Basic Ring Theory - 19 (On Flatness)

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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Basic Ring Theory - 19 (On Flatness)

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Post by Tsakanikas Nickos »

Let $A \longrightarrow B$ be a ring homomorphism and let $N$ be a $B$-module. Show that the following are equivalent:
  1. $N$ is flat over $A$
  2. For all prime ideals $\mathfrak{p}$ of $A$, $N\otimes_{A} A_{\mathfrak{p}}$ is flat over $A_{\mathfrak{p}}$
  3. For all prime ideals $\mathfrak{q}$ of $B$, $N \otimes B_{\mathfrak{q}}$ is flat over $A_{\mathfrak{p}}$, where $\mathfrak{p}$ is the contraction of $\mathfrak{q}$.
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