This post is the second of three parts, culminating in a proof of the fundamental theorem of Galois Theory:
1) Finite Separable Algebras
2) Galois Connections
3) The Fundamental Theorem of Galois Theory
These posts are designed to aid in the third part, so, for the reader who is looking for the easiest route to a proof of the fundamental theorem, I have put a star (*) next to the essential parts of this post.
Definition:* a preorder is a set endowed with a relation
on
, such that
(i)
.
(ii)
.
We may view a preorder as a category whose objects are the elements of , where there is a unique morphism from
to
when
, and no morphism otherwise. A morphism (functor) of preorders
and
is a function of sets
such that
implies
. We may also define a contravariant preorder map as a function of sets
such that
implies
. In lieu of the viewpoint that preorders are particular kinds of categories, two elements
are called isomorphic when
and
.
Definition:* let and
be preorders and
and
contravariant preorder maps. We say that
and
form a Galois connection if
is left adjoint to
. That is, for each
and each
,
if and only if
. We say that the Galois connection is a Galois correspondence if the the units of the adjunction are natural isomorphisms, in which case
and
testify to a categorical equivalence between the preorders
and
. Note that, due to the contravariance of the functors
and
, the adjunction has two units, and no counits.
Theorem:* let and
be preorders and let
and
be contravariant preorder maps forming a Galois connection. Then
(i) . (N.B. this is the unit map of the adjunction).
(ii) . (N.B. this is the counit map of the adjunction).
(iii) and
are isomorphic for all
.
(iv) and
are isomorphic for all
.
(v) and
form a Galois correspondence.
Proof:
(i) For each ,
if and only if
, which is true.
(ii) For each ,
if and only if
, which is true.
(iii) Take . Put
.
by (ii), so
.
by (i), so that
. Hence
.
(iv) Take . Put
.
by (ii), so
.
by (i), so that
. Hence
.
(v) By (iii), for each
, and by (iv),
for each
. The claim follows.
N.B. (iii) and (iv) follow from the triangle identities for a (contravariant) adjunction between categories. For preorders, we see that we get a full isomorphism.
Example: let and
be sets and let
be a relation. Let
be the set of subsets of
, partially ordered by inclusion, and let
be the set of subsets of
, partially ordered by inclusion. Define a preorder map
where
is sent to the subset
and a preorder map
where
is sent to the subset
.
and
form a Galois connection. Indeed, for subsets
and
,
if and only if
, if and only if
.
Example: let be a field,
the polynomial ring over
in
variables. Write
for
,
dimensional affine space. Let
be the set of subsets of
and
be the set of subset of
. Define a relation
where, for
and
,
if
. Define a map
sending a subset
of
to the set
and a map
sending a subset
of
to the set
. Then
and
form a Galois connection. In fact, this is an instance of the above example.
By the proposition above, and
restrict to inclusion reversing bijections
and
. When
is algebraically closed, the Nullstellensatz characterizes the image of
as the radical ideals. The subsets of
contained in the image of
form the closed sets of a topology on
, called the Zariski topology.
Example: be a field and let
be a
-vector space. Let
be the set of subsets of
, ordered by inclusion. Let
be the set of subsets of
, the dual of
as a
-vector space. There is a relation
where
when
. Define a map
sending a subset
of
to the set
, and a map
sending a subset
of
to the set
. This also follows from the first example.
As always, and
restrict to inclusion reversing bijections
and
. We may characterize the image
as the
-vector subspaces of
. The elements of the image
form the closed subsets of a topological space.
Example: Let be an
-module, and form the ring
of endomorphisms of
. Let
be the set of subsets of
. Define a relation
where
when
. Define a map
sending
to the set
.
restricts to an inclusion reversing bijection
, which is is its own inverse. The set
is called the commutant of
.
Exercise: Let be a category with a zero object, kernels, and cokernels. Take an object
in
. Let
be the category of subobjects of
and let
be the category of quotient objects of
. These are each preorders. Define a preorder map
sending
to
and a preorder map
sending
to
. Show that this forms a Galois connection.
How could a Galois connection be called such without the namesake example- a Galois connection between subfields of a field and subgroups of its Galois group? The reader may well object that this example was left out. You won’t have to lament for long, though, since this correspondence is the subject of my next post!