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.
- 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 .
- If are ideals in a Noetherian ring R, then the ring extension 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
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.
- 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.