The extended Rees algebra of I (which some authors[1] refer to as the Rees algebra of I) is defined as
This construction has special interest in
algebraic geometry since the
projective scheme defined by the Rees algebra of an ideal in a ring is the
blowing-up of the spectrum of the ring along the
subscheme defined by the ideal.[2]
Properties
The Rees algebra is an algebra over , and it is defined so that, quotienting by t^{-1}=0 or t=λ for λ any invertible element in R, we get
Thus it interpolates between R and its associated graded ring grIR.
Assume R is
Noetherian; then R[It] is also Noetherian. The
Krull dimension of the Rees algebra is if I is not contained in any prime ideal P with ; otherwise . The Krull dimension of the extended Rees algebra is .[3]
If are ideals in a Noetherian ring R, then the ring extension is
integral if and only if J is a reduction of I.[3]
If I is an ideal in a Noetherian ring R, then the Rees algebra of I is the
quotient of the
symmetric algebra of I by its
torsion submodule.