COHOMOLOGY THEORY OF GALOIS GROUPS HARUZO HIDA Contents 1 Extension of Modules 1 1 Extension groups 1 2 Extension functors 1 3 Cohomology groups of complexes 1 4 Higher extension groups 2 Group Cohomology Theory 2 1 Cohomology of finite groups 2 2 Tate cohomology groups 2 3 Continuous cohomology for profinite groups 2 4 Inflation and restriction sequences 3 Duality in Galois Cohomology 3 1 Class formation and duality of cohomology groups 3 2 Global duality theorems 3 3 Tate Shafarevich groups 3 4 Local Euler characteristic formula 3 5 Global Euler characteristic formula 4 Appendix Categories and Functors 4 1 Categories 4 2 Functors 4 3 Representability 4 4 Abelian categories References 2 2 5 8 10 15 16 20 23 30 33 34 40 44 51 54 57 57 58 58 61 63 In this notes we first describe general theory of forming cohomology groups out of an abelian category and a left exact functor Then we apply the theory to the category of discrete Galois modules and study resulting Galois cohomology groups At the end we would like to give a full proof of the Tate duality theorems and the Euler characteristic formulas of Galois cohomology groups which were essential in the proof of Fermat s last theorem by A Wiles Date March 27 2009 The author is partially supported by the following NSF grant DMS 0753991 1 GALOIS COHOMOLOGY 2 1 Extension of Modules In this section we describe basics of the theory of module extension functors and we relate it to group cohomology in the following section 1 1 Extension groups We fix a ring with identity which may not be commutative We consider the category of modules MOD Thus the objects of MOD are modules and Hom M N for two modules M and N is the abelian group of all linear maps from M into N When is a topological ring we would rather like to consider only modules with continuous action that is continuous modules or we might want to impose further restrictions like compactness or discreteness to the modules we study The totality of such modules makes a subcategory of MOD whose set of morphisms is made of continuous linear maps To accommodate such subcategories in an algebraic way without referring topology we consider subcategories C of MOD satisfying a set of conditions enough to define extension functors First of all since C is a subcategory of MOD Objects of C are made of a collection Ob C of modules We have a set of C morphisms HomC M N Hom M N for M N Ob C The identity map idM M M is in HomC M M for each object M g f HomC M L for f HomC M N and g HomC N L We impose C the following four conditions for our purpose C1 The set HomC X Y Hom X Y is a subgroup C2 If f X Y be a morphism in C Ker f and Coker f are both inside C C3 If X and Y are in C then the direct product X Y is in C C4 The zero module 0 is in C These conditions guarantee that C is an abelian category see 4 4 for formal definitions of abelian categories Hereafter we fix such a category C and work only in C We call linear map X Y for objects X and Y in C a C morphism if it is in HomC X Y Similarly an isomorphism which is also a C morphism is called a C isomorphism For a given pair of modules M and N in C we would like to know all modules E in C which fit into the following exact sequence in C N 0 N E M M 0 We call such E an extension in C of module M by N Two extensions E and E 0 are called isomorphic if we have a C isomorphism E E 0 making the following diagram commutative N E M k k N E 0 M We write E M N EC M N for the set of all isomorphism classes of extensions of M by N When C MOD we write E M N for EC M N Note that M N E M N so E M N 6 An extension E is called split if we have GALOIS COHOMOLOGY 3 a C morphism M M E such that M M idM Then E M N by e 7 M M e e M M e The map M is called a section of M This shows the class M N E M N is the unique split extension class If we have a projection N E N such that N N idN then again E M N by e 7 e N N e N N e because Ker N M by M in this case If Z and M N Z pZ for a prime p then we have at least two extensions Z pZ 2 and Z p2 Z in EZ Z pZ Z pZ Now we would like to study how E M N changes if we change M and N by their homomorphic image or source For a given C morphism M X and N X the fiber product T M X N is a module in C with the following property FP1 We have two projections T M and T N in C making the following diagram commutative M X N y N M y X FP2 If the following diagram in C is commutative 0 Y 0y M y N X then there exist a unique C morphism Y M X N such that 0 and 0 If two fiber products T and T 0 exist in C then we have T 0 T and 0 T T 0 satisfying FP2 for Y T 0 and Y T respectively Then idT and 0 T T satisfy PF2 for Y T and by the uniqueness 0 idT Similarly 0 idT 0 and hence T T 0 Thus the fiber product of M and N is unique in C up to isomorphisms if it exists It is easy to see that M X N m n M N m n satisfies the property FP1 2 for C MOD and the two projections M X N M and M X N N taking m n to m and n respectively For this choice y is given by 0 y 0 y M X N Thus fiber products exist in MOD If further and are C morphisms then by the existence of M N in C the above M X N in MOD is actually the kernel of which is therefore a member of C This shows the existence of the fiber product in C In functorial terms the fiber product represents the functor Y 7 0 0 HomC Y M N 0 0 GALOIS COHOMOLOGY 4 from C to SET S Let N E M be an extension in C For a C morphism M 0 M we look at the fiber product E 0 E M …