Applications of the existence of big Cohen-Macaulay algebras
In their fundamental paper [see Ann. Math., II. Ser. 135, No. 1, 53-89 (1992; Zbl 0753.13003)] the authors have shown the following result: (*) For \(R\) a complete, semilocal, excellent Noetherian domain of positive prime characteristic which is co-equidimensional every system of parameters of \(R\) is a regular sequence with respect to \(R^+\), where \(R^+\) denotes the absolute integral closure, i.e., the integral closure of \(R\) in the algebraic closure of its field of fractions. So \(R^+\) is a balanced big Cohen-Macaulay algebra for \(R\). Here firstly the authors note that a homomorphism \(R \to S\) of domains extends (not uniquely) to a map \(R^+ \to S^+\). So there is a limited kind of functoriality for the construction of balanced big Cohen- Macaulay algebras. It is shown that this (weakly functorial) construction of balanced big Cohen-Macaulay algebras carries over to the case of equi- characteristic zero. Then this is applied to the proof that pure subgrings of regular rings are Cohen-Macaulay rings. A second proof of this result follows by the `Vanishing theorem for maps of Tor' [cf. the authors, Bull. Am. Math. Soc., New Ser. 24, No. 1, 137- 143 (1991; Zbl 0729.13012)] a result shown to be true in the case of equal characteristics. It follows also by the weak functoriality of balanced big Cohen-Macaulay algebras. Further applications concern the behaviour of tight closure of parameter ideals, the Briançon-Skoda theorem, a graded version of \((*)\), etc. In particular, as a variation of the Briançon-Skoda theorem it is shown that the integral closure \(\overline {(I^{d + k})}\) is contained in \(I^{k + 1} R^+ \cap R\) for all \(k \in \mathbb{N}\). Here \(I\) denotes an arbitrary ideal of a Noetherian domain of prime characteristic \(p\) generated by at most \(d = \dim R\) elements. An application of a graded version of \((*)\) yields an interesting result on the low degrees of syzygies.
construction of balanced big Cohen-Macaulay algebras
scientific article; zbMATH DE number 812537