Galois cohomology and class field theory (Q2833183)

The book under review contains eighteen chapters distributed on four parts. It begins by a foreword followed by some notations and conventions, and it ends by an appendix and a list of references. The author, as stated in the foreword, was motived by two shortcomings in the existing literature:NEWLINE {\parindent=0.7cm\begin{itemize}\item[--] There is no book, in French, that deals in detail with the theory of global class fields. \item[--] The English books that give proofs to Poitou-Tate theorems admit, in general, some other results to complete these proofs. NEWLINENEWLINE\end{itemize}} NEWLINEThe goal of the book is to gather in one work, by giving complete proofs, the following notions: {\parindent=0.7cm\begin{itemize}\item[--] Cohomology bases, which are the aim of the part I that contains six chapters: from 1 to 6. \item[--] Local class field theory, it is part II that deals with it, this part contains five chapters: from 7 to 11. Chapter 7 is devoted to recall local fields properties. \item[--] Global class field theory, it is part III that deals with it, this part contains four chapters: from 12 to 15. Chapter 12 is devoted to recall global fields properties. \item[--] Poitou-Tate theorems, it is part IV that deals with them, this part contains three chapters: from 16 to 18. NEWLINENEWLINE\end{itemize}} NEWLINEEach chapter is ended by some exercises to give some applications to its results.NEWLINENEWLINE NEWLINEPart I: Generalities. In the first part, the author gives generalities on cohomological groups and Galois cohomology. He organizes this section as follows.NEWLINENEWLINE NEWLINEChapter 1 presents the basic properties of the cohomology of finite groups: If \(G\) is a finite group and \(A\) is a \(G\)-module, then the \(n\)-th cohomology groups of \(G\) with coefficients in \(A\) are defined as the right derived functors of the functor \(A\mapsto A^G\) and noted by \(H^n(G, A)\). The author also remarks that we can calculate cohomological groups with cochain complexes. He provides the most important basic theorem (Shapiro's lemma, inflation-restriction and Hochschild-Serre) and he finishes the chapter by describing an important relationship between restriction and corestriction maps.NEWLINENEWLINE NEWLINEChapter 2 introduces the Tate cohomology groups and group cohomology of finite cyclic groups: Let \(G\) be a finite group and \(A\) a \(G\)-module and let \(N\) be the endomorphism defined on \(A\) by \(N(x)=\sum_{g\in G}g\cdot x\). \(N\) induces a map \(\hat{N} : A_G \rightarrow A^G\), where \(A_G\) is the group of \(G\)-coinvariants of \(A\). For any \(i\in\mathbb Z\), we define the \(i\)th Tate cohomology group by NEWLINE\[NEWLINE\hat{H}^i(G, A) =\begin{cases} H^i (G, A), & \text{ if }i\geq 1; \\ \hat{H}^0(G, A), & \text{ if }i=0; \\ \hat{H} _0(G, A), & \text{ if }i=-1; \\ H_{-i-1}(G, A), & \text{ if }i\leq -2;\end{cases}NEWLINE\]NEWLINE where \( \hat{H} ^0(G,A)\) (resp., \(\hat{H} _0(G,A)\)) denotes the cokernel (resp., kernel) of the map \(\hat{N}\) and \(H_i(G, A)\) is the \(i\)th homology group of \(G\) (the right derived functors of the functor \(A\mapsto H_0(G, A)\)). In class field theory, the restriction \(H_1(G, \mathbb Z) \rightarrow H_1(H, \mathbb Z)\), where \(H\) is a subgroup of finite index in \(G\), is important and coincides with the transfer map \(\mathrm{Ver}: G/G'\rightarrow H/H'\). If we replace \(G\) by \(G'\) and \(H\) by \(G'\) we find the group theoretical principal ideal theorem. If \(G\) is a finite cyclic group, the author introduces the concept of the Herbrand quotient and cup-product. In this case we have isomorphisms \( \hat{H}^ {2i} (G,A)\cong \hat{H}^ {0} (G,A) \) and \( \hat{H}^ {2i+1} (G,A)\cong \hat{H}^ {1} (G,A) \) for all \(i\in\mathbb Z\).NEWLINENEWLINE Chapter 3 begins by highlighting the interest in studying the cohomology of \(p\)-Sylow subgroups of a finite group \(G\). For example the \(G\)-module \(A\) is cohomologically trivial if and only if it is also for each \(p\)-Sylow subgroup \(G_p\) of \(G\). This chapter ends by stating and demonstrating the Tate-Nakayama Theorem shown by \textit{T. Nakayama} [Ann. Math. (2) 65, 255--267 (1957; Zbl 0198.07404)] as a generalization of a Tate's theorem [\textit{J. Tate}, Ann. Math. (2) 56, 294--297 (1952; Zbl 0047.03703)].NEWLINENEWLINE NEWLINEChapters 4 and 5: A topological group which is the projective limit of finite groups, each given the discrete topology, is called a profinite group. This is the definition given by \textit{J.-P. Serre} [Galois cohomology. Berlin: Springer (1997; Zbl 0902.12004)], we can say that the author tried to summarize the first three sections of the first chapter of the last book. In fact if \(G\) is a profinite group, these two chapters give the necessary results for further (subgroups, indices, pro-\(p\)-groups, Sylow \(p\)-subgroups, discrete \(G\)-modules, cohomology of discrete \(G\)-module, the cohomological \(p\)-dimension of a profinite group and the strict cohomological \(p\)-dimension). It is important to note that this time the author preferred to define the cohomology group of \(G\) in a discrete \(G\)-module \(A\) by \(H^n(G, A) = Z^n(G, A)/B^n(G, A)\), where \(Z^n(G, A)\) (resp. \(B^n(G, A)\)) are the \(n\)-cocyles of \(G\) (resp. the \(n\)-coboundaries of \(G\)) and proved that the cohomology of \(G\) in \(A\) is the inverse limit of cohomology of certain finite quotients: \(H^n(G, A) =\varinjlim H^n(G/U, A^U)\) for each \(n\geq 0\), where \(U\) runs over all open normal subgroups of \(G\).NEWLINENEWLINE NEWLINEChapter 6: In this chapter, the author introduces the Galois cohomology as a particular case of the cohomology of a profinite group by considering the profinite group \(G_k=\mathrm{Gal}(\overline{k}/k)\) where \(\overline{k}\) is a separable closure of some field \(k\), he studies then some properties of this notion. He cites later a result of Artin-Schreier, Hilbert theorem 90, and a theorem of Kummer. The author defines the Brauer group \(Br(k)\) as follows: \(H^ 2(G_k, \overline{k}^*)\) and he recall some of its properties. He also treats the cohomological dimension of a field \(k\). He ends the chapter by studying fields of type \(C_1\).NEWLINENEWLINENEWLINEPart II: Local fields. This part is reserved to the cohomological theory of the class field for local fields which are \(p\)-adic field (\(\mathbb Q_p\) extensions) and the fields of Laurent's series on the finite fields.NEWLINENEWLINE NEWLINEChapter 7: In this chapter, the author recalls the standard results about the local fields, these topics are well detailed in \textit{J.-P. Serre}'s book ``Corps locaux.'' Second edition. Paris: Hermann (1968); first ed. (1962; Zbl 0137.02601) (for the English version, see [Local fields. New York etc.: Springer-Verlag (1979; Zbl 0423.12016)]). He focuses on the following notions: discrete valuation rings, complete fields for a discrete valuation, extensions of complete fields, Galois theory for a complete field and structure of a unit group of a complete field.NEWLINENEWLINE NEWLINEChapter 8 begins the local class field theory which has as a goal the description of the abelian extensions of local fields and their Galois groups. The first step is to calculate the Brauer group of a local field which is the main object of the chapter. For this, the author starts by citing the local class fields axioms, then he calculates the Brauer group of a complete field for a discrete valuation, and he ends the chapter by giving the cohomological dimension of a local field.NEWLINENEWLINE NEWLINEChapter 9 begins the study of the abelian Galois group \(\mathrm{Gal}(K^{\mathrm{ab}}/K)\) of a local field, then the author defines the reciprocity application and gives some of his properties by using the famous Tate-Nakayama theorem. The existence theorem for a \(p\)-adic field is given in the end.NEWLINENEWLINE Chapter 10: The main aim of this chapter is to give a proof of the Tate local duality theorem for a \(p\)-adic field by combining the duality with the results of Chapter 9. The author starts the chapter by giving some properties of a dual module of a profinite group having a finite cohomological dimension. He then states the local duality theorem which allows him to reproof the existence theorem.NEWLINENEWLINE NEWLINEChapter 11: The main task of this chapter is to give an explicit description, by using formal groups, of the maximal abelian extension of a local field. This allows the author to give an easy proof to the existence theorem of a \(p\)-adic field. Moreover, he obtains some explicit formulas to calculate the local symbols \((a, L/K)\) where \(a\in L^*\)and \(L\) is an abelian extension of a local field \(K\). He starts by recalling the abelian law of a formal group of dimension \(1\), then he defines the Lubin-Tate formal group law associated to a formal series. Thereafter, he calculates the reciprocity application and proves the existence theorem for a local field in the general case.NEWLINENEWLINE NEWLINEPart III: Global class field theory. This part is reserved to study the global class field theory, which is the counterpart to number fields and the function fields. The approach uses the results of Part I, in particular the Brauer group, and the local results viewed in Part II.NEWLINENEWLINE NEWLINEChapter 12 is devoted to recall basic notions of the global fields, namely: Dedekind rings, class group, definition of global field, the normalized absolute value, Galois extension of a global field, ideles and the strong approximation theorem, adeles, Minkowski-Hermite theorem and some complement about function fields.NEWLINENEWLINE Chapter 13: the author investigates the cohomological properties of the idele group and the idele class group of a global field. The goal is to prove an analogue to the local class field axiom (proposition 8.2 chapter 8) where the idele class group will replace the multiplicative group of a local field. After defining the cohomology of the idele group, he proves that the Herbrand quotient is \(n=[K:k]\), where \(K\) is a finite Galois extension of a global field \(k\), by proving the first and the second inequalities. He first establishes general results about Kummer extensions, which allow him to prove the first inequality and deduces the global class field axiom. He ends the chapter by proving the class field axiom for function fields.NEWLINENEWLINE NEWLINEChapter 14 begins by the computation of the Brauer group of a global field by using similar method to that used for the local case, the author just replaces unramified extensions by cyclotomic extensions. And then he proves the global reciprocity law associated to the norm symbol. For this, he uses local calculations coming from the Lubin-Tate construction for \(\mathbb Q_p\).NEWLINENEWLINE NEWLINEChapter 15 finishes the description of the abelian Galois group of a global field begun in the previous chapters thanks to an existence theorem similar to that of the local case. Then he explains the old formulations about ideals and ray class fields. He first establishes that the global class field axiom also occurs for any abelian extension (not only for cyclic extensions). He then proves the global existence theorem which is analogous to theorem 11.2. Later he states the existence theorem for function fields. He continues by defining the ray class field, then uses the existence theorem to connect abelian extensions of a global field and ray class fields. He ends the chapter by treating the Galois group of restricted ramification.NEWLINENEWLINE NEWLINEPart IV: Arithmetic duality theorems. In this part, the author deals with arithmetic duality theorems, the goal is to prove Poitou-Tate theorems which are essential tools in the major questions of modern number theory.NEWLINENEWLINE NEWLINEChapter 16 is reserved to the formation of classes which contains a sort of abstract theory of duality. It develops, say, the abstract class field theory. The main object is the general theorem of duality, this allows to generalize the local duality theorem and will be an important tool to prove Poitou-Tate theorems about duality of global fields. The author starts by defining the class formation notion, and gives some properties. He then recalls some standard results of spectral sequences. After that, he establishes the duality theorems for class formations. He ends the chapter by quoting the existence theorem of the duality theorem for a \(p\)-adic field.NEWLINENEWLINE Chapter 17 states and proves the Poitou-Tate difficult theorems which are duality theorems of the Galois cohomology of the global fields. The author proves these theorems by using duality theorems previously stated (theorem 16.2) for a class \(P\)-formations. He starts by quoting some properties of the class \(P\)-formation, he then defines the product \(P^i_S(M)\) (definition 17.5). Next, he enunciates Poitou-Tate theorems, and he finishes the chapter by proving them.NEWLINENEWLINE NEWLINEChapter 18 gives some applications of the above results. He begins by some annihilation results of some groups, and he ends the chapter by talking about a result of the strict cohomological dimension of a number field.NEWLINENEWLINEReviewer's remark: The book contains a list of 55 references, but it lacks other important references like the famous book of \textit{G. Gras} [Class field theory. From theory to practice. Berlin: Springer (2003; Zbl 1019.11032)].