Spectral Spaces

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.

Max Dickmann has been a researcher at the Centre National de la Recherche Scientifique (CNRS), Paris, since 1974, Directeur de Recherche since 1988 and emeritus since 2007. His research interests include the applications of spectral spaces to real algebraic geometry, quadratic forms, and related topics.
Niels Schwartz is Professor of Mathematics at the Universität Passau, Germany, retired since 2016. Many of his publications are concerned with, or use, spectral spaces in essential ways. In particular, he has used spectral spaces to introduce the notion of real closed rings, an important topic in real algebra and geometry.
Marcus Tressl is a mathematician working in the School of Mathematics at the University of Manchester. His research interests include model theory, ordered algebraic structures, ring theory, differential algebra, and non-Hausdorff topology.