TITLE: Is primary decomposition still important?
QUESTION [53 upvotes]: On p.50 of Atiyah and Macdonald's Introduction to Commutative Algebra, in the introduction to the chapter on primary decomposition, it says

In the modern treatment, with its
  emphasis on localization, primary
  decomposition is no longer such a
  central tool in the theory.

However, my understanding is that scheme theory, which fundamentally uses localization and is probably part of the modern treatment they refer to, does not eliminate the need for primary decomposition. Rather, one can and does (e.g. as in Eisenbud and Harris) apply primary decomposition to schemes.
Therefore, is it really true that localization somehow obviates the need for primary decomposition? Are there examples in which primary decomposition can be replaced by other arguments, making primary decomposition merely a classical but obsolete notion? Otherwise, what do the authors mean by this statement?

REPLY [13 votes]: Part of your question is about the relationship between primary decomposition and localization. Here, I think the basic connection is that if $I$ is an ideal in a Noetherian ring $R$ whose minimal primary decomposition is $I = J_1 \cap \cdots \cap J_n$ and $S$ is any multiplicative set in $R$, then a minimal primary decomposition of $(S^{-1} I) \cap R$ consists of those $J_i$ whose associated primes are disjoint from $S$. For example, in Karl's answer, this statement gives two equivalent definitions of the symbolic power by taking $I$ to be a power of a prime ideal $P$ and $S = R \setminus P$.
The point is that one way to look at the usefulness of localization is that it simplifies ideals by replacing them with the intersections of only some of their primary components. From this perspective, primary decomposition seems like a more useful way of analyzing ideals since it gives you each primary component by itself. However, the limitation of primary decomposition is that it is not unique (Eisenbud's book explains this very well if I remember correctly). Thus, primary decomposition gives you a mix of useful information about the original ideal together with some information coming from some arbitrary choices you made during the computation. Which aspects of the primary decomposition are uniquely determined by the original ideal? Well, it's the set of associated primes as well as the intersections between any subset of primary components whose associated primes form an downward-closed subset of poset of the associated primes of the original ideal, i.e. exactly those ideals which show up as $(S^{-1} I) \cap R$ for some multiplicative set $S$. I think that some of the distaste for primary decomposition on the part of, for example, Bourbaki, comes from the non-uniqueness, whereas localization can be seen as a way of capturing exactly the unique part of primary decomposition.
As came up in the comments, primary decomposition is very useful computationally. Here, the non-uniqueness is not a problem since computations typically depend on many choices anyways. I suspect that in most cases, primary decomposition in and of itself is of less interest than other questions which can be answered from it: the irreducible components of the scheme, which of these are generically reduced, the irreducible components of the non-reduced locus, or whether or not the projective scheme corresponding to a homogeneous ideal is reduced (which is not necessarily the same as whether the affine scheme is reduced). Each of these questions could be answered from the set of associated primes together with the localizations at the minimal primes, but the primary decomposition provides a convenient presentation of all of them.