Dimension, Zariski Spaces

Zariski Spaces

A zariski space T is generic, T₀, and noetherian. There is no need to transform T, as described in the previous section, as the result would be homeomorphic to T.

If T is the generic space produced from some other noetherian space S, then T is generic, T₀, and noetherian, hence zariski. The original space S is zariski iff it is generic and T₀, iff the transformation from S to T implements a homeomorphism.

The Closed Points of T

By dcc, every closed set in T has a minimal closed subset, which must be irreducible. Call this minimal set C, and let x be its generic point. Since T is T₀, x is the generic point for C.

If C also contains y, the closure of y lies in C, and is equal to C by minimality. This contradicts x as the generic point for C. Thus the minimal closed sets are points. These are called the closed points of T.

Specialize and Generize

If y is in the closure of x, we say x specializes to y, and y generizes to x. I guess "generize" is a word, although Merriam Webster never heard of it.

This is not a function, rather, it induces a partial ordering on the points of T. Let x ≥ y indicate x specializes to y, or y is in the closure of x. If z is in the closure of y, then the closure of x brings in y, brings in z, so the relation is transitive. If x is in the closure of y and y is in the closure of x, we are violating T₀, hence the relation is antisymmetric. Yes, it establishes a partial ordering.

If x is minimal then its closure brings in nothing else. It is closed and irreducible - one of our closed points. Conversely let x be a closed point in T. It's closure brings in only x, and x is minimal.

Assume x is maximal, and let C be its closure, an irreducible set. If C is contained in a larger irreducible set, with generic point y, then the closure of y includes x, which is a contradiction. The maximal points correspond to the irreducible components of T.

If x is in V and V is closed, the closure of x lies in V. This holds for all x in V. We say that V is stable under specialization.

If U is open and U misses x, U misses a closed set containing x, and U misses the closure of x. Equivalently, if U contains the closure of x it contains x. This means U is stable under generization.