Let \((R, \mfm)\) be a Noetherian local ring.
\(R\) is Cohen–Macaulay if \(\operatorname{depth} R = \dim R\), that is, \(\mfm\) contains a regular sequence of length \(\dim R\).
\(R\) is Gorenstein if \(R\) has finite injective dimension as a module over itself.
\(R\) is a complete intersection if its completion is \(\hat{R} \cong S/(f_1, \ldots, f_c)\) for a regular local ring \(S\) and a regular sequence \(f_1, \ldots, f_c\) in \(S\).
A locally Noetherian scheme is Cohen–Macaulay, Gorenstein, or a local complete intersection if every local ring \(\OO_{X,x}\) has the corresponding property. A closed subscheme \(X \subseteq Y\) of a smooth variety is a local complete intersection when its ideal sheaf is locally generated by \(\codim X\) elements.
For Noetherian local rings, \[ \text{regular} \implies \text{complete intersection} \implies \text{Gorenstein} \implies \text{Cohen--Macaulay}, \] and none of the implications reverses. A Cohen–Macaulay projective scheme \(X\) of pure dimension over a field has a dualizing sheaf \(\omega_X\) with Serre duality in every degree, and \(X\) is Gorenstein exactly when \(\omega_X\) is invertible.
In dimension \(0\) every Noetherian local ring is Cohen–Macaulay. \(k[x,y]/(x^2, y^2)\) is a complete intersection. \(k[x,y]/(x^2, xy, y^2)\) is Cohen–Macaulay but not Gorenstein: its socle \(\{r \st \mfm r = 0\} = (x, y)\) is \(2\)-dimensional, while a zero-dimensional Gorenstein local ring has \(1\)-dimensional socle. The union of two planes in \(\AA^4\) meeting in a point, \(V((x,y) \cap (z,w))\), is not Cohen–Macaulay: removing the point disconnects it, while the punctured spectrum \(\Spec R\sm\ts{\mfm}\) of a Cohen–Macaulay local ring \(R\) of dimension at least \(2\) is connected.
Dependencies
D-DEFREGLRRegular local rings, regular rings, and DVRsusesD-DEFREGSQRegular sequencesuses
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