Lagmental Vicfred

Associated Primes Are Annihilators of Elements by Vicfred

tags: mathematics, primary decomposition, associated primes, support

A prime belongs to Ass M when it is the annihilator of a suitable module element. I want the notation, the mechanism, and the failure mode visible at the same time.

Notation

For a finite \(A\)-module \(M\), the support \(\operatorname{Supp}M\) records primes where localisation is nonzero. Associated primes identify annihilators of individual elements and expose the irreducible pieces of zero divisors.

$$ \operatorname{Ass}_A(M)=\{\mathfrak p:\mathfrak p=\operatorname{Ann}_A(m)\text{ for some }m\in M\} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \operatorname{Ass}_A(M)\subseteq\operatorname{Supp}_A(M) $$

Stress the formula

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ M=A/(xy),\qquad \operatorname{Ann}(\bar x)=(y),\quad\operatorname{Ann}(\bar y)=(x),\quad\operatorname{Ass}(M)=\{(x),(y)\} $$

Interpretation

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{Ass}_A(M)=\{\mathfrak p:\mathfrak p=\operatorname{Ann}_A(m)\text{ for some }m\in M\},\\[5pt] \mathsf{C}\;&:\quad \operatorname{Ass}_A(M)\subseteq\operatorname{Supp}_A(M). \end{aligned} $$

Limit of the argument

Primary decomposition is not unique term by term. Under Noetherian hypotheses, the isolated associated primes are canonical even when embedded components move.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Ass}_A(M)\subseteq\operatorname{Supp}_A(M) \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Tue 23 February 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.