The Spectrum of a Module CategoryAmerican Mathematical Soc., 2001 - Broj stranica: 125 These notes present an introduction into the spectrum of the category of modules over a ring. We discuss the general theory of pure-injective modules and concentrate on the isomorphism classes of indecomposable pure-injective modules which form the underlying set of this spectrum. The interplay between the spectrum and the category of finitely presented modules provides new insight into the geometrical and homological properties of the category of finitely presented modules. Various applications from representation theory of finite dimensional algebras are included. |
Sadržaj
Chapter 0 Introduction | 1 |
Chapter 1 The functor category | 11 |
Chapter 2 Definable subcategories | 15 |
Chapter 3 Left approximations | 24 |
Chapter 4 Duality | 31 |
Chapter 5 Ideals in the category of finitely presented modules | 37 |
Chapter 6 Endofinite modules | 43 |
Chapter 7 KrullGabriel dimension | 54 |
Chapter 11 Rings of definable scalars | 73 |
Chapter 12 Reflective definable subcategories | 82 |
Chapter 13 Sheaves | 89 |
Chapter 14 Tame hereditary algebras | 98 |
Chapter 15 Coherent rings | 105 |
Appendix A Locally coherent Grothendieck categories | 111 |
Appendix B Dimensions | 116 |
Appendix C Finitely presented functors and ideals | 119 |
Chapter 8 The infinite radical | 58 |
Chapter 9 Functors between module categories | 64 |
Chapter 10 Tame algebras | 68 |
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abelian category artin algebra assertion follows bijection category mod closed under taking coherent functor Coker H collection of maps COROLLARY definable scalars definable subcategory denote direct limits direct summand endofinite modules endolength epimorphism exact functor exact sequence finite dimensional algebra finite length finitely presented objects finitely presented R-module following are equivalent fp-idempotent ideals full subcategory functor f functor Mod Gabriel spectrum Grothendieck category Hom R(X Hom Rop Hom(X Homx(M ideals in mod induces an equivalence injective envelope injective objects KGdim Krull dimension Krull-Gabriel dimension left adjoint left almost split localizing subcategory locally finitely presented maps in mod Mod R).p mod R*P monomorphism morphism non-zero ordinal preceding lemma presheaf product-complete PROOF pure-injective modules pure-reflexive quotient functor R-module representation type ring homomorphism ring of definable section functor Serre subcategory sheaf split map subcategory of Mod subobject Suppose Theorem Zariski topology Ziegler closure Ziegler-closed subset