# UCLA MATH 205C - COHOMOLOGY THEORY OF GALOIS GROUPS (63 pages)

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## COHOMOLOGY THEORY OF GALOIS GROUPS

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## COHOMOLOGY THEORY OF GALOIS GROUPS

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Lecture Notes

Pages:
63
School:
University of California, Los Angeles
Course:
Math 205c - Number Theory

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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

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