Although the theory of spectral spaces has been an active topic of research for more than eighty years and appeared in more than 1000 research articles, this is the first monograph devoted to the topic. The authors ambitiously pursue a variety of goals in the book, with a foundational goal of giving a coherent and reasonably comprehensive treatment of the topological theory of spectral spaces. Additional important goals include accessibility of the material, particularly including introductory material at a level accessible to graduate students and material of potential interest to other researchers with overlapping interests in some of the various specialized topics treated. Accessibility is also enhanced by inclusion of a wide variety of examples and concrete descriptions of various constructions. As the overall theory is scattered through a wide variety of scholarly writings, the authors have also sought to find connections, fill in gaps, and provide a more comprehensive approach, thus enriching, not just recording earlier work. But beyond all this they present material resulting from their unified approach that belongs to the frontiers of current research. Thus in the end the book has more the flavor of a research monograph and reference source. The latter is important, given the growth of the field and the previous lack of such a source.
Although spectral spaces form a special class of topological spaces, their historical roots lie in algebraic settings. Indeed their name “spectral spaces” derives from the fact that they are precisely the spaces that arise as Zariski spectra of commutative rings, first famously shown by M. Hochster [Trans. Am. Math. Soc. 142, 43–60 (1969; Zbl 0184.29401)]. They first appeared in the work of M. H. Stone [Čas. Mat. Fys. 67, 1–25 (1937; JFM 63.0830.01)], in which he showed the duality of bounded distributive lattices and spectral spaces, a follow-up to his famous earlier work on what we know as the Stone duality of Boolean lattices. The whole theory received a major boost in the 1960s through Alexander Grothendieck’s introduction in algebraic geometry of affine schemes, which, loosely speaking, allowed a commutative ring to be viewed as a ring of functions on its Zariski spectrum, an idea that has had far reaching consequences.
A spectral space is a compact sober space with a basis of compact, open subsets closed under finite intersection. (Here the reviewer replaces the authors’ “quasi-compact” with the more common “compact.”) The topology of spectral spaces reflects the fact that they are \(T_0\), typically not Hausdorff, and hence one has important features such as the order of specialization, a naturally associated partial order. With respect to this order one has topologies with the reversed order of specialization and “patch” topologies generated by a topology and one with reversed order. An important one with reversed order is the inverse topology, obtained by taking the open compact subsets as the closed sets. The join or patch of these two topologies together with the partial order of specialization yields with what is called a Priestley space, a special compact Hausdorff partially ordered space. It turns out that spectral spaces and Priestley spaces are the same structures viewed from different perspectives, and both are important. Thus in some sense spectral spaces live in both the \(T_0\)-world and the Hausdorff world. Finally the spectral spaces form the objects of a category, with the morphisms all continuous maps between objects with the property that the inverse image of a compact open set is again compact. All this and more are treated in the first introductory chapter.
The book breaks down roughly into two parts with the first part, Chapters 1 through 6, devoted to a wide variety of topological properties and constructions in the category of spectral spaces. (Chapter 3 is something of an exception as it reviews and elaborates on the Stone duality of bounded distributive lattices with dual category the category of spectral spaces.) The second part, consisting of the remaining chapters, looks at various other settings in which spectral spaces appear, and treats the theory in those various contexts and settings.
Chapter 7 treats intrinsic topologies on a partially ordered set, topologies defined directly from the partial order. An important one, especially in connection with spectral spaces, is the Scott topology in which the closed sets of a partially ordered set are the lower or down sets that are also closed under taking directed suprema. The authors identify the conditions for the Scott topology to be spectral and include other considerations involving intrinsic topologies and spectral spaces. Chapter 10 treats the construction of infinite colimits in the category of spectral spaces.
From Chapter 8 through 13, with the exception of Chapter 10, various special classes of spectral spaces are considered, and these chapters will be of interest according to the readers’ background and research interests. These special classes are frequently coming from various other mathematical settings and represent the diverse appearance of frames. Chapter 8 considers special subclasses of spectral spaces such as Noetherian spaces arising as spectra of Noetherian rings and Heyting spaces arising as spectra of Heyting algebras. Chapter 9 treats localic spaces, spaces arising as spectra of locales. Chapter 11 considers spectral reflections of topological spaces and other relations between the category of spectral spaces and other topological categories and categories of partially ordered sets. Chapter 12 develops the theory of the Zariski spectrum of a ring, which as mentioned previously is always a spectral space, and Chapter 13 turns to the real spectrum and connections with real algebraic geometry. Chapter 14 closes the book with appearance of spectral spaces in Model Theory.
This quick overview should convince the reader that this book is a valuable resource for anyone seriously interested in the theory of spectral spaces and represents a substantial addition to the literature on the subject.