Algebra in a Localic Topos with Applications to Ring Theory
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1038 Francis Borceux Gilberte Van den Bossche
Algebra in a Localic Topos with Applications to Ring Theory
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Authors
Francis Borceux Gilberte Van den Bossche Departernent de Mathernatique, Universite de Louvain 2, chemin du Cyclotron, 1348 Louvain-Ia-Neuve, Belgium
AMS Subject Classifications (1980): 18F20, 18C10, 16A64, 16A90 ISBN 3-540-12711-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12711-9 Springer-Verlag New York Heidelberg Berlin Tokyo
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O.
INTRODUCTIa-J
Sheaves of algebras on a topological space appear in many problems in mathematics and their interest has no longer to be demonstrated.
The purpose of this pu-
blication is to study the localizations of the category of sheaves of If-algebras, where 'II is a finitary algebraic theory, and the extent to which it characterizes the topological base space.
The techniques developed to solve these problems,
applied to the case of modules on a ring R, provide new results on pure ideals and the representation of rings.
As a matter of fact we develop our study in the more
general and more natural context of a theory If internally defined with respect to a topos of sheaves on a frame (i.e. a complete Heyting algebra; for example the algebra of open subsets of a topological space). We shall normally use the letter H to denote a frame and, unless stated otherwise, If will denote a finitary
theory in the topes of sheaves on H.
chapter 1, we recall some basic
In
of the categories Pre H, If) and
She H, If) of presheaves and sheaves of If-algebras on H (limits, colimi ts, generators, associated sheaf, and so on ... ). texts.
Reference is made largely to classical
In chapter 2, we first study the Heyting subobjects of a fixed object in She H, If) : these are the subobjects wich satisfy properties analogous to the properties of any subobject in a topos. segments of She H, If).
This allows us to describe the formal initial
If ui- is any initial segment of Hand lfUi- the restriction
of If to u- , Sh(ui-, lf ) is a subcategory of She H, If) satisfying very special prom We then define "formal initial segments" to be subcategories of She H, If)
perties.
satisfying analogous properties.
The Heyting subobjects of a fixed algebraic sheaf
constitute a frame and the same holds for the formal initial segment of She H, If). Chapter 3 applies the results developed in chapter 2 to classify the localizations of She H, If) when the the
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