A commutative Noetherian ring is Cohen-Macaulay when every localization at a prime has depth of a Noetherian local ring equal to its Krull dimension. A regular local ring is Cohen-Macaulay, and quotienting one by a nonzero divisor preserves Cohen-Macaulayness while lowering dimension and depth by one. In particular a hypersurface in a regular ambient variety satisfies the depth condition in the Serre criterion for normality.
A Cohen-Macaulay ring is a type of commutative ring with specific geometric and algebraic properties, often used in algebraic geometry and commutative algebra.
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