Central algebraic cohomology of Galois gerbes

Abstract

We study central algebraic cohomology of Galois gerbes over local fields of characteristic zero and number fields, with coefficients in a connected reductive group and an algebraic central subgroup, not necessarily finite or connected. For gerbes admitting a comparison morphism from Kaletha's rigid gerbe, we obtain fiber product formulas for abelianized cohomology in terms of algebraic fundamental groups of finite central quotients and homomorphisms from the band to the central subgroup. Such comparisons exist whenever the band is of finite type. Over number fields, independently of the comparison hypothesis, the full cohomology set is recovered from its abelianization and its real localizations through a cartesian square. Over non-Archimedean local fields and totally imaginary number fields, abelianization is bijective. We also specialize the formulas to Kottwitz's local and global gerbes.