The concept of torsion is fundamental in algebra, geometry and topology. The
main reason is that torsion-theoretic methods allow us to isolate and therefore to
study better, important phenomena having a local structure. The proper framework
for the study of torsion is the context of torsion theories in a homological or
homotopical category. In essence torsion theories provide a successful formalization
of the localization process. The notion of torsion theory in an abelian category
was introduced formally by Dickson [41], although the concept was implicit in the
work of Gabriel and others from the late fifties, see the books of Stenstr¨om [100]
and Golan [56] for a comprehensive treatment. Since then the use of torsion theories
became an indispensable tool for the study of localization in various contexts.
As important examples of localization we mention the localization of topological
spaces or spectra, the localization theory of rings and abelian categories, the local
study of an algebraic variety, the construction of perverse sheaves in the analysis
of possibly singular spaces, and the theory of tilting in representation theory.
The omnipresence of torsion suggests a strong motivation for the development of a
general theory of torsion and localization which unifies the above rather unrelated
concrete examples