Rees algebra

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

$R[It]=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{I}^n{t}^n\subseteq R[t].$

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

$R[It,t^{-1}]=\underset{n=-\mathrm{\infty }}{\overset{\mathrm{\infty }}{⨁}}{I}^n{t}^n\subseteq R[t,t^{-1}].$

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).

Properties

The Rees algebra is an algebra over $\mathbb{Z}[t^{-1}]$, and it is defined so that, quotienting by $t^{-1}=0$ or t=λ for λ any invertible element in R, we get

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

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

Relationship with other blow-up algebras

The associated graded ring of I may be defined as

${\mathrm{gr}}_I(R)=R[It]/IR[It].$

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

${ℱ}_I(R)=R[It]/𝔪R[It].$

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

External links

• Template:MathWorld
• What Is the Rees Algebra of a Module?
• Geometry behind Rees algebra (deformation to the normal cone)