Selmer Complexes

This work gives a unified treatment of commutative Iwasawa theory centered on a small number of sufficiently general principles. It should be regarded as a ‘Grothendification’ of the subject into a landscape of arithmetic geometry. The contents of each chapter are as follows. The background material from homological algebra is collected together in chapter 1. In chapter 2, the formalism of Grothendieck duality theory over complete local rings is covered. Chapter 3 contains a description of the formalism of continuous cohomology and chapter 4 deals with finiteness results for continuous cohomology of pro-finite groups. Results for big Galois representations from the classical duality results for Galois cohomology of finite Galois modules over local and global fields are deduced in chapter 5.

In chapter 6, the author introduces Selmer complexes in an axiomatic setting and proves a duality theorem for them. A study of generalization of unramified cohomology is contained in chapter 7. The next two chapters contain duality results in Iwasawa theory (deduced from those over number fields) and their applications to classical Iwasawa theory. Various incarnations of generalized Cassels-Tate pairings are constructed and studied in chapter 10. The last two chapters contain a construction of generalized p-adic height pairing and applications of ideas developed in chapter 10 to big Galois representations arising from Hida families of Hilbert modular forms, and to anticyclotomic Iwasawa theory of CM points on Shimura curves.

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