Big Cohen-Macaulay and seed algebras in equal characteristic zero via ultraproducts

TL;DR

This paper extends the theory of seed algebras and big Cohen-Macaulay algebras from prime characteristic to equal characteristic zero using ultraproducts, establishing similar properties and stability results.

Contribution

It introduces a method to transfer properties of seed algebras from prime characteristic to characteristic zero via ultraproducts, expanding the theory's applicability.

Findings

Properties of seeds are extended to characteristic zero.

Ultraproducts enable transfer of algebraic properties.

Directed systems of big Cohen-Macaulay algebras exist in characteristic zero.

Abstract

Let $R$ be a commutative, local, Noetherian ring. In a past article, the first author developed a theory of $R$-algebras, termed seeds, that can be mapped to balanced big Cohen-Macaulay $R$-algebras. In prime characteristic $p$, seeds can be characterized based on the existence of certain colon-killers, integral extensions of seeds are seeds, tensor products of seeds are seeds, and the seed property is stable under base change between complete, local domains. As a result, there exist directed systems of big Cohen-Macaulay algebras over complete, local domains. In this work, we will show that these properties can be extended to analogous results in equal characteristic zero. The primary tool for the extension will be the notion of ultraproducts for commutative rings as developed by Schoutens and Aschenbrenner.