Cataproducts

Abstract

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.