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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

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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).

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