Primary Ideals, Finitely Many Saturations

Finitely Many Saturations

Let J be a proper ideal and assume a primary decomposition for J. Thus J is the intersection of primary ideals H_i, below primes P_i.

If T is a multiplicatively closed set containing 1, what can we say about the saturation of J with respect to T?

Look at J as an intersection. Some u drives x into J iff it drives x into each H_i. In other words, saturation commutes with intersection. Find the saturation of each H_i with respect to T, and take their intersection.

If H_i is P_i primary, ask whether T intersects P_i. If they intersect in u, a power of u winds up in H_i, T intersects H_i, and the saturation is all of R.

If T and P_i do not intersect, ratchet T up to the complement of P_i, which can only make the saturation larger. (Since T contains 1, the saturation is at least H_i.) Now P_i is minimal over H_i, and the saturation through the complement of P_i gives the smallest primary ideal between H_i and P_i. This has to be H_i. Thus the saturation of H_i with respect to T yields H_i.

Everything depends on whether T intersects P_i. The saturation of J becomes the intersection of some of the primes in σ(J), specifically, those primes that miss T.

There are finitely many cases, finitely many ways to miss some of the primes in σ(J). If σ(J) has n primes, there are at most 2ⁿ different saturations of J.

Sometimes it is helpful to change T in some way. As long as T intersects the same prime ideals in σ(J), the saturation has not changed.