EGA Every Day: Irreducible and Noetherian Spaces
2.1 Irreducible Spaces
Definition: A topological space X is irreducible if it’s nonempty and can’t be written as a union of two proper closed subsets.
Equivalent characterizations:
- Any two nonempty open sets intersect
- Every nonempty open is dense
- Every proper closed subset is nowhere dense
- Every open subset is connected
A note for classical topologists: These definitions are stated for arbitrary topological spaces, but they’re basically vacuous for Hausdorff spaces. In a Hausdorff space with two distinct points, you can find disjoint open neighborhoods—violating “any two nonempty opens intersect.” So the only irreducible Hausdorff spaces are singletons! This machinery is really built for non-Hausdorff topologies like Zariski, where closed sets are “small” (algebraic subvarieties) and points aren’t separated.
Generic Points
A subspace Y is irreducible iff its closure is irreducible. In particular, the closure of any singleton {x} is irreducible.
This leads to the key definitions:
- Specialization: y is a specialization of x if y ∈ closure({x})
- Generization: x is a generization of y (the reverse relation)
- Generic point: If X = closure({x}), we call x a generic point of X
Every nonempty open contains the generic point, and in a T₀ space, generic points are unique when they exist.
What’s a generic point, geometrically? At the generic point of an integral scheme, the local ring is the function field K(X). You’ve localized at the zero ideal—every nonzero function is invertible. No polynomial has vanished, no special condition has been imposed. You’re hovering over the whole variety at once without landing anywhere specific.
Closed points, by contrast, are specializations of the generic point. You start generic, impose conditions, and “specialize” down to particular points.
Key Properties
Continuous images: Continuous maps send irreducible sets to irreducible sets, and generic points to generic points. If Y is irreducible with unique generic point y, then f(X) is dense in Y iff f sends the generic point of X to y. Checking density becomes a one-point check.
Open subsets: Nonempty opens of an irreducible space are irreducible, with the same generic point.
Checking via covers: If you have an open cover of X by irreducibles, where every pair intersects, then X is irreducible. Conversely, any open cover of an irreducible space has these properties.
Irreducible Components
Every irreducible subspace is contained in a maximal one, which is necessarily closed. These maximal irreducible subspaces are called irreducible components.
Unlike connected components, irreducible components can overlap—but their intersection is nowhere dense in each.
Example: V(XY) in A² is the union of the two coordinate axes. Two irreducible components: V(X) with generic point (X), and V(Y) with generic point (Y). They overlap at exactly the origin (X,Y)—a single closed point, nowhere dense in each line. The generic points are distinct and each lives only on its own component.
If you write X as a finite union of closed irreducible sets, the irreducible components are just the maximal ones among them.
2.2 Noetherian Spaces
Definition: A topological space X is Noetherian if its open sets satisfy ACC (equivalently, closed sets satisfy DCC). X is locally Noetherian if every point has a Noetherian neighborhood.
Topological vs algebraic Noetherian: The topological condition is weaker. Spec(R) is a Noetherian space iff R has DCC on radical ideals—implied by R being Noetherian, but not equivalent.
Example: $k[x, x^{1/2}, x^{1/4}, …]$ has an infinite ascending chain $(x) ⊂ (x^{1/2}) ⊂ (x^{1/4}) ⊂ \ldots$, so it’s not a Noetherian ring. But Spec has only two prime ideals: (0) and the maximal ideal. Two points! The non-Noetherian behavior hides “underneath” the closed point, invisible to the topology.
Noetherian Induction
If E is a poset with DCC, and P is a property such that “P holds for all x < a implies P(a),” then P holds everywhere.
Proof: If P fails somewhere, the failure set has a minimal element a. But then P holds for all x < a, so P(a) holds. Contradiction.
In practice: to prove something about closed sets in a Noetherian space, show “if it’s true for all proper closed subsets of Z, then it’s true for Z.”
Key Properties
- Subspaces of Noetherian spaces are Noetherian
- Finite unions of Noetherian spaces are Noetherian
- Noetherian spaces are quasi-compact (every open cover has a finite subcover)
- Noetherian spaces have finitely many irreducible components (by Noetherian induction)
Next time: Section 3, sheaf technicalities.