Relative patching properties

Abstract

INTRODUCTION A diagram of commutative rings R ------+ R, I I R,- R, BRR2=R’ is said to have the Milnor patching property if the projective modules over R are precisely those modules that may be constructed by giving projective modules P, and over R, and R2, and a “patching” over R, @ R R2. If we write P(R) for the category of projective R-modules, then this amounts to saying that P(R)= P(R,) x rCR,) IFo(R,). Diagrams having the Milnor patching property were investigated by S. Landsburg in [S, 91. Of course, Milnor’s original result says that if one of the morphisms R, -+ R’ or R, + R’ is surjective, then the above diagram has Milnor patching property. It is well known that any diagram (*) which has the Milnor patching property induces exact sequences of the form K,(R) + K,(R,)OK,(R,) -+ K,(R’) -+ G(R) --f &(R,)O&(&) -+ KdR’) resp. 1 + U(R) -+ U(R,) x U(R,) + U(R’) + Pit(R) -+ Pic(R,) x Pic(R,) -+ Pic(R’). On the other hand, if we consider Krull domains, it is natural to ask for similar exact sequences, but involving reflexive instead of projective modules and class groups instead of Picard groups. It appears that in 474