On the class group of a graded domain

This paper studies the class group of a graded integral domain R = ⊕_α∈Γ R_α. We prove that if the extension R₀ ⊂ R is inert, then Cl(R) = HCl(R) if and only if R is almost normal. As an application, we state a decomposition theorem for class groups of semigroup rings, namely, Cl(A[Γ]) ≅ Cl(A) ⊕ HCl(K[Γ]) if and only if A[Γ] is integrally closed. This recovers the well-known results developed for the classic contexts of polynomial rings and Krull semigroup rings. Further, we obtain an interesting result on the natural homomorphism φ: Cl(A) → Cl(A[Γ]), that is, Cl(A[Γ]) = Cl(A) if and only if A and Γ are integrally closed and Cl(Γ) = 0. Our results are backed by original examples.