# Rees algebra

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

The **extended Rees algebra** of *I* (which some authors[1] refer to as the 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.[2]

## Properties

- Assume
*R*is Noetherian; then*R[It]*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 .[3] - If are ideals in a Noetherian ring
*R*, then the ring extension 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

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*.

## References

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

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