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