Lagmental Vicfred

A Primary Decomposition Separates Radical Components by Vicfred

tags: mathematics, primary decomposition, associated primes, support

In a Noetherian ring, many ideals can be written as finite intersections of primary ideals. 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.

$$ I=Q_1\cap\cdots\cap Q_r $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \sqrt I=\sqrt{Q_1}\cap\cdots\cap\sqrt{Q_r} $$

Stress the formula

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ (xy,xz)= (x)\cap(y,z)\subseteq k[x,y,z],\qquad V(xy,xz)=V(x)\cup V(y,z) $$

Interpretation

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad I=Q_1\cap\cdots\cap Q_r,\\[5pt] \mathsf{C}\;&:\quad \sqrt I=\sqrt{Q_1}\cap\cdots\cap\sqrt{Q_r}. \end{aligned} $$

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \sqrt I=\sqrt{Q_1}\cap\cdots\cap\sqrt{Q_r} \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Wed 13 June 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.