= Vanishing of a finitely generated module near a prime
{title2=$M_P=0\iff\operatorname{Ann}_R(M)\not\subseteq P$}
For a <finitely generated module> $M=(m_1,\ldots,m_t)$ and a <prime ideal> $P$, if $M_P=0$, choose $s_i\notin P$ annihilating $m_i$. Their product $f\notin P$ annihilates all of $M$. Conversely such an $f$ becomes a <unit> in $R_P$, forcing $M_P=0$. Thus
$$
\{P:M_P=0\}=\bigcup_{f\in\operatorname{Ann}_R(M)}D(f)
=\operatorname{Spec}R\setminus V(\operatorname{Ann}_R(M)).
$$
This identifies the <support of a module> $M$ with the <Zariski-closed set> $V(\operatorname{Ann}_R(M))$. For $M=0$, use $f=1$. No <Noetherian> assumption on $R$ is required.
\Comment[[
Exact target: reduced-locus-of-a-ring
Existing parent: Reduced ring
Intended file: /home/ciro/bak/git/codex-wiki/commutative-algebra.bigb
]]
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