Extended plus closure and colon-capturing

Three results concerning the colon-capturing property for extended plus closures in excellent mixed characteristic rings are demonstrated. It is shown that the extended plus closure has the colon-capturing property for arbitrary sets of three parameters. It is also seen that establishing the colon-capturing property for the extended plus closure is sufficient to guarantee the existence of balanced big Cohen–Macaulay algebras. Also, for three-dimensional complete domains, the second local cohomology of R+ is annihilated by arbitrarily small powers of every non-unit.