Direct summands of syzygy modules of the residue class field

Abstract

Let $R$ be a commutative Noetherian local ring. This paper deals with the problem asking whether $R$ is Gorenstein if the $n$th syzygy module of the residue class field of $R$ has a non-trivial direct summand of finite G-dimension for some $n$. It is proved that if $n$ is at most two then it is true, and moreover, the structure of the ring $R$ is determined essentially uniquely.