Lagmental Vicfred

Support Is the Closed Set Defined by an Annihilator by Vicfred

tags: mathematics, primary decomposition, associated primes, support

For a finitely generated module, support is exactly the variety of its annihilator. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

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.

$$ \operatorname{Supp}_A(M)=\{\mathfrak p:M_{\mathfrak p}\ne0\} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ M\ \text{finite}\Longrightarrow\operatorname{Supp}M=V(\operatorname{Ann}M) $$

Stress the formula

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \operatorname{Supp}(A/I)=V(I)=\{\mathfrak p\in\operatorname{Spec}A:I\subseteq\mathfrak p\} $$

Interpretation

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{Supp}_A(M)=\{\mathfrak p:M_{\mathfrak p}\ne0\},\\[5pt] \mathsf{C}\;&:\quad M\ \text{finite}\Longrightarrow\operatorname{Supp}M=V(\operatorname{Ann}M). \end{aligned} $$

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] M\ \text{finite}\Longrightarrow\operatorname{Supp}M=V(\operatorname{Ann}M) \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Thu 04 February 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.