Affine-scheme properties are ring-theoretic: reduced, irreducible, integral, connected, Noetherian, and one-point all read off $A$.Suggest a slogan
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.
When is $\Spec A$ reduced, irreducible, integral?
What is the generic point of an irreducible scheme?