Consider the following definition. It seems to be standard, although I have not found a place where it is actually spelled out in this way.
Definition. A pointed scheme is local if
is contained in every nonempty closed subset of
.
Example. If is a local ring, then
is a local scheme. Indeed,
is contained in every nonempty closed subset
, because every strict ideal
is contained in
.
We prove that this is actually the only example.
Lemma. Let be a local scheme. Then
is affine, and
is a local ring whose maximal ideal corresponds to the point
.
Proof. Let be an affine open neighbourhood of
. Then the complement
is a closed set not containing
, hence
. Thus,
is affine. Let
. Let
be a maximal ideal of
; then
. Since this contains
, we must have
, i.e.
corresponds to the (necessarily unique) maximal ideal
.