On the class group of a graded domain

✍ Scribed by S. El Baghdadi; L. Izelgue; S. Kabbaj

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 130 KB
• Volume: 171
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(01)00146-3

✦ Synopsis

This paper studies the class group of a graded integral domain R = ∈ R . 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[ ]) ∼ = Cl(A) ⊕ HCl(K[ ]) if and only if A[ ] 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 : Cl(A) → Cl(A[ ]), that is, Cl(A[ ]) = Cl(A) if and only if A and are integrally closed and Cl( ) = 0. Our results are backed by original examples.