Infinite integral extensions and big Cohen-Macaulay algebras.

*(English)*Zbl 0753.13003Here are two main results of this paper in a simplified, and so weakened form:

1. The integral closure \(R^ +\) of an excellent local domain \(R\) of finite characteristic in the algebraic closure of its fraction field is a balanced big Cohen-Macaulay \(R\)-algebra, that is any system of parameters for \(R\) is a regular sequence on \(R^ +\).

2. A locally excellent Noetherian domain of finite characteristic that is a direct summand of every module-finite extension ring is Cohen- Macaulay.

Graded versions of the results are presented in the paper, as well as a discussion of the zero characteristic case. One of the applications is a new proof of a connectedness theorem of Faltings. Connections with tight closure are presented. The paper provides motivation for studying rings of the form \(R^ +\). A theorem of M. Artin that the sum of two prime ideals in a domain of the form \(R^ +\) is either prime or the unit ideal is generalized for quadratically closed rings and proved in a simple fashion.

1. The integral closure \(R^ +\) of an excellent local domain \(R\) of finite characteristic in the algebraic closure of its fraction field is a balanced big Cohen-Macaulay \(R\)-algebra, that is any system of parameters for \(R\) is a regular sequence on \(R^ +\).

2. A locally excellent Noetherian domain of finite characteristic that is a direct summand of every module-finite extension ring is Cohen- Macaulay.

Graded versions of the results are presented in the paper, as well as a discussion of the zero characteristic case. One of the applications is a new proof of a connectedness theorem of Faltings. Connections with tight closure are presented. The paper provides motivation for studying rings of the form \(R^ +\). A theorem of M. Artin that the sum of two prime ideals in a domain of the form \(R^ +\) is either prime or the unit ideal is generalized for quadratically closed rings and proved in a simple fashion.

Reviewer: M.Roitman (Haifa)

##### MSC:

13C14 | Cohen-Macaulay modules |

13B22 | Integral closure of commutative rings and ideals |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |