Some criteria for regular and Gorenstein local rings via syzygy modules

Let R be a Cohen–Macaulay local ring. We prove that the nth syzygy module of a maximal Cohen–Macaulay R-module cannot have a semidualizing direct summand for every n ≥ 1. In particular, it follows that R is Gorenstein if and only if some syzygy of a canonical module of R has a nonzero 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.