Relative patching properties (Q1070290)

A diagram of commutative rings \(R\to R_ 1\to R',\quad R\to R_ 2\to R'\) with \(R'=R_ 1\otimes_ RR_ 2\) 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 \(P_ 1\) and \(P_ 2\) over \(R_ 1\) and \(R_ 2\), and a ''patching'' over \(R_ 1\otimes_ RR_ 2\). If we write \({\mathbb{P}}(R)\) for the category of projective R-modules, then this amounts to saying that \({\mathbb{P}}(R)={\mathbb{P}}(R_ 1)\times_{{\mathbb{P}}(R')}{\mathbb{P}}(R_ 2)\). It is well known that if one of the morphisms \(R_ 1\to R'\) or \(R_ 2\to R'\) is surjective then the above diagram has the Milnor patching property and that any diagram (*) which has the Milnor patching property induces exact sequences of the form \[ K_ 1(R)\to K_ 1(R_ 1)\oplus K_ 1(R_ 2)\to K_ 1(R')\to K_ 0(R)\to K_ 0(R_ 1)\oplus (K_ 0(R_ 2)\to K_ 0(R'), \] respectively \[ 1\to U(R)\to U(R_ 1)\times U(R_ 2)\to U(R')\to Pic(R)\to Pic(R_ 1)\times Pic(R_ 2)\to Pic(R'). \] 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 general we cannot expect such sequences to exist, even if all morphisms in the above cartesian square satisfy PDE. We will see, however, that under reasonable restrictions exact Mayer-Vietoris sequences may be constructed. The note is organized as follows. In the first section we introduce the notion of a torsion couple, which is the natural object to study in order to obtain ''relative'' functoriality. In particular, a morphism of torsion couples is the good generalization of a morphism of Krull domains satisfying condition PDE. In the second section we consider cartesian diagrams of torsion couples and study patching properties that do not involve global finiteness assumptions. Some local-global properties of diagrams having the Milnor patching property are deduced. In the last section we first consider briefly patching properties that involve finiteness conditions. We then focus on cartesian diagrams of Krull domains and show how our methods may be applied in this case to deduce exact Mayer-Vietoris sequences of reflexive K-groups and of class groups.