On the depth of the Rees algebra of an ideal module

We study the Rees algebra R(E):=S(E)/τR(S(E)) of an ideal module E⊂G≃Re. We use the technique of Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos (2003) [22] to relate the Rees algebra of a module E to the Rees algebra of an ideal I=I(E)⊂R″, where R″ is a Nagata extension of R. We shall prove that depthR(E)=depthR(I)+e−1 and use it to deduce formulae for the depth of R(E) for ideal modules having small reduction number.