Going down for flat extensions: Difference between revisions

Revision as of 21:14, 9 March 2008

Statement

Suppose $φ : R \to S$ is a flat extension i.e. $R$ is a subring of a commutative unital ring $S$ and $S$ is flat as a $R$ -module. Then, if $P_{1} \supset P_{2}$ are prime ideals of $R$ , and $Q_{1} \in S p e c (S)$ contracts to $P_{1}$ , there exists $Q_{2} \in S p e c (S)$ such that $Q_{1} \supset Q_{2}$ , and $Q_{2}$ contracts to $P_{2}$ .

References

Book:Eisenbud, Page 239

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 Suppose <math>\varphi:R \to S</math> is a flat extension i.e. <math>R</math> is a [[subring]] of a [[commutative unital ring]] <math>S</math> and <math>S</math> is flat as a <math>R</math>-module. Then, if <math>P_1 \supset P_2</math> are [[prime ideal]]s of <math>R</math>, and <math>Q_1 \in Spec(S)</math> contracts to <math>P_1</math>, there exists <math>Q_2 \in Spec(S)</math> such that <math>Q_1 \supset Q_2</math>, and <math>Q_2</math> contracts to <math>P_2</math>.
+==References==
+* {{booklink|Eisenbud}}, Page 239