A local global theorem for extended ideals (Q2907423)

Let \(R\) be a commutative ring and \(I\) an ideal in \(R\). Let \(n\geq1\). The elementary symplectic group \(\text{ESp}_{2n}(R)\) acts on the set \(\text{Um}_{2n}(R)\) of unimodular rows of length \(2n\) over \(R\). And the relative elementary symplectic group \(\text{ESp}_{2n}(R,I)\) acts on the set \(\text{Um}_{2n}(R,I)\) of relative unimodular rows of length \(2n\) over \(R\). The bigger group \(E_{2n}(R,I)\) also acts. The authors show that the orbit sets \(\text{Um}_{2n}(R)/\text{E}_{2n}(R,I)\) and \(\text{Um}_{2n}(R)/\text{ESp}_{2n}(R,I)\) are the same and that they depend only on \(I\), not \(R\) (excision). To this end they first establish a relative version of a `Quillen local global principle' for the action of \(\text{ESp}_{2n}\) on \(\text{Um}_{2n}\). Then they develop the symplectic version of an excision result of the reviewer.