A scheme is reduced, or locally noetherian, exactly when the rings of one affine open cover are reduced, or noetherian.
Properties of $\Spec A$ read off $A$ (Tag PR-6TCSJ)
For a ring \(A\):
| \(\Spec A\) | \(A\) |
|---|---|
| reduced | no nonzero nilpotents |
| irreducible | the nilradical is prime |
| integral | a domain |
| connected | no nontrivial idempotents |
| Noetherian | Noetherian |
| a single point | local artinian |
More generally, a scheme \(X\) is integral if and only if it is reduced and irreducible [@Har10a, Proposition II.3.1].
An irreducible scheme has a unique generic point, dense in it; for \(\Spec A\) integral it is the prime \((0)\), with residue field \(\Frac(A)\).
“Integral is reduced plus irreducible” follows immediately from the table: no nilpotents plus a prime nilradical means \((0)\) is prime.
Generic points are the visible difference from the classical picture, and the useful slogan is that a statement holds generically exactly when it holds at the generic point. The function field of an integral scheme is the residue field there, which is how \(k(X)\) becomes an object rather than a construction.
A scheme is integral exactly when it is reduced and irreducible: on an affine open \(\Spec A\), a reduced ring whose nilradical is prime has \((0)\) prime. “Noetherian” for a scheme means a finite cover by spectra of Noetherian rings, which is strictly stronger than the topological Noetherian condition of the dictionary: \(\Spec k[x_1,x_2,\ldots]/(x_1,x_2,\ldots)^2\) is a one-point space, and its ring is not noetherian.
Generic points
An irreducible closed subset \(Z\) of a scheme has a unique generic point \(\eta_Z\), with \(\overline{\{\eta_Z\}}=Z\). For \(X\) integral the generic point \(\eta\) has residue field \(k(X)\), the function field, and a rational map is a morphism defined on some neighbourhood of \(\eta\).
For \(X\) integral and \(\mcf\) coherent, \(\mcf_\eta\) is a finite-dimensional \(k(X)\)-vector space, so \(\mcf\) is locally free on a dense open subset of \(X\). A property that holds on an open subset of an irreducible scheme \(X\) holds on a dense open subset exactly when it holds at \(\eta\).
Singularity classes
Cohen--Macaulay, Gorenstein and local complete intersection schemes (Tag D-SCHCMGOR)
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.