Cohen–Macaulay ring

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Published on May 6, 2025 12:00:00 AM ● Video Link: https://www.youtube.com/watch?v=TjywfYbw3gY

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In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.
They are named for, who proved the unmixedness theorem for polynomial rings, and for, who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property.
For Noetherian local rings, there is the following chain of inclusions.

Source: https://en.wikipedia.org/wiki/Cohen–Macaulay_ring
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