Algebraic spaces.
(Champs algébriques.)

*(English)*Zbl 0945.14005
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 39. Berlin: Springer. xii, 208 p. (2000).

The book under review is the first monograph, in the entire literature of algebraic geometry, that is exclusively devoted to the subject of algebraic stacks. The authors’ aim is to provide a systematic, comprehensive, rigorous and general exposition of the theory of algebraic stacks which, over the past thirty years, has become a powerful and indispensible tool for constructing and investigating moduli schemes in algebraic geometry. The concept of algebraic stack is still far from being generally familiar to, or even used by the majority of active researchers in the classification theory of algebro-geometric objects, and many features of algebraic stacks are only used intuitively, superficially, reluctantly, in an awkward ad-hoc manner, or sometimes even in a sloppy folklore style.

On the other hand, algebraic stacks arise quite naturally from A. Grothendieck’s functorial approach to algebraic geometry, and they proved their ubiquity in many concrete situations, long before they were formally introduced and recognized as objects that are just as important as schemes and sheaves themselves. According to Grothendieck’s re-foundation of algebraic geometry, the category of schemes can be interpreted in two ways, namely

(1) as full subcategory of the category of ringed spaces (i.e., from the geometric viewpoint), or

(2) as full subcategory of the category of covariant functors from the category of rings to the category of sets (i.e., from the functorial viewpoint).

The second point of view is particularly useful in those situations where schemes with certain universal properties are to be established (e.g., Hilbert schemes, Picard varieties, moduli schemes, etc.). Based on Grothendieck’s ideas and techniques developed along these lines, which had led him to introduce objects such as étale topologies, sheaves of categories, sites, and topoi, Mumford (1963), Deligne-Mumford (1969), and M. Artin (1974) extended the concept of a sheaf of categories to the one of an “algebraic stack” and used it in the moduli theory of algebraic curves and singularities. – More precisely, Deligne and Mumford used their algebraic stack of stable curves to construct a compactification of the moduli space of smooth curves of given genus \(g\), and M. Artin applied his version of an algebraic stack to create construction techniques for algebraic spaces and versal deformations of singularities.

The past twenty-five years have seen various applications of these approaches to moduli problems via algebraic stacks, and also a few attempts to develop a general theory of stacks and their intersection theory, but up to now, no systematic, comprehensive, or at least compiling treatise on that subject had emerged.

The authors of the book under review have filled this painful gap in a thorough, masterly and rewarding manner. They focus on precisely that approach to a theory of algebraic stacks, which has been initiated by Deligne-Mumford and M. Artin, and they present it in its most general, deep-going and methodically perfect form. – The text consists of nineteen chapters, one appendix, an up-to-date bibliography, and a carefully arranged terminological index.

Chapter 1 resumes briefly the basic facts from the theory of algebraic spaces. As for more details, the reader is referred to the original works of M. Artin, D. Knutson, and B. Moishezon. Chapter 2 provides the generalities on the 2-category of groupoids over a given base scheme \(S\). – This is used, in chapter 3, to introduce the category of \(S\)-stacks, \(S\) being a fixed base scheme, the notion of an 1-morphism between representable stacks, and the first properties of the so-called “gerbes” (i.e., stack-like fibre spaces). – Chapter 4 discusses diverse properties of algebraic stacks (à la Deligne-Mumford-Artin) and their 1-morphisms. – Chapter 5 turns to the topological aspects of algebraic stacks and their 1-morphisms, culminating in a stack-theoretic generalization of Chevalley’s theorem on constructible sets. – Chapter 6 deals with the local structure of algebraic stacks and provides various results on the existence of schemes over stacks.

Valuation criteria, specialisation properties, and special morphisms between algebraic stacks are the subject of chapter 7, while chapter 8 gives characterizations for algebraic spaces and for Deligne-Mumford stacks. Chapter 9 adds some remarks on the most important Grothendieck topologies, which are used, in chapter 10, to discuss M. Artin’s algebraicity criterion for general \(S\)-stacks. Chapter 11 gives the definitions and properties of algebraic points, residual sheaves, residual gerbes, and gerbes on algebraic stacks. Then, in chapter 12, the authors begin the study of general sheaves over the so-called smooth-étale site of an algebraic stack, and chapter 13 extends this study to quasi-coherent module sheaves on algebraic stacks. This includes such functorial properties of quasi-coherent modules, which lead to a general variant of the theory of faithfully flat descent for them.

Chapter 14 deals with local constructions over algebraic stacks such as generalized vector bundles and generalized projective bundles, on the one hand, and with the concepts of Picard stacks and the Deligne correspondence between Picard stacks, on the other hand. Chapter 15 is devoted to the study of functorial properties of coherent sheaves over locally noetherian algebraic stacks. The totality of the preceding concepts, methods and results is utilized in chapters 16 to 18, where the deep results in the theory of stacks are proved. Chapter 16 describes a (partial) generalization of “Zariski’s main theorem” to 1-morphisms of algebraic stacks, a variant of “Chow’s lemma” for Deligne-Mumford stacks of finite type, and some applications to algebraic spaces. Chapter 17 turns to a stack-theoretic generalization of the cotangent complex of a morphism. This extends L. Illusie’s construction for morphisms of ringed topoi to the 2-category of algebraic stacks and represents the probably deepest result and, without any doubt, the hardest part of the entire book. – Chapter 18 gives an account on constructible sheaves over the smooth-étale site of an algebraic stack, mainly with a view towards their functorial and cohomological properties. The material presented here generalizes, to a major extent, related results obtained by P. Deligne in 1977 [cf. P. Deligne, “Cohomologie étale”, in: SGA \(4{1\over 2}\), Lect. Notes in Math. 569 (1977; Zbl 0349.14008-14013)]. The concluding chapter 19 is devoted to a very brief survey on some more recent results in the theory of algebraic stacks. As for details and complete proofs, the authors refer to the corresponding original papers listed in the bibliography. However, their brief comments are very inspiring, and illustrate the steadily growing role of algebraic stacks in algebraic geometry, particularly in view of their applications to the intersection theory of cycles in moduli spaces. – Finally, in a short appendix, the authors have compiled (with full proofs) some complementary results on algebraic spaces, which were used in the course of the text.

Altogether, this comprehensive treatise on algebraic stacks is not at all easy to read. It is written in the abstract style that the matter dictates, and the reader is required to be reasonably familiar with Grothendieck’s EGA [“Élements de géométrie algébrique”] as well as with most of the volumes of “Seminaire de géométrie algébrique du Bois Marie” (SGA 1-7). The style is concise, but extremely elegant and efficient, and the reader, who is willing to work hard while studying the material, will profit a great deal from this effort. Also, the authors have provided the mathematical community with a standard source book on algebraic stacks, which is unique, so far, and is therefore unchallenged and indispensible.

On the other hand, algebraic stacks arise quite naturally from A. Grothendieck’s functorial approach to algebraic geometry, and they proved their ubiquity in many concrete situations, long before they were formally introduced and recognized as objects that are just as important as schemes and sheaves themselves. According to Grothendieck’s re-foundation of algebraic geometry, the category of schemes can be interpreted in two ways, namely

(1) as full subcategory of the category of ringed spaces (i.e., from the geometric viewpoint), or

(2) as full subcategory of the category of covariant functors from the category of rings to the category of sets (i.e., from the functorial viewpoint).

The second point of view is particularly useful in those situations where schemes with certain universal properties are to be established (e.g., Hilbert schemes, Picard varieties, moduli schemes, etc.). Based on Grothendieck’s ideas and techniques developed along these lines, which had led him to introduce objects such as étale topologies, sheaves of categories, sites, and topoi, Mumford (1963), Deligne-Mumford (1969), and M. Artin (1974) extended the concept of a sheaf of categories to the one of an “algebraic stack” and used it in the moduli theory of algebraic curves and singularities. – More precisely, Deligne and Mumford used their algebraic stack of stable curves to construct a compactification of the moduli space of smooth curves of given genus \(g\), and M. Artin applied his version of an algebraic stack to create construction techniques for algebraic spaces and versal deformations of singularities.

The past twenty-five years have seen various applications of these approaches to moduli problems via algebraic stacks, and also a few attempts to develop a general theory of stacks and their intersection theory, but up to now, no systematic, comprehensive, or at least compiling treatise on that subject had emerged.

The authors of the book under review have filled this painful gap in a thorough, masterly and rewarding manner. They focus on precisely that approach to a theory of algebraic stacks, which has been initiated by Deligne-Mumford and M. Artin, and they present it in its most general, deep-going and methodically perfect form. – The text consists of nineteen chapters, one appendix, an up-to-date bibliography, and a carefully arranged terminological index.

Chapter 1 resumes briefly the basic facts from the theory of algebraic spaces. As for more details, the reader is referred to the original works of M. Artin, D. Knutson, and B. Moishezon. Chapter 2 provides the generalities on the 2-category of groupoids over a given base scheme \(S\). – This is used, in chapter 3, to introduce the category of \(S\)-stacks, \(S\) being a fixed base scheme, the notion of an 1-morphism between representable stacks, and the first properties of the so-called “gerbes” (i.e., stack-like fibre spaces). – Chapter 4 discusses diverse properties of algebraic stacks (à la Deligne-Mumford-Artin) and their 1-morphisms. – Chapter 5 turns to the topological aspects of algebraic stacks and their 1-morphisms, culminating in a stack-theoretic generalization of Chevalley’s theorem on constructible sets. – Chapter 6 deals with the local structure of algebraic stacks and provides various results on the existence of schemes over stacks.

Valuation criteria, specialisation properties, and special morphisms between algebraic stacks are the subject of chapter 7, while chapter 8 gives characterizations for algebraic spaces and for Deligne-Mumford stacks. Chapter 9 adds some remarks on the most important Grothendieck topologies, which are used, in chapter 10, to discuss M. Artin’s algebraicity criterion for general \(S\)-stacks. Chapter 11 gives the definitions and properties of algebraic points, residual sheaves, residual gerbes, and gerbes on algebraic stacks. Then, in chapter 12, the authors begin the study of general sheaves over the so-called smooth-étale site of an algebraic stack, and chapter 13 extends this study to quasi-coherent module sheaves on algebraic stacks. This includes such functorial properties of quasi-coherent modules, which lead to a general variant of the theory of faithfully flat descent for them.

Chapter 14 deals with local constructions over algebraic stacks such as generalized vector bundles and generalized projective bundles, on the one hand, and with the concepts of Picard stacks and the Deligne correspondence between Picard stacks, on the other hand. Chapter 15 is devoted to the study of functorial properties of coherent sheaves over locally noetherian algebraic stacks. The totality of the preceding concepts, methods and results is utilized in chapters 16 to 18, where the deep results in the theory of stacks are proved. Chapter 16 describes a (partial) generalization of “Zariski’s main theorem” to 1-morphisms of algebraic stacks, a variant of “Chow’s lemma” for Deligne-Mumford stacks of finite type, and some applications to algebraic spaces. Chapter 17 turns to a stack-theoretic generalization of the cotangent complex of a morphism. This extends L. Illusie’s construction for morphisms of ringed topoi to the 2-category of algebraic stacks and represents the probably deepest result and, without any doubt, the hardest part of the entire book. – Chapter 18 gives an account on constructible sheaves over the smooth-étale site of an algebraic stack, mainly with a view towards their functorial and cohomological properties. The material presented here generalizes, to a major extent, related results obtained by P. Deligne in 1977 [cf. P. Deligne, “Cohomologie étale”, in: SGA \(4{1\over 2}\), Lect. Notes in Math. 569 (1977; Zbl 0349.14008-14013)]. The concluding chapter 19 is devoted to a very brief survey on some more recent results in the theory of algebraic stacks. As for details and complete proofs, the authors refer to the corresponding original papers listed in the bibliography. However, their brief comments are very inspiring, and illustrate the steadily growing role of algebraic stacks in algebraic geometry, particularly in view of their applications to the intersection theory of cycles in moduli spaces. – Finally, in a short appendix, the authors have compiled (with full proofs) some complementary results on algebraic spaces, which were used in the course of the text.

Altogether, this comprehensive treatise on algebraic stacks is not at all easy to read. It is written in the abstract style that the matter dictates, and the reader is required to be reasonably familiar with Grothendieck’s EGA [“Élements de géométrie algébrique”] as well as with most of the volumes of “Seminaire de géométrie algébrique du Bois Marie” (SGA 1-7). The style is concise, but extremely elegant and efficient, and the reader, who is willing to work hard while studying the material, will profit a great deal from this effort. Also, the authors have provided the mathematical community with a standard source book on algebraic stacks, which is unique, so far, and is therefore unchallenged and indispensible.

Reviewer: Werner Kleinert (Berlin)