Title Spectral spaces / Max Dickmann, Niels Schwartz, Marcus Tressl

Published Cambridge, United Kingdom ; New York, NY, USA : Cambridge University Press, [2019]

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Description 1 online resource Series New mathematical monographs ; 35 New mathematical monographs ; 35. Contents Cover; Half-title; Series information; Title page; Copyright information; Contents; Preface; An Outline of the History of Spectral Spaces; 1 Spectral Spaces and Spectral Maps; 1.1 The Definition of Spectral Spaces; 1.2 Spectral Maps and the Category of Spectral Spaces; 1.3 Boolean Spaces and the Constructible Topology; 1.4 The Inverse Topology; 1.5 Specialization and Priestley Spaces; 1.6 Examples; 1.7 Further Reading; 2 Basic Constructions; 2.1 Spectral Subspaces; 2.2 Products of Spectral Spaces; 2.3 Spectral Subspaces of Products; 2.4 Finite Coproducts; 2.5 Zariski, Real, and Other Spectra 3 Stone Duality3.1 The Spectrum of a Bounded Distributive Lattice; 3.2 Stone Duality; 3.3 Spectral Spaces via Prime Ideals and Prime Filters; 3.4 The Boolean Envelope of a Bounded Distributive Lattice; 3.5 Inverse Spaces and Inverse Lattices; 3.6 The Spectrum of a Totally Ordered Set; 3.7 Further Reading; 4 Subsets of Spectral Spaces; 4.1 Quasi-Compact Subsets, Closure, and Generalization; 4.2 Directed Subsets and Specialization Chains; 4.3 Rank and Dimension; 4.4 Minimal Points and Maximal Points; 4.5 Convexity and Locally Closed Sets and Points; 5 Properties of Spectral Maps 5.1 Images of Proconstructible Sets under Spectral Maps5.2 Monomorphisms and Epimorphisms; 5.3 Closed and Open Spectral Maps; 5.4 Embeddings; 5.5 Irreducible Maps and Dominant Maps; 5.6 Extending Spectral Maps; 6 Quotient Constructions; 6.1 Spectral Quotients Modulo Relations; 6.2 Saturated Relations; 6.3 Spectral Orders and Spectral Relations; 6.4 Quotients Modulo Equivalence Relations and Identifying Maps; 6.5 Spectral Quotients and Lattices; 6.6 The Space of Connected Components; 7 Scott Topology and Coarse Lower Topology; 7.1 When Scott is Spectral 7.2 Fine Coherent Posets and Complete Lattices7.3 The Coarse Lower Topology on Root Systems and Forests; 7.4 Finite and Infinite Words; 8 Special Classes of Spectral Spaces; 8.1 Noetherian Spaces; 8.2 Spectral Spaces with Scattered Patch Space; 8.3 Heyting Spaces; 8.4 Normal Spectral Spaces; 8.5 Spectral Root Systems and Forests; 9 Localic Spaces; 9.1 Frames and Completeness; 9.2 Localic Spaces -- Spectra of Frames; 9.3 Localic Maps; 9.4 Localic Subspaces; 9.5 Localic Points; 10 Colimits in Spec; 10.1 Coproducts; 10.2 Fiber Sums; 10.3 Colimits; 10.4 Constructions with Fiber Sums 11 Relations of Spec with Other Categories11.1 The Spectral Reflection of a Topological Space; 11.2 The Sobrification; 11.3 Spectral Reflections of Continuous Maps; 11.4 Properties of Topological Spaces and their Spectral Reflections; 11.5 How Localic Spaces are Located in the Category of Spectral Spaces; 11.6 The Categories Spec and PoSets; 11.7 The Subcategory BoolSp of Spec; 12 The Zariski Spectrum; 12.1 The Zariski Spectrum -- Topology on the Set of Prime Ideals of a Ring; 12.2 Functoriality; 12.3 Locally Closed Points and the Nullstellensatz; 12.4 The Spectrum of a Noetherian Ring Summary Spectral spaces are a class of topological spaces. They are a tool linking algebraic structures, in a very wide sense, with geometry. They were invented to give a functional representation of Boolean algebras and distributive lattices and subsequently gained great prominence as a consequence of Grothendieck's invention of schemes. There are more than 1,000 research articles about spectral spaces, but this is the first monograph. It provides an introduction to the subject and is a unified treatment of results scattered across the literature, filling in gaps and showing the connections between different results. The book includes new research going beyond the existing literature, answering questions that naturally arise from this comprehensive approach. The authors serve graduates by starting gently with the basics. For experts, they lead them to the frontiers of current research, making this book a valuable reference source Notes Online resource; title from PDF title page (EBSCO, viewed April 1, 2019) Subject Topological spaces. Geometry, Algebraic. Commutative rings. MATHEMATICS -- Topology. Geometría algebraica Topología Commutative rings Geometry, Algebraic Topological spaces Form Electronic book Author Schwartz, Niels, 1950- author. Tressl, Marcus, 1964- author. ISBN 9781108609586 1108609589 9781316543870 1316543870

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