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There are some necessary basic facts about flatness. First of all, let $R$ be a local Noetherian, flat $A$-algebra, and let $x_1,\dots,x_c \in \mathfrak{m}_R$ be a collection of elements. Then, by the Local Flatness Criterion, if the images $\overline{x}_1,\dots,\overline{x}_c \in \mathfrak{m}_R/\mathfrak{m}_AR$ form a regular sequence in ...

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