Cohomology for Linearized Boundary-Value Problems on General Riemannian Structures

We develop a framework for casting the solvability and uniqueness conditions of linearized geometric boundary-value problems in cohomological terms. The theory is designed to be applicable without assumptions on the underlying Riemannian structure and provides tools to study the emergent cohomology explicitly. To achieve this generality, we extend Hodge theory to sequences of Douglas--Nirenberg systems that interact via Green's formulae, overdetermined ellipticity, and a condition we call the order-reduction property, replacing the classical requirement that the sequence form a cochain complex. This property typically arises from linearized constraints and gauge equivariance, as demonstrated by several examples, including the linearized Einstein equations with sources, where the cohomology encodes geometric and topological data.