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 $(R,m)$ maps to a balanced big Cohen-Macaulay $R$-algebra that is an absolutely integrally closed, $m$-adically separated, quasilocal domain.