Rees algebra

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

$R [I t] = ⨁_{n = 0}^{\infty} I^{n} t^{n} \subseteq R [t] .$

The extended Rees algebra of I (which some authors^[1] refer to as the Rees algebra of I) is defined as

$R [I t, t^{- 1}] = ⨁_{n = - \infty}^{\infty} I^{n} t^{n} \subseteq R [t, t^{- 1}] .$

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 (see Template:Section link.^[2]

Properties

The Rees algebra is an algebra over

ℤ [t^{- 1}]

, and it is defined so that, quotienting by

t^{- 1} = 0

or t=λ for λ any invertible element in R, we get

${gr}_{I} R \leftarrow R [I t] \to R .$

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

Assume R is Noetherian; then R[It] is also Noetherian. The Krull dimension of the Rees algebra is $\dim R [I t] = \dim R + 1$ if I is not contained in any prime ideal P with $\dim (R / P) = \dim R$ ; otherwise $\dim R [I t] = \dim R$ . The Krull dimension of the extended Rees algebra is $\dim R [I t, t^{- 1}] = \dim R + 1$ .^[3]
If $J \subseteq I$ are ideals in a Noetherian ring R, then the ring extension $R [J t] \subseteq R [I t]$ 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.

The associated graded ring of I may be defined as

${gr}_{I} (R) = R [I t] / I R [I t] .$

If R is a Noetherian local ring with maximal ideal

𝔪

, then the special fiber ring of I is given by

$ℱ_{I} (R) = R [I t] / 𝔪 R [I t] .$

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