Download 1 Preliminaries

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Preliminaries
[s:prelims]
1.1
Noetherian topological spaces
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A topological space is Noetherian if any (and hence all) of the following conditions hold:
• Open sets satisfy the ascending chain condition
• Any non-empty collection of open sets admits a maximal element
• Closed sets satisfy the descending chain condition
• Any non-empty collection of closed sets admits a minimal element
• Any open set is quasi-compact1
The spectrum of a Noetherian commutative ring (with identity) is Noetherian in the above sense.
1.2
Irreducible topological spaces
A topological space is irreducible if any (and hence all) of the following conditions hold:
• The space is not the union of two proper non-empty closed sets
• Any two non-empty open sets intersect
• Any non-empty open set is dense
A subset of a topological space irreducible if it is irreducible with respect to the subspace topology.
A subset is irreducible if and only if its closure is. In particular, the closure of a singleton is
irreducible.
Continuous images of irreducible spaces are irreducible. Irreducible spaces are connected.
Maximal irreducible subsets, called irreducible components exist (by a Zorn argument). In
fact, the argument applies moreover to show that, given an element, there is an irreducible
component containing it. The irreducible components thus cover the space.
Irreducible components are closed, for the closure of an irreducible set is also irreducible.
The irreducible closed sets of the spectrum of a commutative ring are of type V (p) for a
prime ideal p. In particular, the irreducible components are precisely those of the form V (p) as
p varies over minimal prime ideals.2
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A topological space is quasi-compact if every open cover has a finite subcover. The term compact is reserved
for spaces that are also separated.
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Minimal primes always exist (by a Zorn argument). The subset of a commutative ring with identity consisting
of the nilpotent elements is clearly an ideal. It is the intersection of all prime ideals, equivalently, all minimal
prime ideals. Indeed, every prime ideal contains all nilpotent elements, by the definition of primality; on the other
hand, given an element that is not nilpotent, we can find, following Krull, a prime not containing it: pull back a
maximal ideal from the localization at powers of that element.
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