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arXiv · 1611.03263

Some criteria for regular and Gorenstein local rings via syzygy modules

Dipankar Ghosh
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Abstract

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.

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Dipankar Ghosh. 2016-11-10. Some criteria for regular and Gorenstein local rings via syzygy modules. https://doi.org/10.1142/s021949881950097x

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