Some criteria for regular and Gorenstein local rings via syzygy modules
2019 ◽
Vol 18
(05)
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pp. 1950097
Keyword(s):
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.
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1992 ◽
Vol 111
(1)
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pp. 47-56
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2016 ◽
Vol 16
(09)
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pp. 1750163
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1966 ◽
Vol 27
(1)
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pp. 223-230
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2018 ◽
Vol 17
(02)
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pp. 1850023
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2008 ◽
Vol 07
(01)
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pp. 109-128
2014 ◽
Vol 14
(01)
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pp. 1550004
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