Some criteria for regular and Gorenstein local rings via syzygy modules

Let [Formula: see text] be a Cohen–Macaulay local ring. We prove that the [Formula: see text]th syzygy module of a maximal Cohen–Macaulay [Formula: see text]-module cannot have a semidualizing direct summand for every [Formula: see text]. In particular, it follows that [Formula: see text] is Gorenstein if and only if some syzygy of a canonical module of [Formula: see text] 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.