## Purity, Spectra and LocalisationThe central aim of this book is to understand modules and the categories they form through associated structures and dimensions, which reflect the complexity of these, and similar, categories. The structures and dimensions considered arise particularly through the application of model-theoretic and functor-category ideas and methods. Purity and associated notions are central, localization is an ever-present theme and various types of spectrum play organizing roles. This book presents a unified, coherent account of material which is often presented from very different viewpoints and clarifies the relationships between these various approaches. |

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Ab)fP abelian category absolutely pure artin algebra closed subset condition Corollary corresponding countable deﬁnable category deﬁnable scalars deﬁnable subcategory deﬁned deﬁnition denote direct limits direct sum direct summand dual duality endomorphism ring epimorphism equivalent exact sequence example ﬁeld ﬁnite length ﬁnite type ﬁnite-dimensional ﬁnitely presented functors ﬁnitely presented modules ﬁrst ﬂat follows free realisation free variables functor category Grothendieck hence hereditary torsion theory ideal indecomposable modules indecomposable pure-injective inﬁnite injective hull isomorphism lattice left modules Lemma localisation locally coherent locally ﬁnitely presented Math model theory module category morphism noetherian non-Zero notation ofﬁnite open sets pp-deﬁnable subgroups pp-pair pp-type preadditive category presheaf projective Proof Proposition Puninski pure-injective hull pure-injective module quotient R-mod R-module result right R-modules ring of deﬁnable satisﬁes Section Serre subcategory simple subcategory of Mod-R subfunctor subobjects Suppose Theorem topology torsionfree triangulated category tuple uniserial Weyl algebra Ziegler spectrum