The Use of Ultraproducts in Commutative Algebra

Unbeknownst to the majority of algebraists, ultraproducts have been around in model-theory for more than half a century, since their first appearance in a paper by Łoś ([65]), although the construction goes even further back, to work of Skolem in 1938 on non-standard models of Peano arithmetic. Through Kochen’s seminal paper [61] and his joint work [9] with Ax, ultraproducts also found their way into algebra. They did not leave a lasting impression on the algebraic community though, shunned perhaps because there were conceived as non-algebraic, belonging to the alien universe of set-theory and non-standard arithmetic, a universe in which most mathematicians did not, and still do not feel too comfortable.

In this chapter, W denotes an infinite set, always used as an index set, on which we fix a non-principal ultrafilter.¹ Given any collection of (first-order) structures indexed by W, we can define their ultraproduct. However, in this book, we will be mainly concerned with the construction of an ultraproduct of rings, an ultraring for short, which is then defined as a certain residue ring of their Cartesian product. From this point of view, the construction is purely algebraic, although it is originally a model-theoretic one (we only provide some supplementary background on the model-theoretic perspective). We review some basic properties (deeper theorems will be proved in the later chapters), the most important of which is Łoś Theorem, relating properties of the approximations with their ultraproduct. When applied to algebraically closed fields, we arrive at a result that is pivotal in most of our applications: the Lefschetz Principle (Theorem 2.4.3), allowing us to transfer many properties between positive and zero characteristic.

To effectively apply ultraproducts to commutative algebra, we will use, as our main tool, flatness. Since it is neither as intuitive nor as transparent as many other concepts from commutative algebra, we review quickly some basic facts, and then discuss some flatness criteria that will be used later on. Flatness is an extremely important and versatile property, which underlies many deeper results in commutative algebra and algebraic geometry. In fact, I dare say that many a theorem or conjecture in commutative algebra can be recast as a certain flatness result; an instance is Proposition 6.4.6. With David Mumford, the great geometer, we observe:

In this chapter, we will discuss our first application of ultraproducts: the existence of uniformbounds over polynomial rings. Themethod goes back to A. Robinson, but really gained momentum by the work of Schmidt and van den Dries in [86], where they brought in flatness as an essential tool. Most of our applications will be concerned with affine algebras over an ultra-field. For such an algebra, we construct its ultra-hull as a certain faithfully flat ultra-ring. As we will also use this construction in our alternative definition of tight closure in characteristic zero in Chapter 6, we study it in detail in §4.3. In particular, we study transfer between the affine algebra and its approximations. We conclude in §4.4 with some applications to uniform bounds, in the spirit of Schmidt and van den Dries.

In this chapter, p is a fixed prime number, and all rings are assumed to have characteristic p, unless explicitly mentioned otherwise. We review the notion of tight closure due toHochster and Huneke (as a general reference, we will use [59]). The main protagonist in this elegant theory is the p-th power Frobenius map. We will focus on five key properties of tight closure, which will enable us to prove, virtually effortlessly, several beautiful theorems. Via these five properties, we can give a more axiomatic treatment, which lends itself nicely to generalization, and especially to a similar theory in characteristic zero (see Chapters 6 and 7).

We will develop a tight closure theory in characteristic zero which is different from the Hochster-Huneke approach discussed briefly in §5.6. In this chapter we treat the affine case, that is to say, we develop the theory for algebras of finite type over an uncountable algebraically closed field K of characteristic zero; the general local case will be discussed in Chapter 7. Recall that under the Continuum Hypothesis, any uncountable algebraically closed field K of characteristic zero is a Lefschetz field, that is to say an ultraproduct of fields of positive characteristic, by Theorem 2.4.3 and Remark 2.4.4. In particular, without any set-theoretic assumption, ℂ, the field of complex numbers, is a Lefschetz field. The idea now is to use the ultra-Frobenius, that is to say, the ultraproduct of the Frobenii (see Definition 2.4.21), in the same manner in the definition of tight closure as in positive characteristic. However, the ultra-Frobenius does not act on the affine algebra but rather on its ultra-hull, so that we have to introduce a more general setup. It is instructive to do this first in an axiomatic manner (§6.1) and then specialize to the situation at hand (§6.2). We briefly discuss a variant construction in §6.3, and conclude in §6.4 with another example how ultraproducts can be used to transfer constructions from positive to zero characteristic, to wit, the balanced big Cohen-Macaulay algebras of Hochster and Huneke.

The goal of this chapter is to extend the tight closure theory from the previous chapter to include all Noetherian rings containing a field. However, the theory becomes more involved, especially if one wants to maintain full functoriality. We opt in these notes to forego this cumbersome route (directing the interested reader to the joint paper [6] with Aschenbrenner), and only develop the theory minimally as to still obtain the desired applications. In particular, we will only focus on the local case.

One of the main obstacles in the study of ultra-rings is the absence of the Noetherian property, forcing us to modify several definitions from Commutative Algebra. This route is further pursued in [101]. However, there is another way to circumvent these problems: the cataproduct A _#, the first of our chromatic products. We will mainly treat the local case, which turns out to yield always a Noetherian local ring. The idea is simply to take the separated quotient of the ultraproduct with respect to the maximal adic topology. The saturatedness property of ultraproducts—well-known to model-theorists—implies that the cataproduct is in fact a complete local ring. Obviously, we do no longer have the full transfer strength of Łoś Theorem, although we shall show that many algebraic properties still persist, under some mild conditions. We conclude with some applications to uniform bounds. Whereas the various bounds in Chapter 4 were expressed in terms of polynomial degree, we will introduce a different notion of degree here,1 in terms of which we will give the bounds. Conversely, we can characterize many local properties through the existence of such bounds.

In Chapter 4, we used ultraproducts to derive uniform bounds for various algebraic operations, where the bounds are given in terms of the degrees of the polynomials involved. This was done by constructing a faithfully flat embedding of the polynomial ring A into an ultraproduct U(A) of polynomial rings, called its ultra-hull. Moreover, A is characterized as the subring of U(A) of all elements of finite degree. In this chapter, we want to put these uniformity results in a more general context, by replacing the degree on A by what we will call a proto-grading.

In this final chapter, we discuss some of the homological conjectures. Although now theorems in equal characteristic, many remain conjectures in mixed characteristic. Whereas there may be no consensus as to which conjectures count as ‘homological’, an extensive list of them together with their interconnections, can be found in Hochster’s authoritative treatise [43].