On the star class group of a pullback

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, φ:T→k the natural projection, and let R=φ−1(D). For each star operation * on R, we define the star operation *φ on D, i.e., the “projection” of * under φ, 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*φ(D)→Cl*(R)→Cl(*)T(T)→0 is split exact. In particular, when *=tR, we deduce that the sequence 0→CltD(D)→CltR(R)→Cl(tR)T(T)→0 is split exact. The relation between (tR)T and tT (and between Cl(tR)T(T) and CltT(T)) is also investigated.