Big Cohen-Macaulay algebras and seeds

In this article, we delve into the properties possessed by algebras, which we have termed seeds , that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke’s “weakly functorial" existence result for big Cohen-Macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic. We also show that every seed over a positive characteristic ring ( R , m ) (R,m) maps to a balanced big Cohen-Macaulay R R -algebra that is an absolutely integrally closed, m m -adically separated, quasilocal domain.