A commutative noetherian ring is Cohen-Macaulay precisely when the levels of all its perfect complexes are finite with respect to G_C(R) for any semidualizing module C.
Reiten , The converse to a theorem of Sharp on Gorenstein modules, Proc
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A characterization of Cohen-Macaulay rings in terms of levels of perfect complexes
A commutative noetherian ring is Cohen-Macaulay precisely when the levels of all its perfect complexes are finite with respect to G_C(R) for any semidualizing module C.