Primary Decomposition - Applications

Primary decomposition provides powerful tools for understanding module structure, zero divisors, and geometric properties.

TheoremZero Divisors and Associated Primes

The set of zero divisors in $R/I$ is the union of all associated primes: $\mathcal{Z}(R/I) = \bigcup_{\mathfrak{p} \in \text{Ass}(R/I)} \mathfrak{p}$

An element $\overline{a} \in R/I$ is a zero divisor if and only if $a$ belongs to some associated prime of $I$ .

This theorem provides an explicit description of zero divisors, connecting the algebraic notion to the geometric notion of functions vanishing on subvarieties.

ExampleZero Divisors in Quotients

For $I = (x^2, xy) \subset k[x,y]$ with $\text{Ass}(k[x,y]/I) = \{(x), (x,y)\}$ :

Zero divisors are elements of $(x) \cup (x,y) = (x)$
Indeed, multiplying any element of $(x)$ by appropriate elements yields zero in $k[x,y]/I$
Non-zero-divisors are polynomials with non-zero constant term

TheoremPrimary Decomposition of Modules

Let $M$ be a finitely generated module over a Noetherian ring $R$ . Then:

$M$ has finitely many associated primes $\text{Ass}(M)$
The set of zero divisors on $M$ is $\bigcup_{\mathfrak{p} \in \text{Ass}(M)} \mathfrak{p}$
$\text{Supp}(M) = V(\text{ann}(M)) = \bigcup_{\mathfrak{p} \in \text{Ass}(M)} V(\mathfrak{p})$

where $\text{Supp}(M) = \{\mathfrak{p} : M_\mathfrak{p} \neq 0\}$ .

Remark

For a submodule $N \subseteq M$ , primary decomposition of $N$ decomposes $M/N$ into "irreducible" pieces, generalizing ideal theory to the module setting. The geometric interpretation involves coherent sheaves on schemes.

TheoremMinimal Primes and Dimension

The minimal associated primes of $I$ are precisely the minimal primes containing $I$ . Their number equals the number of irreducible components of $V(I)$ .

The Krull dimension $\dim(R/I)$ equals the maximum height of minimal primes over $I$ , connecting algebraic and geometric dimensions.

ExampleDimension Computation

For $I = (x,y) \cap (z) \subset k[x,y,z]$ :

Minimal primes: $(x,y)$ and $(z)$
$\dim(k[x,y,z]/(x,y)) = 1$ and $\dim(k[x,y,z]/(z)) = 2$
$\dim(k[x,y,z]/I) = \max(1,2) = 2$

The dimension is the maximum dimension of irreducible components.

TheoremSymbolic Powers and Separation

For distinct primes $\mathfrak{p}_1, \ldots, \mathfrak{p}_n$ , the symbolic powers separate in the sense that: $\mathfrak{p}_1^{(m_1)} \cap \cdots \cap \mathfrak{p}_n^{(m_n)}$ is $\mathfrak{p}_i$ -primary at each $\mathfrak{p}_i$ and provides canonical primary decompositions for ideals of this form.

ExampleIntersection Multiplicities

In $k[x,y]$ , the ideal $I = (x^2) \cap (y^3)$ represents coordinate axes with multiplicities. Primary decomposition:

$(x^2)$ is $(x)$ -primary with multiplicity 2
$(y^3)$ is $(y)$ -primary with multiplicity 3
Generators: $I = (x^2y^3, x^2, y^3)$ mixes both components

TheoremAssociated Primes Under Extension

For $R \to S$ a ring homomorphism and $I \subseteq R$ an ideal:

Associated primes of $IS$ relate to associated primes of $I$ via contraction
For flat extensions, $\text{Ass}(S/IS)$ consists of primes lying over primes in $\text{Ass}(R/I)$
This principle governs base change in algebraic geometry

Remark

Depth and Cohen-Macaulay rings are intimately connected to primary decomposition. A ring $R$ is Cohen-Macaulay if every system of parameters forms a regular sequence, which constrains associated primes significantly.

These applications demonstrate how primary decomposition bridges ideal theory, module theory, and algebraic geometry, providing both computational tools and theoretical insights.