Rees algebra
In commutative algebra, the Rees algebra of an ideal I in a commutative ring R is defined to be
[math]\displaystyle{ R[It]=\bigoplus_{n=0}^{\infty} I^n t^n\subseteq R[t]. }[/math]
The extended Rees algebra of I (which some authors[1] refer to as the Rees algebra of I) is defined as
[math]\displaystyle{ R[It,t^{-1}]=\bigoplus_{n=-\infty}^{\infty}I^nt^n\subseteq R[t,t^{-1}]. }[/math]
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 [math]\displaystyle{ \dim R[It]=\dim R+1 }[/math] if I is not contained in any prime ideal P with [math]\displaystyle{ \dim(R/P)=\dim R }[/math]; otherwise [math]\displaystyle{ \dim R[It]=\dim R }[/math]. The Krull dimension of the extended Rees algebra is [math]\displaystyle{ \dim R[It, t^{-1}]=\dim R+1 }[/math].[3]
- If [math]\displaystyle{ J\subseteq I }[/math] are ideals in a Noetherian ring R, then the ring extension [math]\displaystyle{ R[Jt]\subseteq R[It] }[/math] 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
[math]\displaystyle{ \operatorname{gr}_I(R)=R[It]/IR[It]. }[/math]
If R is a Noetherian local ring with maximal ideal [math]\displaystyle{ \mathfrak{m} }[/math], then the special fiber ring of I is given by
[math]\displaystyle{ \mathcal{F}_I(R)=R[It]/\mathfrak{m}R[It]. }[/math]
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
- ↑ 3.0 3.1 Swanson, Irena; Huneke, Craig (2006). Integral Closure of Ideals, Rings, and Modules. Cambridge University Press. ISBN 9780521688604.
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