Rees algebra

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

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

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

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

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 Ideal sheaf § Algebraic geometry).^[2]

Properties

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

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

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

Relationship with other blow-up algebras

The associated graded ring of I may be defined as

${\displaystyle \operatorname {gr} _{I}(R)=R[It]/IR[It].}$

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

${\displaystyle {\mathcal {F}}_{I}(R)=R[It]/{\mathfrak {m}}R[It].}$

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