Splitting results in module-finite extension rings and Koh's conjecture
The objective of this paper is to study a question raised by \textit{J. H. Koh} in his thesis [``The direct summand conjecture and behavior of codimension in graded extensions'' (Thesis), Univ. Michigan 1983)] which is a generalization of the direct summand conjecture. The direct summand conjecture asserts that a Noetherian regular ring \(R\) is a direct summand (as an \(R\)-module) of every module-finite extension ring \(S \supset R\). Hochster has proved that the direct summand conjecture holds when \(R\) contains a field and also for regular rings of dimension \(< 3\). He has also proved that this conjecture implies most of the homological conjectures, e.g., the improved new intersection conjecture and the Evans-Griffith syzygy conjecture. -- Koh has raised the question of whether \(R \subset S\) splits when \(R\) is a ring not necessarily regular and \(S\) is a module-finite extension algebra of finite projective dimension over \(R\). In section 1 we treat several special cases. We prove Koh's conjecture for cyclic extension algebras of \(R\), i.e., algebras of the form \(S = R [\theta]\), where \(R\) is Noetherian. We also prove \(R \subset S\) splits, even when \(S\) does not have finite projective dimension over \(R\), provided that there is a presentation \(R^n \to R^m \to S \to 0\) of \(S\), where \(n\) is ``small'' compared with \(m\). We will also prove Koh's conjecture for module-finite extensions of certain classes of Gorenstein rings of dimension 1. -- In section 2 we will show that Koh's conjecture is false in general by constructing a counterexample in prime characteristic 2, and in mixed characteristic 2. A crucial step in the proof depends on a computation done with the program Macaulay. So far it has not been possible to bypass the use of computers.
scientific article; zbMATH DE number 763202