2014/5/19 17:41
Chapter 3
The Boolean Spectrum
3.1 Pierce’s representation
As we noted in Section 2.1 of Chapter 2, the presence of idempotents sig-
nificantly complicates the Galois theory of separa ble projective algebras over
commutative rings. Galo is theory is not unique in this respec t. A similar thing
happens in the theory of commutative rings which are regular in the sense of
von Neumann: that is, rings where every element is an idempotent times a
unit. An example is a product, finite or infinite, of fields . Since commutative
von Neumann regular rings differ from fields only in that they have idempo-
tents, their theory should be much like that of fields. An instructive case to
consider is the commutative von Neumann regular rings C(X, Z/Z2) of con-
tinuous Z/Z2 valued functions on a profinite space X. That these are commu-
tative von Neumann regular rings follows fro m the fact that every element is
idempotent (so the only unit is 1). Commutative rings in which every element
is idempotent are called Boolean rings. The Stone Representation Theorem
shows that any such ring R is the ring of continuous Z/Z2 valued functions
on its maximal ideal space X, and that the latter is pr ofinite.
R.S. Pierce in his study “Modules over commutative regular rings,” Mem.
Amer. Math Soc. 70 (1967) introduced a powerful generalization of the Stone
Representation: he shows that every commutative ring R (not necessarily
regular in the sense of von Neumann) ca n be regarded as the ring of global
section of a sheaf R of rings over a profinite space X such that for each x X
the stalk R
x
has no idempotents except 0 and 1, and this she af is produced
69
2014/5/19 17:41
70 CHAPTER 3. THE BOOLEAN SPECTRUM
in a minimal way. Of course the us ual representation of a commutative ring
as the ring of global sections o f a sheaf of local rings over its prime sp ectrum
has the property that the stalks have no nontrivial idempotents, but this is
not, in general, the minimal way to do this. The minimality” referr ed to here
means tha t the stalks are the weakest such that they ar e connected. As Pierce
showed, and as we will see, the repres e ntation has the property that every
element of every stalk is restriction of a global section. Among other reasons ,
this follows from the fact that the base space of the sheaf is profinite. Pierce’s
construction mimics that of the usual construction, except that instead of
the prime ideal spectrum and localizations at prime ideals of the ring, Pierce
works with the corresponding notions for the Boole an algebra of idempotents
of the ring.
Pierce was also interested in non-commutative rings, so his co nstruction
is more ge neral than what has been outlined so far. If commutative rings
alone are to be studied, then there is an easier construction, which Villamayor
and Zelinsky pointed out in their paper “Galois theory with infinitely many
idempotents,” Nagoya Math. J. 35(1969): let R be a commutative ring, let
Spec(R) be its prime spectrum and let O
R
be the usual sheaf of local rings on
Spec(R): fo r P Spec(R), O
R,P
= R
P
. Let q : Spec(R) Comp(Spec(R))
be the map which identifies connected components to points. Then Pierce’s
sheaf is (Comp(Spec(R)), q
(O
R
)).
Villamayo r and Zelinsky went on to show further that the basic properties
of the repres entation which figures in Galois theory could easily be stated and
prove d without the language of sheaves. This appr oach has obvious expository
advantages and will be adopted here.
3.2 Topology of the Boolean spectrum
We begin with the definition of the Boolean spectrum a nd a look at its topol-
ogy.
Definition 15. Let R be a commutative ring, and let Spec(R) denote the set of
prime ideals of R, with the topology which makes V (I) = {P Spec(R)|P I}
closed for every ideal I of R. Then the Boolean Spectrum of R is the space of
components Comp(Spec(R)) (see Definition 8). The Boolean spectrum of R is
denoted X(R).

Get The Separable Galois Theory of Commutative Rings, 2nd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.