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Fix notation in proof of proposition in dpa

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aisejohan committed Aug 11, 2019
1 parent 1fcf0ae commit da121b70b463592bba2510a9bb65b35fe05ed993
Showing with 5 additions and 5 deletions.
  1. +5 −5 dpa.tex
10 dpa.tex
@@ -1830,8 +1830,8 @@ \section{Local complete intersection rings}
\ref{more-algebra-lemma-embed-map-Noetherian-complete-local-rings}.
Let $I = \Ker(R \to A)$ and $J = \Ker(S \to B)$.
Note that since $R/I = A \to B = S/J$ is flat the map
$J/I \otimes_R R/\mathfrak m_R \to J/J \cap \mathfrak m_R S$
is an isomorphism. Hence a minimal system of generators of $J/I$
$J/IS \otimes_R R/\mathfrak m_R \to J/J \cap \mathfrak m_R S$
is an isomorphism. Hence a minimal system of generators of $J/IS$
maps to a minimal system of generators of
$\Ker(S/\mathfrak m_R S \to B/\mathfrak m_A B)$.
Finally, $S/\mathfrak m_R S$ is a regular local ring.
@@ -1840,11 +1840,11 @@ \section{Local complete intersection rings}
Assume (1) holds, i.e., $J$ is generated by a regular sequence.
Since $A = R/I \to B = S/J$ is flat we see
Lemma \ref{lemma-perfect-map-ci} applies and we deduce
that $I$ and $J/I$ are generated by regular sequences.
that $I$ and $J/IS$ are generated by regular sequences.
We have $\dim(B) = \dim(A) + \dim(B/\mathfrak m_A B)$ and
$\dim(S/IS) = \dim(A) + \dim(S/\mathfrak m_R S)$
(Algebra, Lemma \ref{algebra-lemma-dimension-base-fibre-equals-total}).
Thus $J/I$ is generated by
Thus $J/IS$ is generated by
$$
\dim(S/J) - \dim(S/IS) = \dim(S/\mathfrak m_R S) - \dim(B/\mathfrak m_A B)
$$
@@ -1862,7 +1862,7 @@ \section{Local complete intersection rings}
(see above), using flatness of $R/I \to S/IS$,
and using Grothendieck's lemma
(Algebra, Lemma \ref{algebra-lemma-grothendieck-regular-sequence})
we find that $J/I$ is generated by a regular sequence in $S/IS$.
we find that $J/IS$ is generated by a regular sequence in $S/IS$.
Thus Lemma \ref{lemma-perfect-map-ci} tells us that $J$
is generated by a regular sequence, whence (1) holds.
\end{proof}

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