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# stacks/stacks-project

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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