On spectral sequences for Iwasawa adjoints à la Jannsen for families

Abstract

Jannsen established several spectral sequences for (global and local) Iwasawa modules over (not necessarily commutative) Iwasawa algebras (mainly of $p$-adic Lie groups) over $\mathbb{Z}_p$, which are very useful for determining certain properties of such modules in arithmetic applications. Slight generalizations of said results are also obtained by Nekovář (for abelian groups and more general coefficient rings), by Venjakob (for products of not necessarily abelian groups, but with $\mathbb{Z}_p$-coefficients), and by Lim-Sharifi. Unfortunately, some of Jannsen's spectral sequences for families of representations as coefficients for (local) Iwasawa cohomology are still missing. We explain and follow the philosophy that all these spectral sequences are consequences or analogues of local cohomology and duality à la Grothendieck (and Tate for duality groups).