The Stacks project

Lemma 15.38.1. Let $K$ be a field of characteristic $0$ and $A = K[[x_1, \ldots , x_ n]]$. Let $L$ be a field of characteristic $p > 0$ and $B = L[[x_1, \ldots , x_ n]]$. Let $\Lambda $ be a Cohen ring. Let $C = \Lambda [[x_1, \ldots , x_ n]]$.

  1. $\mathbf{Q} \to A$ is formally smooth in the $\mathfrak m$-adic topology.

  2. $\mathbf{F}_ p \to B$ is formally smooth in the $\mathfrak m$-adic topology.

  3. $\mathbf{Z} \to C$ is formally smooth in the $\mathfrak m$-adic topology.

Proof. By the universal property of power series rings it suffices to prove:

  1. $\mathbf{Q} \to K$ is formally smooth.

  2. $\mathbf{F}_ p \to L$ is formally smooth.

  3. $\mathbf{Z} \to \Lambda $ is formally smooth in the $\mathfrak m$-adic topology.

The first two are Algebra, Proposition 10.152.9. The third follows from Algebra, Lemma 10.154.7 since for any test diagram as in Definition 15.36.1 some power of $p$ will be zero in $A/J$ and hence some power of $p$ will be zero in $A$. $\square$


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