Preface

Summary

Spectral spaces constitute a class of topological spaces used in various branches of mathematics. They were introduced in the 1930s by M. H. Stone and have been used extensively ever since. There was a marked growth of interest following A. Grothendieck's revolution of algebraic geometry. It was realized that spectral spaces can be associated with many mathematical structures. Numerous publications are devoted to various properties of spectral spaces and to a growing number of diverse applications. The area is extremely active and is growing at a fast pace.

With this book we provide the first comprehensive and coherent treatment of the basic topological theory of spectral spaces. It is possible to study spectral spaces largely with algebraic tools, namely using bounded distributive lattices, or, in more abstract form, using category theory, model theory, or topos theory. However, our focus is clearly on the topology, which provides geometric tools and intuition for applications that, a priori, do not have geometric meaning. Also, in our experience, the topological techniques are very flexible towards possible extensions of techniques and results to wider classes of spaces, where a corresponding algebraic framework does not exist.

We start with a careful analysis of the definition of spectral spaces, describe fundamental structural features, and discuss elementary properties. Numerous examples, counterexamples, and constructions, listed in an index of examples, show how one can work with spectral spaces in concrete situations or illustrate results. We exhibit methods illustrating how spectral spaces can be associated with different classes of structures and describe some of the most important applications.

It was our original intention to assemble basic material about spectral spaces in one place to make it more easily accessible. Collecting the material and preparing a coherent presentation proved to be more laborious than anticipated: the terminology and notation differ from publication to publication. The Zariski spectrum of commutative unital rings is undoubtedly the most widely used construction of spectral spaces. Therefore, many results on spectral spaces are found in publications about rings and are expressed in the corresponding language. These needed to be translated into topological language to make them compatible with our intentions and presentation.