On the construction of cousin complexes

Abstract

This research aims to construct Cousin complexes that exhibit geometric and algebraic properties. The geometric construction begins with a topological space X endowed with a decreasing filtration of closed subsets Zₙ ⊆ ⋯ ⊆ Z^2⊆ Z^1⊆ X , and employs sheaves and their sections with support in these subsets. By applying the global section functor Γ(X,-), its variants Γ_Z (X,-),Γ_(z_1/z_2 ) (X,-) and their derived functors Hⁱ(X,-),Hⁱ_z (X,-) and Hⁱ_(z_1/z_2 ) (X,-), long exact sequences in cohomology are obtained, from which the geometric version of the Cousin complex is constructed. The algebraic analogue is constructed using an A-module over a commutative ring A. Beginning with an A-module M and defining the initial morphisms d^(-2): M^(-2)→ M^(-1) with M^(-2)= 0 and M^(-1)= M, the construction proceeds inductively to produce a Cousin complex 0 → M →ᵈ^(-1) M^0→ᵈ^0 M^1→ᵈ^1⋯ .