On the class group of a graded domain

This paper studies the class group of a graded integral domain R=circled plusαset membership, variantΓ Rα. We prove that if the extension R0subset ofR 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])congruent withCl(A)circled plusHCl(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.