Rees algebra explained

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

R[It] = o plus ^{inf ty} _{n =0} 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[It,t ^{-1} ] = o plus ^{inf ty} _{n =-inf ty} I ^{nt} ^{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.^[2]

Properties

The Rees algebra is an algebra over

Z [t ^{-1} ] , and it is defined so that, quotienting by t^=0 or t=λ for λ any invertible element in R, we get

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 .^[3]

• 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

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

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

l{F} _{I(R)=R[It]/ak{m}R[It].}

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

References

1. Book: Eisenbud, David. Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag. 1995. 978-3-540-78122-6.
2. Eisenbud-Harris, The geometry of schemes. Springer-Verlag, 197, 2000
3. Book: Swanson, Irena. Integral Closure of Ideals, Rings, and Modules. Irena Swanson . Cambridge University Press. 2006. 9780521688604. Huneke. Craig.

External links

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