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]


  • 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.


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