Generalizations of the Direct Summand Theorem over UFD-s for some Bigenerated Extensions and an Asymptotic Version of Koh's Conjecture

Abstract

This article deals with two different problems in commutative algebra. In the first part, we give a proof of generalized forms of the Direct Summand Theorem (DST (or DCS)) for module-finite extension rings of mixed characteristic $R\subset S$ satisfying the following hypotheses: The base ring $R$ is a Unique Factorization Domain of mixed characteristic zero. We assume that $S$ is generated by two elements which satisfy, either radical quadratic equations, or general quadratic equations under certain arithmetical restrictions. In the second part of this article, we discuss an asymptotic version of Koh's Conjecture. We give a model theoretical proof using "non-standard methods".

Updated version after the result of Y. Andre regarding the DSC

References in corpus (1)

• Heitmann's proof of the direct summand conjecture in dimension 3