d
PETER H. KRAUSS AND DAVID M. CLARK
the Jacobson radical) it again does not tell us that much about semi-simple
commutative rings, and ring theorists are known to complain about this short-
coming. Although direct sums of rings exist, interesting direct sum represen-
tations for important classes of rings apparently
donft.
This dilemma appears
to have provided a major motivation for the development of sheaf representa-
tions. Structures of global sections of sheaves are again special subdirect
products which are "tight" enough to allow significant conclusions to be drawn
from the properties of the factors (or stalks).
Both in the case of direct sum representations as well as in the case of
sheaf representations the crucial issue of course is to explicate the under-
lying notion of "tightness" which renders these constructions interesting.
Now we shall see that direct sum representations are sheaf representations in
a perfectly natural and direct sense, so that from this point of view the
sheaf construction is a natural generalization of the direct sum construction.
However, what exactly is it that makes these subdirect representations "tight"?
It appears that this question has never been explicitly pursued in the
literature. Sheaf representations apparently "just have nice properties".
Implicitly the so-called "patchwork property" plays a central role in sheaf
representation. Structures of global sections patch over the equalizer topo-
logy, and it is plausible that this accounts for the "tightness" of the sub-
direct representation.
Unrestricted patching over the equalizer topology actually characterizes
structures of global sections, as was noticed independently by Weispfenning
[38]. However, unrestricted patching is a condition which is difficult to
verify in applications. In fact, in applications unrestricted patching is
usually reduced to finite patching via some kind of compactness argument.
Therefore we first introduce the more general notion of (finite) patching
over _a dual ring of subsets of the index set of a subdirect product, and
then we give a uniform procedure for "tightening" a subdirect product which
patches finitely to a sheaf representation which patches unrestrictedly.
This procedure is called globalization. It is carried out in the setting of
universal algebra and applies to all sheaf constructions in algebra which
start from finite patching. Many discrete sheaf constructions in algebra
directly start from finite patching and it appears to us that all discrete
sheaf constructions in algebra start from finite patching in some way or
other. We shall amply support this view with examples. As a practical con-
sequence of our analysis many major papers on the sheaf construction can be
substantially shortened and simplified with our methods, in fact some of them
collapse to triviality. Moreover, in many instances of sheaf constructions
known from the literature we shall show that universal algebra yields all the
Previous Page Next Page