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Infinite integral extensions and big Cohen-Macaulay algebras. (English) Zbl 0753.13003
Here 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.
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.)
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