Le groupe de Picard de l'anneau des polynômes à valeurs entières

✍ Scribed by Jean-Luc Chabert

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 777 KB
• Volume: 150
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(05)80057-2

✦ Synopsis

We study the Picard group of the ring B of integer-valued polynomials over a noetherian one dimensional domain A with finite residue fields. This problem can be reduced to the case of a local ring A; let m be the maximal ideal and A* the m-adic completion of A. There is a natural injective homomorphism from Pic(B) to the class group ~(A, :(A ))/:(A ) of locally constant functions from A to the group .~(A) of principal fractional ideals of A. This homomorphism is an isomorphism if and only if the spectrum of B/mB is homeomorphic with A* and that is equivalent to the integrity of A*.