Lemma 10.63.11 (05BW)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.63: Associated primes
Lemma 10.63.11 (cite)

previous lemma
next remark

numberstags

history
statistics
1 tag refers to this tag

Lemma 10.63.11. Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$ -module. Then $\mathop{\mathrm{Spec}}(\varphi )(\text{Ass}_ S(M)) \subset \text{Ass}_ R(M)$ .

Proof. If $\mathfrak q \in \text{Ass}_ S(M)$ , then there exists an $m$ in $M$ such that the annihilator of $m$ in $S$ is $\mathfrak q$ . Then the annihilator of $m$ in $R$ is $\mathfrak q \cap R$ . $\square$

Comments (0)

There are also:

16 comment(s) on Section 10.63: Associated primes

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment