Some criteria for regular and Gorenstein local rings via syzygy modules

Let $R$ be a Cohen-Macaulay local ring. We prove that the $n$th syzygy module of a maximal Cohen-Macaulay $R$-module cannot have a semidualizing direct summand for every $n\ge 1$. In particular, it follows that $R$ is Gorenstein if and only if some syzygy of a canonical module of $R$ has a non-zero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen-Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.