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
[edit]The Rees algebra is an algebra over , and it is defined so that, quotienting by t^{-1}=0 or t=λ for λ any invertible element in R, we get
Thus it interpolates between R and its associated graded ring grIR.
- 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
[edit]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
[edit]- ^ 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
- ^ a b Swanson, Irena; Huneke, Craig (2006). Integral Closure of Ideals, Rings, and Modules. Cambridge University Press. ISBN 9780521688604.