to be proven that in the flat case being a flat deformation characterizes $B$.

So the question I really want to understand is:

Question 2: In the above lemma of Kato, when $S$ is flat over $R$ and $B'$ is a flat $A$-algebra such that $B'\otimes_AR\cong S$, why $B\cong B'$?

I'll be content with the case where $A$ is the quotient of a DVR by a power of the maximal ideal and $R$ is the residue field.


Question 1 can be deduced from EGA $0_{IV}.$

  • $\begingroup$ Regarding question 2, I understand you do not include any smoothness condition in the uniqueness statement, only flatness? Also, I'm not sure of having understood the base change condition in Kato's lemma; is there a link to that paper? $\endgroup$ – Vinteuil May 8 '15 at 15:37
  • $\begingroup$ Thank you Vinteuil. The situation treated in EGA seems to be much more complicated than needed and not easy to follow. So could you explain a bit more about how the arguments in EGA are applied? $\endgroup$ – Yong Hu May 11 '15 at 10:23
  • $\begingroup$ To Vinteuil: The paper of Kato I cited is published in J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 1, 31–43. I don't have a link. But here is what I meant by the base change condition: if h: R to S denotes the structural map, then the square with the two vertical maps given by h and the two horizontal maps given by the frobenius of R and S, is cocartesian. $\endgroup$ – Yong Hu May 11 '15 at 10:27

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