Spectral Spaces

Offers a comprehensive presentation of spectral spaces focussing on their topology and close connections with algebra, ordered structures, and logic.

Max Dickmann (Author), Niels Schwartz (Author), Marcus Tressl (Author)

9781107146723, Cambridge University Press

Hardback, published 21 March 2019

650 pages
23.5 x 16 x 4 cm, 1.04 kg

'The book covers a substantial amount of material that had not been considered before. It also contains material available nowhere else in book form.' Tomasz Kubiak, Mathematical Reviews Clippings

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.

Outline of the history of spectral spaces
1. Spectral spaces and spectral maps
2. Basic constructions
3. Stone duality
4. Subsets of spectral spaces
5. Properties of spectral maps
6. Quotient constructions
7. Scott topology and coarse lower topology
8. Special classes of spectral spaces
9. Localic spaces
10. Colimits in Spec
11. Relations of Spec with other categories
12. The Zariski spectrum
13. The real spectrum
14. Spectral spaces via model theory
Appendix. The poset zoo
References
Index of categories and functors
Index of examples
Symbol index
Subject index.

Subject Areas: Algebraic topology [PBPD], Algebra [PBF], Mathematical logic [PBCD]