Cohen–Macaulay scheme | Wikipedia audio article
Channel: wikipedia tts
Date: 06/06/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Cohen%E2%80%93Macaulay_ring


00:01:07 1 Definition
00:01:53 2 Examples
00:06:56 3 Geometric consequences
00:10:23 4 Miracle flatness or Hironaka's criterion
00:12:42 5 Properties
00:15:20 6 The unmixedness theorem
00:16:27 7 Counterexamples
00:16:39 8 Grothendieck duality
00:17:46 9 Notes
00:18:19 10 References
00:18:30 11 External links



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SUMMARY
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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 Francis Sowerby Macaulay (1916), who proved the unmixedness theorem for polynomial rings, and for Irvin Cohen (1946), 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.

Universally catenary rings ⊃  Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings