The Class Group of Pullbacks

Abstract

Let T be an integral domain, I a nonzero ideal of T, ϕ: T → T/I the natural ring epimorphism, D an integral domain which is a proper subring of T/I, and R = ϕ⁻¹(D). Let k = qf(D) be the quotient field of D such that k ⊆ T/I, S = ϕ⁻¹(k), and k* = k\{0}. We prove that if k ⊊ T/I and if the map φ: U(T) → U(T/I)/U(D), given by u ↦ ϕ(u)U(D), is surjective, then Cl(R) = Cl(D) ⊕ Cl(S) and Pic(R) = Pic(D) ⊕ Pic(S). We also prove that the map :U(T) → U(T/I)/k*, given by u ↦ ϕ(u)k*, is surjective if and only if {x ∈ T | ϕ(x) ∈ U(T/I)} = U(T)(S\I), if and only if the map λ:Cl(S) → Cl(T), defined by [H] ↦ [(HT) _t ], is injective.

Key Words:

• A + XB[X]
• A + X ² B[X]
• (t-)Class group
• Picard group
• Pullback

2000 Mathematics Subject Classification:

• 13A15
• 13C20
• 13G05

ACKNOWLEDGMENT

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (Grant No. R05-2003-000-10180-0).

Notes

Communicated by J. Kuzmanovich.