GLOBAL SUBDIRECT PRODUCTS 3
information the authors obtain in the special setting of some particular al-
gebra, and in many other instances we shall show that this information consists
of universal algebra in conjunction with a minute amount of special algebra.
Now in this process of globalization frequently something rather peculiar
is happening. As mentioned above, the main shortcomings of Birkhoff's Theorem
are twofold: First, too little is known about the factors of the subdirect
representation and secondly, the subdirect product is not "tight" enough to
establish a useful connection to the factors. In our view one deficiency is
just as serious as the other. From this point of view it is interesting to
notice that many sheaf constructions actually start from a subdirect repre-
sentation where a good deal is known about the factors and where the factors
are indeed "simpler" than the subdirect product in some tangible and algebrai-
cally meaningful sense (e.g. a subdirect product of prime rings). However,
the subdirect representation is not deemed "tight" enough. To obtain a sheaf
representation the crucial step is to establish that the given subdirect re-
presentation patches over a suitable dual ring of subsets of the index set
(e.g. the dual ring of hull-kernel open sets). If equalizers are open and the
index space is compact (with respect to the topology whose basis is the dual
ring) then such a subdirect product is essentially already a structure of
global sections of a sheaf. This is well-known, however there is a fly in the
oatmeal: Frequently equalizers are not open (e.g. in the hull-kernel topo-
logy) or the index space is not compact (e.g. with the equalizer topology) or
both. In these cases the typical sheaf construction amounts to a simple trade-
off: The subdirect representation is "tightened" at the expense of almost com-
pletely losing track of the factors. It turns out that this process of
"tightening" the subdirect representation can be uniformly described in the
setting of universal algebra as globalization. Moreover, in this setting it
will become clear how in this process the factors usually are blown up and
proliferated into obscurity. This accounts for the striking fact (easily
accessible in the literature) that many architects of sheaf constructions know
practically nothing about the factors (or stalks) of the subdirect represen-
tation they erect. We rather doubt the merits of this procedure because it
appears to us that the algebraically interesting information comes directly
from the original subdirect representation (which is "tight" in a somewhat
less rigid but still rather significant fashion and where something substantial
is known about the factors) and not from the final sheaf representation. We
shall more explicitly discuss this point as we develop our exposition in
detail.
Sheaves of structures are rather complicated mathematical constructions
which involve two topological spaces and a local homeomorphism. However, in
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