Spectrum is sober
Statement
Topological statement
The spectrum of a commutative unital ring, with the usual topology, is a sober space: in other words, any irreducible closed subset is the closure of a one-point subset.
Ring-theoretic statement
Any irreducible closed subset in the spectrum of a commutative unital ring corresponds to a prime ideal under the Galois correspondece between a ring and its spectrum.
Proof
Given: A commutative unital ring , its spectrum
, an irreducible closed subset
of
. Let
, for an ideal
, denote the set of prime ideals containing
.
To prove: There is a prime ideal such that
is precisely the set of all prime ideals containing
Proof: By definition of closed set, there exists a unique radical ideal such that
is precisely the set of prime ideals containing
. We need to show that
is in fact prime. Suppose not. Then there exist ideals
and
of
such that
but neither
nor
is contained in
.
We now argue the following:
-
: If
is a prime ideal containing
, then
. But primeness of
forces
or
, thus forcing
.
- Both are closed subsets: This is by definition
- Both are proper subsets: If
, then
being a radical ideal, must be the radical of
. But that'd force
, a contradiction to assumption. Hence
must be a proper subset of
. A similar argument holds for
.
Thus, we have expressed as a union of two proper closed subsets, a contradiction. Hence, our original assumption was false, and
is prime.