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Purity, Spectra and Localisation

Book Description

It is possible to associate a topological space to the category of modules over any ring. This space, the Ziegler spectrum, is based on the indecomposable pure-injective modules. Although the Ziegler spectrum arose within the model theory of modules and plays a central role in that subject, this book concentrates specifically on its algebraic aspects and uses. The central aim 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, localisation is an ever-present theme and various types of spectrum play organising 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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Introduction
    1. Conventions and notations
      1. Selected notations list
  9. Part I: Modules
    1. 1. Pp conditions
      1. 1.1 Pp conditions
        1. 1.1.1 Pp-definable subgroups of modules
        2. 1.1.2 The functor corresponding to a pp condition
        3. 1.1.3 The lattice of pp conditions
        4. 1.1.4 Further examples
      2. 1.2 Pp conditions and finitely presented modules
        1. 1.2.1 Pp-types
        2. 1.2.2 Finitely presented modules and free realisations
        3. 1.2.3 Irreducible pp conditions
      3. 1.3 Elementary duality of pp conditions
        1. 1.3.1 Elementary duality
        2. 1.3.2 Elementary duality and tensor product
        3. 1.3.3 Character modules and duality
        4. 1.3.4 Pp conditions in tensor products
        5. 1.3.5 Mittag–Leffler modules
    2. 2. Purity
      1. 2.1 Purity
        1. 2.1.1 Pure-exact sequences
        2. 2.1.2 Pure-projective modules
        3. 2.1.3 Purity and tensor product
      2. 2.2 Pure global dimension
      3. 2.3 Absolutely pure and flat modules
        1. 2.3.1 Absolutely pure modules
        2. 2.3.2 Flat modules
        3. 2.3.3 Coherent modules and coherent rings
        4. 2.3.4 Von Neumann regular rings
      4. 2.4 Purity and structure of finitely presented modules
        1. 2.4.1 Direct sum decomposition of finitely presented modules
        2. 2.4.2 Purity over Dedekind domains and RD rings
        3. 2.4.3 Pp conditions over serial rings
      5. 2.5 Pure-projective modules over uniserial rings
    3. 3. Pp-pairs and definable subcategories
      1. 3.1 Pp conditions and morphisms between pointed modules
      2. 3.2 Pp-pairs
        1. 3.2.1 Lattices of pp conditions
        2. 3.2.2 The category of pp-pairs
      3. 3.3 Reduced products and pp-types
        1. 3.3.1 Reduced products
        2. 3.3.2 Pp conditions in reduced products
        3. 3.3.3 Realising pp-types in reduced products
      4. 3.4 Definable subcategories
        1. 3.4.1 Definable subcategories
        2. 3.4.2 Duality and definable subcategories
        3. 3.4.3 Further examples of definable subcategories
        4. 3.4.4 Covariantly finite subcategories
    4. 4. Pp-types and pure-injectivity
      1. 4.1 Pp-types with parameters
      2. 4.2 Algebraic compactness
        1. 4.2.1 Algebraically compact modules
        2. 4.2.2 Linear compactness
        3. 4.2.3 Topological compactness
        4. 4.2.4 Algebraically compact Z-modules
        5. 4.2.5 Algebraic compactness of ultraproducts
      3. 4.3 Pure-injectivity
        1. 4.3.1 Pure-injective modules
        2. 4.3.2 Algebraically compact = pure-injective
        3. 4.3.3 Pure-injective hulls
        4. 4.3.4 Pure-injective extensions via duality
        5. 4.3.5 Hulls of pp-types
        6. 4.3.6 Indecomposable pure-injectives and irreducible pp-types
        7. 4.3.7 Pure-injective hulls of finitely presented modules
        8. 4.3.8 Krull–Schmidt rings
        9. 4.3.9 Linking and quotients of pp-types
        10. 4.3.10 Irreducible pp-types and indecomposable direct summands
      4. 4.4 Structure of pure-injective modules
        1. 4.4.1 Decomposition of pure-injective modules
        2. 4.4.2 Σ-pure-injective modules
        3. 4.4.3 Modules of finite endolength
        4. 4.4.4 Characters
        5. 4.4.5 The ascending chain condition on pp-definable subgroups
      5. 4.5 Representation type and pure-injective modules
        1. 4.5.1 Pure-semisimple rings
        2. 4.5.2 Finite length modules over pure-semisimple rings
        3. 4.5.3 Rings of finite representation type
        4. 4.5.4 The pure-semisimplicity conjecture
        5. 4.5.5 Generic modules and representation type
      6. 4.6 Cotorsion, flat and pure-injective modules
    5. 5. The Ziegler spectrum
      1. 5.1 The Ziegler spectrum
        1. 5.1.1 The Ziegler spectrum via definable subcategories
        2. 5.1.2 Ziegler spectra via pp-pairs: proofs
        3. 5.1.3 Ziegler spectra via morphisms
      2. 5.2 Examples
        1. 5.2.1 The Ziegler spectrum of a Dedekind domain
        2. 5.2.2 Spectra over RD rings
        3. 5.2.3 Other remarks
      3. 5.3 Isolation, density and Cantor–Bendixson rank
        1. 5.3.1 Isolated points and minimal pairs
        2. 5.3.2 The isolation condition
        3. 5.3.3 Minimal pairs and left almost split maps
        4. 5.3.4 Density of (hulls of) finitely presented modules
        5. 5.3.5 Neg-isolated points and elementary cogenerators
        6. 5.3.6 Cantor–Bendixson analysis of the spectrum
        7. 5.3.7 The full support topology
      4. 5.4 Duality of spectra
      5. 5.5 Maps between spectra
        1. 5.5.1 Epimorphisms of rings
        2. 5.5.2 Representation embeddings
      6. 5.6 The dual-Ziegler topology
    6. 6. Rings of definable scalars
      1. 6.1 Rings of definable scalars
        1. 6.1.1 Actions defined by pp conditions
        2. 6.1.2 Rings of definable scalars and epimorphisms
        3. 6.1.3 Rings of definable scalars and localisation
        4. 6.1.4 Duality and rings of definable scalars
        5. 6.1.5 Rings of definable scalars over a PI Dedekind domain
        6. 6.1.6 Rings of type-definable scalars
    7. 7. M-dimension and width
      1. 7.1 Dimensions on lattices
      2. 7.2 M-dimension
        1. 7.2.1 Calculating m-dimension
        2. 7.2.2 Factorisable systems of morphisms and m-dimension
      3. 7.3 Width
        1. 7.3.1 Width and superdecomposable pure-injectives
        2. 7.3.2 Existence of superdecomposable pure-injectives
    8. 8. Examples
      1. 8.1 Spectra of artin algebras
        1. 8.1.1 Points of the spectrum
        2. 8.1.2 Spectra of tame hereditary algebras
        3. 8.1.3 Spectra of some string algebras
        4. 8.1.4 Spectra of canonical algebras
      2. 8.2 Further examples
        1. 8.2.1 Ore and RD domains
        2. 8.2.2 Spectra over HNP rings
        3. 8.2.3 Pseudofinite representations of sl2
        4. 8.2.4 Verma modules over sl2
        5. 8.2.5 The spectrum of the first Weyl algebra and related rings
        6. 8.2.6 Spectra of V-rings and differential polynomial rings
        7. 8.2.7 Spectra of serial rings
        8. 8.2.8 Spectra of uniserial rings
        9. 8.2.9 Spectra of pullback rings
        10. 8.2.10 Spectra of von Neumann regular rings
        11. 8.2.11 Commutative von Neumann regular rings
        12. 8.2.12 Indiscrete rings and almost regular rings
    9. 9. Ideals in mod-R
      1. 9.1 The radical of mod-R
        1. 9.1.1 Ideals in mod-R
        2. 9.1.2 The transfinite radical of mod-R
        3. 9.1.3 Powers of the radical and factorisation of morphisms
        4. 9.1.4 The transfinite radical and Krull–Gabriel/m-dimension
      2. 9.2 Fp-idempotent ideals
      3. Appendix A: Model theory
        1. A.1 Model theory of modules
  10. Part II: Functors
    1. 10. Finitely presented functors
      1. 10.1 Functor categories
        1. 10.1.1 Functors and modules
        2. 10.1.2 The Yoneda embedding
        3. 10.1.3 Representable functors and projective objects in functor categories
      2. 10.2 Finitely presented functors in (mod-R, Ab)
        1. 10.2.1 Local coherence of (mod-R, Ab)
        2. 10.2.2 Projective dimension of finitely presented functors
        3. 10.2.3 Minimal free realisations and local functors
        4. 10.2.4 Pp conditions over rings with many objects
        5. 10.2.5 Finitely presented functors = pp-pairs
        6. 10.2.6 Examples of finitely presented functors
        7. 10.2.7 Free abelian categories
        8. 10.2.8 Extending functors along direct limits
      3. 10.3 Duality of finitely presented functors
      4. 10.4 Finitistic global dimension
    2. 11. Serre subcategories and localisation
      1. 11.1 Localisation in Grothendieck categories
        1. 11.1.1 Localisation
        2. 11.1.2 Finite-type localisation in locally finitely generated categories
        3. 11.1.3 Elementary localisation and locally finitely presented categories
        4. 11.1.4 Finite-type localisation in locally coherent categories
        5. 11.1.5 Pp conditions in locally finitely presented categories
      2. 11.2 Serre subcategories and ideals
        1. 11.2.1 Annihilators of ideals of mod-R
        2. 11.2.2 Duality of Serre subcategories
    3. 12. The Ziegler spectrum and injective functors
      1. 12.1 Making modules functors
        1. 12.1.1 The tensor embedding
        2. 12.1.2 Injectives in the category of finitely presented functors
        3. 12.1.3 The Ziegler spectrum revisited yet again
      2. 12.2 Pp-types, subfunctors of (RRn, –), finitely generated functors
      3. 12.3 Definable subcategories again
      4. 12.4 Ziegler spectra and Serre subcategories: summary
      5. 12.5 Hulls of simple functors
      6. 12.6 A construction of pp-types without width
      7. 12.7 The full support topology again
      8. 12.8 Rings of definable scalars again
    4. 13. Dimensions
      1. 13.1 Dimensions
        1. 13.1.1 Dimensions via iterated localisation
        2. 13.1.2 Dimensions on lattices of finitely presented subfunctors
      2. 13.2 Krull–Gabriel dimension
        1. 13.2.1 Definition and examples
        2. 13.2.2 Gabriel dimension and Krull–Gabriel dimension
      3. 13.3 Locally simple objects
      4. 13.4 Uniserial dimension
    5. 14. The Zariski spectrum and the sheaf of definable scalars
      1. 14.1 The Gabriel–Zariski spectrum
        1. 14.1.1 The Zariski spectrum through representations
        2. 14.1.2 The Gabriel–Zariski and rep-Zariski spectra
        3. 14.1.3 Rep-Zariski = dual-Ziegler
        4. 14.1.4 The sheaf of locally definable scalars
      2. 14.2 Topological properties of ZarR
      3. 14.3 Examples
        1. 14.3.1 The rep-Zariski spectrum of a PI Dedekind domain
        2. 14.3.2 The sheaf of locally definable scalars of a PI Dedekind domain
        3. 14.3.3 The presheaf of definable scalars of a PI HNP ring
        4. 14.3.4 The presheaf of definable scalars of a tame hereditary artin algebra
        5. 14.3.5 Other examples
      4. 14.4 The spectrum of a commutative coherent ring
    6. 15. Artin algebras
      1. 15.1 Quivers and representations
        1. 15.1.1 Representations of quivers
        2. 15.1.2 The Auslander–Reiten quiver of an artin algebra
        3. 15.1.3 Tubes and generalised tubes
      2. 15.2 Duality over artin algebras
      3. 15.3 Ideals in mod-R when R is an artin algebra
      4. 15.4 mdim ≠ 1 for artin algebras
      5. 15.5 Modules over group rings
    7. 16. Finitely accessible and presentable additive categories
      1. 16.1 Finitely accessible additive categories
        1. 16.1.1 Representation of finitely accessible additive categories
        2. 16.1.2 Purity in finitely accessible categories
        3. 16.1.3 Conjugate and dual categories
      2. 16.2 Categories generated by modules
      3. 16.3 Categories of presheaves and sheaves
        1. 16.3.1 Categories of presheaves
        2. 16.3.2 Finite-type localisation in categories of presheaves
        3. 16.3.3 The category Mod-Ox: local finite presentation
        4. 16.3.4 The category Mod-Ox: local finite generation
        5. 16.3.5 Pp conditions in categories of sheaves
    8. 17. Spectra of triangulated categories
      1. 17.1 Triangulated categories: examples
      2. 17.2 Compactly generated triangulated categories
        1. 17.2.1 Brown representability
        2. 17.2.2 The functor category of a compactly generated triangulated category
      3. 17.3 Purity in compactly generated triangulated categories
        1. 17.3.1 The Ziegler spectrum of a compactly generated triangulated category
        2. 17.3.2 The Ziegler spectrum of D(R)
      4. 17.4 Localisation
      5. 17.5 The spectrum of the cohomology ring of a group ring
      6. Appendix B: Languages for definable categories
        1. B.1 Languages for finitely accessible categories
          1. B.1.1 Languages for modules
        2. B.2 Imaginaries
      7. Appendix C: A model theory/functor category dictionary
  11. Part III: Definable categories
    1. 18. Definable categories and interpretation functors
      1. 18.1 Definable categories
        1. 18.1.1 Definable subcategories
        2. 18.1.2 Exactly definable categories
        3. 18.1.3 Recovering the definable structure
        4. 18.1.4 Definable categories
      2. 18.2 Functors between definable categories
        1. 18.2.1 Interpretation functors
        2. 18.2.2 Examples of interpretations
        3. 18.2.3 Tilting functors
        4. 18.2.4 Another example: lattices over groups
        5. 18.2.5 Definable functors and Ziegler spectra
      3. Appendix D: Model theory of modules: an update
      4. Appendix E: Some definitions
  12. Main examples
  13. Bibliography
  14. Index