On Spectral Sequences for Iwasawa Adjoints \`a la Jannsen for Families

TL;DR

This paper explores spectral sequences related to Iwasawa modules and cohomology, extending existing frameworks and emphasizing their connection to local cohomology and duality theories in arithmetic geometry.

Contribution

It provides new insights into spectral sequences for Iwasawa modules, addressing gaps in Jannsen's spectral sequences for families of representations.

Findings

Established connections between spectral sequences and local cohomology.

Extended existing spectral sequence frameworks to broader classes of Iwasawa modules.

Clarified the role of duality in the structure of these spectral sequences.

Abstract

In \citenospec{MR1097615} several spectral sequences for (global and local) Iwasawa modules over (not necessarily commutative) Iwasawa algebras (mainly of $p$-adic Lie groups) over $\Z_p$ are established, which are very useful for determining certain properties of such modules in arithmetic applications. Slight generalizations of said results can be found in \citenospec{MR2333680} (for abelian groups and more general coefficient rings), \citenospec{MR1924402} (for products of not necessarily abelian groups, but with $\Z_p$-coefficients), and \citenospec{MR3084561}. 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 \`a la Grothendieck (and Tate for duality groups).