A note on cohomology and algebraic geometric codes on the curves over rings

Let A be a local Artinian ring with residue field k(A). Let X be a curve over A and let be X′ = X ×_spec A spec k(A) the fiber of X over k(A). Consider ℒ an invertible sheaf on X and ℒ ′ = ϕ*ℒ ∈ Pic(X′), where ϕ : X′ → X is the natural map. Let C and C′ be the algebraic geometric codes constructed using the groups of cohomology Γ(X, ℒ) and Γ(X′, ℒ ′) respectively. In this note, we first give the complete relation between Γ(X, ℒ) and Γ(X′, ℒ ′) without any condition and finally, we provide relations between C ⊗ _A k(A) and C′ using exact sequences and dimensional theory. Therefore we extend, some results of Walker [18, 20] giving the characterization of WAG codes over rings.