Splitting in integral extensions, Cohen-Macaulay modules and algebras (Q1109826)

This paper concerns the direct summand problem and the existence of Cohen-Macaulay modules. A Noetherian domain is called a ``splinter'' if R is a R-module direct summand of every module-finite ring extension. If R is a splinter, then R is normal. And if R contains the rationals the converse is true via the trace argument. This paper studies the case where \(char(R)=p>0\) and R is normal, especially where R is a complete local domain. It is shown that in some cases the problem can be passed on to the associated graded ring. On the negative side, it is shown that for \(R=k[X_ 1,...,X_ n]/(f)\) (k a field with \(char(k)=p>0)\) where f is homogeneous of degree \(\geq n\), then R is not a splinter. It is shown that the coordinate ring of a product of smooth projective curves over an algebraically closed field has a finitely generated Cohen-Macaulay module. Examples of big Cohen-Macaulay algebras with identities are given.