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Fix typo in algebra
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aisejohan committed Apr 11, 2024
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Showing 1 changed file with 3 additions and 2 deletions.
5 changes: 3 additions & 2 deletions algebra.tex
Expand Up @@ -37946,10 +37946,11 @@ \section{Smoothness and differentials}
of $R$-algebras we can fill in the dotted arrow by some $R$-algebra
map $\tau : S \to P/J^n$ making the diagram commute. This induces an
$R$-algebra map $\overline{\tau} : S/I^n \to P/J^n$ which is equal to
$\sigma_{n - 1}$ modulo $J^n$. By construction the map $\Psi_n$ is surjective
$\sigma_{n - 1}$ modulo $J^{n - 1}$. By construction the map $\Psi_n$
is surjective
and now $\overline{\tau} \circ \Psi_n$ is an $R$-algebra endomorphism
of $P/J^n$ which maps $x_i$ to $x_i + \delta_{i, n}$ with
$\delta_{i, n} \in J^{n -1}/J^n$. It follows that $\Psi_n$ is an
$\delta_{i, n} \in J^{n - 1}/J^n$. It follows that $\Psi_n$ is an
isomorphism and hence it has an inverse $\sigma_n$.
This proves the lemma.
\end{proof}
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