Primary Decomposition - Key Properties

The uniqueness properties of primary decomposition reveal both what is canonical and what depends on choices.

TheoremFirst Uniqueness Theorem

In a minimal primary decomposition $I = Q_1 \cap \cdots \cap Q_n$ with associated primes $\mathfrak{p}_1, \ldots, \mathfrak{p}_n$ :

The set $\{\mathfrak{p}_1, \ldots, \mathfrak{p}_n\}$ is uniquely determined by $I$
The primary components $Q_i$ corresponding to minimal primes are uniquely determined
The primary components corresponding to embedded primes may not be unique

An associated prime $\mathfrak{p}$ is minimal if it is minimal in $\text{Ass}(R/I)$ under inclusion.

ExampleNon-Uniqueness

In $k[x,y,z]$ , consider $I = (x,y) \cap (x,z) \cap (x^2, xz, y)$ . The ideals $(x,y)$ and $(x,z)$ correspond to minimal primes and are unique. However, the component $(x^2, xz, y)$ can be replaced by other $(x)$ -primary ideals containing $I$ without changing the intersection.

This illustrates that embedded components are not uniquely determined.

DefinitionIsolated and Embedded Components

An associated prime $\mathfrak{p}$ is isolated if $\mathfrak{p}$ is minimal among associated primes (under inclusion). It is embedded if it properly contains another associated prime.

Geometrically, isolated primes correspond to irreducible components of dimension equal to the variety's dimension, while embedded primes correspond to lower-dimensional components.

TheoremMinimal Primes and Radicals

The minimal associated primes of $I$ are precisely the minimal primes containing $I$ . Moreover: $\sqrt{I} = \bigcap_{i=1}^n \mathfrak{p}_i = \bigcap_{\mathfrak{p} \text{ minimal over } I} \mathfrak{p}$

The radical captures the "support" of the ideal, forgetting multiplicities.

ExampleGeometric Interpretation

For $I = (x^2, xy) \subset k[x,y]$ , we have:

$I = (x) \cap (x^2, y)$ as a primary decomposition
Associated primes: $(x)$ (minimal) and $(x,y)$ (embedded)
$\sqrt{I} = (x)$
Geometrically: $V(I)$ is the $y$ -axis with an embedded point at the origin

Remark

The length of a primary decomposition (number of components) is not an invariant. However, the number of isolated components equals the number of minimal primes, which is intrinsic.

TheoremPrimary Decomposition and Localization

If $I = Q_1 \cap \cdots \cap Q_n$ with associated primes $\mathfrak{p}_i$ , then for any multiplicative set $S$ : $S^{-1}I = \bigcap_{\mathfrak{p}_i \cap S = \emptyset} S^{-1}Q_i$

Localization "removes" components whose associated primes meet $S$ . At a prime $\mathfrak{p}$ , localization isolates the $\mathfrak{p}$ -primary component.

DefinitionSymbolic Powers

The $n$ -th symbolic power of a prime ideal $\mathfrak{p}$ is: $\mathfrak{p}^{(n)} = \mathfrak{p}^n R_\mathfrak{p} \cap R$

This is the $\mathfrak{p}$ -primary component of $\mathfrak{p}^n$ in any primary decomposition. Unlike ordinary powers, symbolic powers capture the "correct" geometric notion of multiple components.

ExampleSymbolic vs. Ordinary Powers

In $k[x,y,z]$ with $\mathfrak{p} = (x,y)$ , we have $\mathfrak{p}^2 \subsetneq \mathfrak{p}^{(2)}$ . Specifically, $z \in \mathfrak{p}^{(2)}$ but $z \notin \mathfrak{p}^2$ when extended to certain configurations.

The failure of $\mathfrak{p}^n = \mathfrak{p}^{(n)}$ relates to geometric singularities and depth.

These uniqueness properties distinguish canonical features (associated primes, isolated components) from non-canonical ones (embedded components), guiding both theoretical understanding and computational approaches.