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On some sheaves of special groups Vincent Astier

Summary: On some sheaves of special groups
Vincent Astier
1 Introduction and basic notions
We consider sheaves of special groups (mainly over Boolean spaces). These
are connected to the sheaves of abstract Witt rings considered by Marshall
in [9], used therein in particular to classify spaces of orderings with a finite
number of accumulation points. Our approach allows us to show that the
so-called "question 1" (see [11, 1]) has a positive answer for these spaces. We
conclude these notes by computing the behaviour of the Boolean hull functor
when applied to a sheaf of special groups. The author would like to thank
the referee for some very helpful comments.
Our references are [3] for special groups and the Boolean hull functor, and
[10] for spaces of orderings. The functorial link between (reduced) special
groups and spaces of orderings ­a categorical duality­ can be found in [3,
Chapter 3]. If G is a (reduced) special group we denote by (XG, G) (or
simply XG) its associated space of orderings. Conversely, if Y is a space of
orderings, we denote by GY its associated reduced special group.
Definition 1.1 Let G be a special group and let Satf (G) be the set of satu-
rated subgroups of finite index in G. We say that G is pure in the product of
its finite quotients (ppfq, for short) if the canonical map


Source: Astier, Vincent - Department of Mathematics, University College Dublin


Collections: Mathematics