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## Tag: 07NS

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Remark 14.27.6. The assertion of Lemma 14.27.5 is quite strong. Namely, suppose that we have a diagram $$\xymatrix{ & B \\ A \ar[r] & A' \ar[u] }$$ of local homomorphisms of Noetherian complete local rings where $A \to A'$ induces an isomorphism of residue fields $k = A/\mathfrak m_A = A'/\mathfrak m_{A'}$ and with $B \otimes_{A'} k$ formally smooth over $k$. Then we can extend this to a commutative diagram $$\xymatrix{ C \ar[r] & B \\ A \ar[r] \ar[u] & A' \ar[u] }$$ of local homomorphisms of Noetherian complete local rings where $A \to C$ is formally smooth in the $\mathfrak m_C$-adic topology and where $C \otimes_A k \cong B \otimes_{A'} k$. Namely, pick $A \to C$ as in Lemma 14.27.5 lifting $B \otimes_{A'} k$ over $k$. By formal smoothness we can find the arrow $C \to B$, see Lemma 14.25.5. Denote $C \otimes_A^\wedge A'$ the completion of $C \otimes_A A'$ with respect to the ideal $C \otimes_A \mathfrak m_{A'}$. Note that $C \otimes_A^\wedge A'$ is a Noetherian complete local ring (see Algebra, Lemma 9.93.9) which is flat over $A'$ (see Algebra, Lemma 9.94.10). We have moreover
1. $C \otimes_A^\wedge A' \to B$ is surjective,
2. if $A \to A'$ is surjective, then $C \to B$ is surjective,
3. if $A \to A'$ is finite, then $C \to B$ is finite, and
4. if $A' \to B$ is flat, then $C \otimes_A^\wedge A' \cong B$.
Namely, by Nakayama's lemma for nilpotent ideals (see Algebra, Lemma 9.18.1) we see that $C \otimes_A k \cong B \otimes_{A'} k$ implies that $C \otimes_A A'/\mathfrak m_{A'}^n \to B/\mathfrak m_{A'}^nB$ is surjective for all $n$. This proves (1). Parts (2) and (3) follow from part (1). Part (4) follows from Algebra, Lemma 9.94.1.

\begin{remark}
\label{remark-what-does-it-mean}
The assertion of Lemma \ref{lemma-lift-fs} is quite strong. Namely,
suppose that we have a diagram
$$\xymatrix{ & B \\ A \ar[r] & A' \ar[u] }$$
of local homomorphisms of Noetherian complete local rings where
$A \to A'$ induces an isomorphism of residue fields
$k = A/\mathfrak m_A = A'/\mathfrak m_{A'}$ and with
$B \otimes_{A'} k$ formally smooth over $k$.
Then we can extend this to a commutative diagram
$$\xymatrix{ C \ar[r] & B \\ A \ar[r] \ar[u] & A' \ar[u] }$$
of local homomorphisms of Noetherian complete local rings
where $A \to C$ is formally smooth in the $\mathfrak m_C$-adic
topology and where $C \otimes_A k \cong B \otimes_{A'} k$.
Namely, pick $A \to C$ as in Lemma \ref{lemma-lift-fs}
lifting $B \otimes_{A'} k$ over $k$. By formal smoothness we
can find the arrow $C \to B$, see
Lemma \ref{lemma-lift-continuous}.
Denote $C \otimes_A^\wedge A'$ the completion of
$C \otimes_A A'$ with respect to the ideal $C \otimes_A \mathfrak m_{A'}$.
Note that $C \otimes_A^\wedge A'$ is a Noetherian complete local
ring (see Algebra, Lemma \ref{algebra-lemma-completion-Noetherian})
which is flat over $A'$ (see
Algebra, Lemma \ref{algebra-lemma-flat-module-powers}).
We have moreover
\begin{enumerate}
\item $C \otimes_A^\wedge A' \to B$ is surjective,
\item if $A \to A'$ is surjective, then $C \to B$ is surjective,
\item if $A \to A'$ is finite, then $C \to B$ is finite, and
\item if $A' \to B$ is flat, then $C \otimes_A^\wedge A' \cong B$.
\end{enumerate}
Namely, by Nakayama's lemma for nilpotent ideals (see
Algebra, Lemma \ref{algebra-lemma-NAK}) we see that
$C \otimes_A k \cong B \otimes_{A'} k$ implies that
$C \otimes_A A'/\mathfrak m_{A'}^n \to B/\mathfrak m_{A'}^nB$
is surjective for all $n$. This proves (1). Parts (2) and (3) follow
from part (1). Part (4) follows from
Algebra, Lemma \ref{algebra-lemma-mod-injective}.
\end{remark}


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