The Asymptotic Behavior of Solid Closure in Mixed Characteristic

We study how solid closure in mixed characteristic behaves after taking ultraproducts. The ultraproduct will be chosen so that we land in equal characteristic, and therefore can make a comparison with tight closure. As a corollary we get an asymptotic version of the Hochster-Roberts invariant theorem in dimension three: if R is a mixed characteristic (cyclically) pure 3-dimensional local subring of a regular local ring S, then R is Cohen-Macaulay, provided the ramification of S is large with respect to its dimension and residual characteristic, and with respect to the multiplicity of R. 1. ULTRAVERSUS CATASolid closure was introduced by Hochster in [5, 6] as a potential substitute for tight closure in mixed characteristic. In this note, we comment on some of its properties, but as the title indicates, only ‘asymptotically’, that is to say, after taking an ultraproduct (see §3.7 for an elaboration on the terminology). More precisely, let Aw be a sequence of (commutative) rings (with identity), indexed by an infinite index set endowed with a non-principal ultrafilter, which, for technical reasons, we also assume to be countably incomplete. The ultraproduct of the Aw is again a ring A, realized as the quotient of the product of the Aw modulo the ideal of all sequences almost all of whose entries are zero (with almost all one means for all indices in some member of the ultrafilter). We sometimes refer to the Aw as components of A, although they are not uniquely defined by A. If P is a property of rings, then we say that A has property ultra-P if almost all Aw have property P . If a property P is first-order, then ultra-P is the same as P by Łos’ Theorem. For instance, being local is a first-order property, so that ultra-local is the same as local. However, most properties are not first-order (mostly because they require quantification over ideals or involve infinitely many statements). For instance, an ultra-Noetherian local ring is an ultraproduct of Noetherian local rings, and in general is no longer Noetherian (in fact, its prime spectrum is infinite and can be quite complicated; for some instances of this, see [13, 14, 15]). We will only be concerned with a certain subclass of ultra-Noetherian local rings, those of finite embedding dimension. An ultra-Noetherian local ring has embedding dimension n if, and only if, almost all of its components have embedding dimension n (because having embedding dimension n is a first-order property). Ultra-Noetherian local rings of finite embedding dimension already appeared as an essential tool in the earlier Date: February 1, 2008. Partially supported by a PSC-CUNY grant. 1Suffice it here to say that this set-theoretic notion can be realized on any infinite index set and holds automatically when the index set is countable. Moreover, it is consistent with ZFC (=’usual set theory’) that every ultrafilter is countably incomplete. 2We will call a ring R local if it has a unique maximal ideal m, and we denote this by (R, m) (in the literature one sometimes uses the term quasi-local in the non-Noetherian case). Note that R is local if, and only if, the sum of any two non-units is a non-unit, indeed a first-order property.