Rees algebra

In commutative algebra, the Rees algebra or Rees ring of an ideal I in a commutative ring R is defined to be

The extended 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.

Properties

The Rees algebra is an algebra over, and it is defined so that, quotienting by or t=λ for λ any invertible element in R, we get

Thus it interpolates between R and its associated graded ring gr_IR.

Assume R is Noetherian; then R 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.
If are ideals in a Noetherian ring R, then the ring extension is integral if and only if J is a reduction of I.
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.

The associated graded ring of I may be defined as

If R is a Noetherian local ring with maximal ideal, then the special [fiber ring] of I is given by

The Krull dimension of the special fiber ring is called the analytic spread of I.