Patching and birationality (Q1320186)

Consider the commutative square \[ \begin{tikzcd} R \ar[r,"i_ R"]\ar[d,"\varphi"'] & R^\prime\ar[d,"\varphi^\prime"] \\S \ar[r,"i_ S"] & S' \end{tikzcd}\tag{*} \] of commutative rings with \(R\) Noetherian. A collection of patching data for (*) is a triple \[ (P_{R'}, P_ S, \alpha:P_{R'} \otimes_{R'} S'@>>\approx> S' \otimes_ SP_ S), \] where \(P_{R'}\) is s finitely generated projective \(R'\)-module, \(P_ S\) is a finitely generated projective \(S\)- module, and \(\alpha\) is an isomorphism of \(S'\)-modules. The category of finitely generated projective \(R\)-modules maps functorially to the category of patching data via \[ F:P \mapsto \left( P \otimes_ R R', S \otimes_ R P, \alpha \right) \] with \(\alpha\) the obvious identification induced by \((p \otimes 1) \otimes 1 \mapsto 1 \otimes (1 \otimes p)\). We say that (*) is a Milnor patching diagram if \(F\) is an equivalence of categories, with inverse equivalence given by \[ G:(P_{R'}, P_ S, \alpha) \mapsto \bigl\{ (p_{R'}, p_ S) \mid p_{R'} \in P_{R'}, p_ S \in P_ S, \text{ and } \alpha (p_{R'} \otimes 1) = 1 \otimes p_ S \bigr\}. \] Some properties of a Milnor patching diagram for a domain \(R\) and a subring \(R'\) of its quotient field are investigated (theorem 1.3.). As a main result the Milnor patching diagrams are characterized in the case when: (1) \(R'=R[I/g]\) is the subring of the quotient field of \(R\) generated by \(R\) and all \(f/g\) with \(0 \neq g \in R\) and \(f \in I\), \(I \subseteq R\) an ideal, (2) \(\varphi (g)\) is not a zero divisor of \(S\) and \(S' = S[I/g]\) is induced from \(R'\) via \(\varphi\) in the natural way (theorem 3.3).