Algebraic spaces and stacks (Q2797859)

In the course of the past fifty years, both algebraic spaces and stacks have turned into central, utmost important concepts in modern algebraic geometry. While the theory of stacks can be traced back to the late 1950s, when A. Grothendieck developed his ideas on effective descent data and obstructions to the existence of fine moduli spaces, the notion of algebraic space was introduced by M. Artin around 1970 with a view toward algebraic deformation theory and specific local moduli problems. Roughly speaking, the category of stacks extends the category of algebraic spaces, whereas algebraic spaces already form a suitable generalization of ordinary algebraic schemes. In this context, both algebraic spaces and stacks provide an appropriate framework to formalize many important constructions in moduli theory beyond the smaller category of schemes or varieties, respectively.NEWLINENEWLINENEWLINEThe book under review is the first systematic textbook providing an introduction to algebraic spaces and stacks simultaneously. As such the text is largely self-contained, and basically geared toward graduate students and researchers with a profound knowledge of the principles of advanced algebraic geometry at the level of a graduate course. Actually, this textbook grew out of a course the author taught at Berkeley in 2007, and also therefore it differs from the various, partly very abstract and sophisticated monographs on the subjects. In fact, as the author points out, several of these standard texts served as main sources for the present (more down-to-earth) primer, and they are referred to for further, more detailed reading throughout the book. Among those fundamental references are such standard texts as [\textit{M. Artin}, Invent. Math. 27, 165--189 (1974; Zbl 0317.14001); \textit{D. Knutson}, Algebraic spaces. York: Springer-Verlag (1971; Zbl 0221.14001); \textit{G. Laumon} and \textit{L. Moret-Bailly}, Champs algébriques. Berlin: Springer (2000; Zbl 0945.14005)], and [\textit{A. Vistoli}, ``Grothendieck topologies, fibered categories and descent theory'', in: B. Fantechi (ed.) et al., Fundamental algebraic geometry: Grothendieck's FGA explained. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1085.14001)].NEWLINENEWLINENEWLINEAs for the contents of the current book, there are thirteen chapters and an appendix.NEWLINENEWLINENEWLINEChapter 1 provides some necessary background material from basic algebraic geometry, including flatness properties, étale and smooth morphisms, schemes from the functorial point of view as well as Hilbert and Quot schemes.NEWLINENEWLINENEWLINEChapter 2 develops some foundational material on Grothendieck topologies and sites, with focus on their sheaves and sheaf cohomology. In this chapter, the author also discusses simplicial topoi as a crucial technical tool in the sequel.NEWLINENEWLINENEWLINEChapter 3 discusses the basic formalism of fibered categories, with particular emphasis on categories fibered in groupoids. This provides the necessary framework for defining the notion of stack later on.NEWLINENEWLINENEWLINEChapter 4 turns to descent theory, together with some basic facts on torsors, principal homogeneous spaces, and the conditions for a fibered category to be a so-called stack.NEWLINENEWLINENEWLINEChapter 5 introduces algebraic spaces and describes some of their basic properties. This is done in a slightly more general form as in the classical approaches, and appears to be somewhat more natural than the latter. NEWLINENEWLINENEWLINEChapter 6 deals with some basic topics concerning invariant theory and quotients of schemes by finite flat equivalence relations. This is crucial for developing the theory of algebraic spaces beyond the basics, on the one hand, and for the discussion of coarse moduli spaces on the other. Also, it is explained how certain algebraic spaces can be stratified by ordinary schemes. NEWLINENEWLINENEWLINEChapter 7 is devoted to the study of quasi-coherent sheaves on algebraic spaces. This includes affine morphisms and Stein factorization, Chow's lemma for algebraic spaces, and the finiteness of cohomology in this context.NEWLINENEWLINENEWLINEChapter 8 gives the basic definitions and properties concerning algebraic stacks (Artin stacks) and Deligne-Mumford stacks, together with various important, concrete examples such as the moduli stack \(\mathcal{M}_g\) of curves of genus \(g\), group quotients of schemes, and others.NEWLINENEWLINENEWLINEChapter 9 turns to the study of quasi-coherent sheaves on algebraic stacks, thereby working with the so-called lisse-étale site as in the standard monograph on algebraic stacks by Laumon and Moret-Bailly [loc. cit.].NEWLINENEWLINENEWLINEChapter 10 discusses several fundamental constructions and examples in the theory of stacks, with particular focus on proper morphisms of stacks, root stacks, and a stack-theoretic version of relative Spec and Proj.NEWLINENEWLINENEWLINEChapter 11 describes the construction of coarse moduli spaces for algebraic stacks with finite diagonal. The crucial existence theorem in this context is the famous Keel-Mori theorem [\textit{S. Keel} and \textit{S. Mori}, Ann. Math. (2) 145, No. 1, 193--213 (1997; Zbl 0881.14018)], a proof of which is given following an unpublished manuscript of B. Conrad, as the author acknowledges in the introduction to this chapter. Applications to Deligne-Mumford stacks and the local structure of coarse moduli spaces are further contents of this part.NEWLINENEWLINENEWLINEThe following last two chapters discuss further, more specific applications of the general theory of algebraic spaces and stacks.NEWLINENEWLINENEWLINEChapter 12 deals with gerbes, torsors, and their relations to Azumaya algebras, certain cohomology classes of schemes, and Brauer groups of schemes, whereas Chapter 13 discusses various moduli stacks of curves such as the moduli stack of elliptic curves, the Deligne-Mumford compactification \(\overline{\mathcal{M}}_g\) of the moduli stack \(\mathcal{M}_g\) of curves of genus \(g\), and the Deligne-Mumford stack of stable maps of schemes.NEWLINENEWLINENEWLINEFinally, the appendix provides a glossary of basic facts of elementary category theory.NEWLINENEWLINENEWLINEApart from the utmost lucid, detailed, systematic and versatile presentation of the fundamental theories of algebraic spaces, stacks, and their applications, the numerous exercises at the end of each chapter provide a wealth of additional, highly instructive material with respect to further developments of these theories.NEWLINENEWLINENEWLINEAll together, this is an absolutely unique and excellent textbook on modern, highly advanced and abstract topics in algebraic geometry, which has no equal in the current literature, and which finally fills a long-continued gap therein.