# primaryComponent -- find a primary component corresponding to an associated prime

## Synopsis

• Usage:
Q = primaryComponent(I, P)
• Inputs:
• I, an ideal, an ideal in a (quotient of a) polynomial ring R
• P, an ideal, an associated prime of I
• Optional inputs:
• Increment => ..., default value 1
• Strategy => ..., default value 2
• Outputs:
• Q, an ideal, a P-primary ideal of I

## Description

The output Q is topComponents(I + P^m) for sufficiently large m. The criterion that Q is primary is given in Eisenbud-Huneke-Vasconcelos, Invent. Math. 110 (1992) 207-235. However, we use localize(Ideal,Ideal).

The Strategy option value sets the strategy option for localize, and should be one of the following:

• Strategy => 0 -- Uses localize Strategy 0
• Strategy => 1 -- Uses localize Strategy 1
• Strategy => 2 -- Uses localize Strategy 2

The Increment option value should be an integer. The algorithm given in Eisenbud-Huneke-Vasconcelos, Invent. Math. 110 (1992) 207-235, relies on topComponents(I + P^m) for $m$ sufficiently large. The algorithm begins with $m = 1$, and increases $m$ by the value of the Increment option until m is sufficiently large. The default value is 1.