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**Cartiertheorie kommutativer formaler Gruppen. Unter Mitarb. von Harry Reimann.**
*(German)*
Zbl 0578.14039

Teubner-Texte zur Mathematik, Bd. 68. Leipzig: BSB B. G. Teubner Verlagsgesellschaft. 124 S. M 13.00 (1984).

In this book, the Cartier theory of commutative formal groups is presented. Cartier’s theory is based on the concept of deformation theory, and its seems simpler and more general than other theories. The book in intended for students and mathematicians who are interested in algebraic geometry, number theory or arithmetical geometry. Some background on commutative algebra is required. - Throughout the book, K is a commutative ring with identity element.

Chapter I is the introduction to the theory of commutative formal group laws. Discussions are centered around formal power series. Standard theorems and properties are proved. - In chapter II, formal groups are defined as functors from the category of nilpotent, commutative K- algebras to the category of abelian groups: \(\tilde G:\) Nil\({}_ K\to (Ab)\). Representable, prorepresentable functors, and bigebras associated to formal groups, among other things, are discussed. - Chapter III is devoted to the proofs of the two main theorems of Cartier’s theory on commutative formal groups. E.g., the second main theorem: Let \(H_ 1\) and \(H_ 2\) be formal groups and let \(M_ 1\) and \(M_ 2\) be the corresponding Cartier modules. Then there is a canonical bijection \(Hom(H_ 1,H_ 2)\cong Hom_{{\mathbb{E}}}(M_ 1,M_ 2)\) where \({\mathbb{E}}\) is the Cartier ring of K. - In chapter IV, local Cartier theory is discussed in great detail: p-typical formal groups, formal groups of Wittvectors, local version of the main theorems, among others, are illustrated. - In chapter V, as the first step toward classifications of formal groups, the concept of isogenies of formal groups is introduced. In particular, for K an algebraically closed field of finite characteristic p, the classification theorem for p-divisible groups is presented. Deformation of p-divisible groups is also discussed. - In the final chapter VI, structure theorems for isogeny classes of p-divisible groups and formal groups over perfect fields of finite characteristic are presented. Crystals, isocrystals and their Newton polygons which play essential roles here are discussed.

Chapter I is the introduction to the theory of commutative formal group laws. Discussions are centered around formal power series. Standard theorems and properties are proved. - In chapter II, formal groups are defined as functors from the category of nilpotent, commutative K- algebras to the category of abelian groups: \(\tilde G:\) Nil\({}_ K\to (Ab)\). Representable, prorepresentable functors, and bigebras associated to formal groups, among other things, are discussed. - Chapter III is devoted to the proofs of the two main theorems of Cartier’s theory on commutative formal groups. E.g., the second main theorem: Let \(H_ 1\) and \(H_ 2\) be formal groups and let \(M_ 1\) and \(M_ 2\) be the corresponding Cartier modules. Then there is a canonical bijection \(Hom(H_ 1,H_ 2)\cong Hom_{{\mathbb{E}}}(M_ 1,M_ 2)\) where \({\mathbb{E}}\) is the Cartier ring of K. - In chapter IV, local Cartier theory is discussed in great detail: p-typical formal groups, formal groups of Wittvectors, local version of the main theorems, among others, are illustrated. - In chapter V, as the first step toward classifications of formal groups, the concept of isogenies of formal groups is introduced. In particular, for K an algebraically closed field of finite characteristic p, the classification theorem for p-divisible groups is presented. Deformation of p-divisible groups is also discussed. - In the final chapter VI, structure theorems for isogeny classes of p-divisible groups and formal groups over perfect fields of finite characteristic are presented. Crystals, isocrystals and their Newton polygons which play essential roles here are discussed.

Reviewer: N.Yui

### MSC:

14L05 | Formal groups, \(p\)-divisible groups |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14D15 | Formal methods and deformations in algebraic geometry |