On the star class group of a pullback

Abstract

For the domain R arising from the construction T, M, D, we relate the star class groups of R to those of T and D. More precisely, let T be an integral domain, M a nonzero maximal ideal of T, D a proper subring of k := T/M, phi: T -> k the natural projection, and let R = phi(-1) (D). For each star operation * on R, we define the star operation *phi on D, i.e., the "projection" of * under phi, and the star operation (*)(T) on T, i.e., the "extension" of * to T. Then we show that, under a mild hypothesis on the group of units of T, if * is a star operation of finite type, then the sequence of canonical homomorphisms 0 -> Cl*phi (D) -> Cl* (R), Cl-(*)T -> (T) -> 0 is split exact. In particular, when * = t(R), we deduce that the sequence 0 -> Cl-tD (D) -> Cl-tR (R) -> Cl-(tR)T (T) -> 0 is split exact. The relation between (t(R))(T) and t(T) (and between Cl-(tR)T (T) and Cl-tT (T)) is also investigated. (c) 2005 Elsevier Inc. All rights reserved.