Is primary decomposition still important?

Question

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?

Answer 1

I don't know the history (in particular what role primary decomposition held that is now "obsolete"), but I do know a couple places where primary decomposition appears in various other guises.

I realize this isn't exactly an answer to the question, but perhaps it might illuminate some of the relationship between primary decomposition and localization, and ways that primary decomposition still appears more than simply finding the associated primes...

Symbolic powers

The first is when taking the symbolic power of a prime ideal (or more generally any power). Suppose that $Q \subseteq R$ is a prime ideal. If it helps, assume $R$ is regular (or smooth over a field). Now consider $Q^n$ for various integers $n$. It might come as a surprise that $Q^n$ is not $Q$-primary (at least if you haven't thought about this before). In particular, one can take a primary decomposition of $Q^n = P_1 \cap \dots \cap P_s$. Exactly one of those ideals is $Q$-primary, and that is called the $n$th symbolic power of $Q$, and is denoted by $Q^{(n)}$ in commutative algebra

Let me at least give you some indication of a link between symbolic powers and localization. Consider the extension $Q^n R_Q$ of $Q^n$ to the localization $R_Q$, this is $QR_Q$ primary (because $QR_Q$ is maximal). Now, pull that back to $R$. You get $$Q^{(n)} = (Q^n R_Q) \cap R.$$ In other words, the $n$th symbolic power of $Q$ is the largest ($Q$-primary) ideal that agrees with $Q^n$ GENERICALLY.

Comment: I wonder if this sort of thing that Atiyah-Macdonald are referring to. From the geometric perspective, the main information one wants is often the symbolic power (I'll give some evidence of this below where symbolic power is rephrased in other languages), and now we can think of this purely via localization, and thus forget about the primary decomposition perspective completely.

Finally, if you are finite type over say $\mathbb{C}$ and $R$ is regular, then symbolic powers have a natural description. $Q^{(n)}$ is the set of elements of $R$ that vanish to order $n$ at every closed point of $V(Q)$. In the general case, the description is the same, we just can't restrict to closed points. I believe Eisenbud's book has a nice presentation of some of this.

There are a bunch of natural questions one can ask, but probably the most natural is:

Question: When do symbolic powers equal ordinary ideal powers?

This is a hard question that many commutative algebraists have studied extensively (in my opinion, it is one of the most active areas of research in commutative algebra). The big case where they are known to be equal is for ideals defined by regular sequences, ie complete intersections. Generally speaking, symbolic and ordinary powers are far from equal. Indeed, the set of associated primes that come up could also be viewed as some measure of the complexity of the ideal, but now I'm getting into material that I know less well.

Graded symbolic powers

This is a short aside, but it is probably worth mentioning. A classical question in algebraic geometry is the following (see for example some of the work of Chudnovsky, Esnault-Viehweg, and Harbourne).

Question: $\text{ }$ Given a set of points $S$ in $\mathbb{P}^m$ and an integer $k > 0$, what is the smallest degree $N$ of a hypersurface passing through each point in $S$ with multiplicity $k$?

The set $S$ has an associated homogeneous ideal $J$ in $k[x_0, \dots, x_m]$. $J = Q_1 \cap \dots \cap Q_t$ (where $t$ is the number of points in $S$). Consider $J^{(k)} = Q_1^k \cap \dots \cap Q_t^k$ (a variant of the symbolic power, note that the $Q_i$ are complete intersections, so ordinary and symbolic powers of the $Q_i$ coincide). It often happens that $J^{(k)} \neq J^k$ (in general $J^{(k)}$ is a part of $J^k$'s primary decomposition).

In this case, the answer to the above question is $N =$ ``the smallest degree of a non-zero element of $J^{(k)}$.''

Weil divisorial sheaves

This can sort of be viewed as a special case of symbolic power, but it has a somewhat different flavor / set of applications. Suppose that $X$ is a normal variety and $D$ is an effective Weil divisor on $X$. Consider $O_X(-nD)$ for various integers $n$. Is $X$ is smooth, or more generally, $D$ is Cartier, then the algebra:

$$\bigoplus_{n \geq 0} O_X(-nD)$$

is finitely generated. But in general, it is not finitely generated! However, when it is finitely generated many wonderful things happen (for example, this can be helpful when proving the existence of flips in the minimal model program, take sheafy Proj). Interestingly, in varieties with Kawamata log terminal singularities, all such rings are finitely generated (Stefano Urbinati recently pointed this out to me).

What does this have to do with primary decomposition you ask? Well, this is basically just a variant of symbolic powers. Suppose that $X = \text{Spec} R$ and $D = \sum a_i D_i$ for prime divisors $D_i$. Say that $D_i$ is defined by the prime divisor $P_i \subseteq R$. Then

$$O_X(-nD) = \bigcap P_i^{(a_i n)}.$$

(ok, one's a sheaf, the other is an ideal, but you get the point)

If you haven't done this before, you can try the following exercise. Consider the ideal $Q = (x,y) \subseteq k[x,y,z]/(x^2 - y z)$. This ideal defines a divisor (its a ruling of the cone). Show that the rings $\bigoplus_{n \geq 0} Q^{(n)}$ and $\bigoplus_{n \geq 0} Q^n$ are both finitely generated rings, but they are different rings...

Associated primes as a measure of complexity of modules

Consider a local cohomology module, $H^i_J(R)$ (here $J \subseteq R$ is an ideal). Such modules are useful for computing sheaf cohomology of open sets of varieties since it is basically the algebraic version of cohomology with support. These also show up (often as primary examples) when studying $D$-modules.

If the module has only finitely many associated primes, then its certainly arguable that it's a much simpler module since there are only finitely many building blocks. It was a long standing question asked by Huneke (I think) whether every local cohomology module has only finitely many associated primes. This was shown by Huneke and Sharp for regular rings (for any ideal).

However, Anurag Singh gave a couterexample a few years ago, that showed there could be infinitely many associated primes (Moty Katzman then gave a equi-characteristic example a little later). People are still studying when it happens that local cohomology modules have only finitely many associated primes.

Log canonical centers

David Speyer requested some examples of applications of non-minimal terms from a primary decomposition in application. Here's one example that I know of, or at least an example when non-minimal associated primes come up (as Dustin points out above, the non-minimal primary components of a decomposition are not really unique).

Suppose that $\pi : Y \to X$ is a resolution of singularities of some normal algebraic variety over $\mathbb{C}$. It turns out that $X$ has rational singularities if $R^i O_Y = 0$ for all $i > 0$. Recently Valery Alexeev -- Christopher Hacon and Sándor Kovács defined the notion of an irrational center to be any associated prime of these modules $R^i O_Y$. In particular, these centers measure a failure of a variety to have rational singularities.

Both sets of authors were interested in these primes because their existence or non-existence imply various things about the depth of $O_X$ and $\omega_X$ (the motivation comes from moduli theory of higher dimensional varieties). Interestingly, these irrational centers centers are also always log canonical centers.

Log canonical centers are special subvarieties which have been used quite a lot in higher dimensional algebraic geometry in the past 15 years (they are discussed some in the latter sections of Rob Lazarsfeld's book). In particular, there are some nice extension theorems for global sections from log canonical centers to their ambient spaces (see for example Kawamata's papers on subadjunction). They are also excellent tools for performing induction on dimension with. Many questions about the ambient variety can be reduced to questions on a log canonical center.

Answer 2

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.