Rees algebra

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

$R[It]=\bigoplus_{n=0}^{\infty} I^n t^n\subseteq R[t].$

The extended Rees algebra of I (which some authors^[1] refer to as the Rees algebra of I) is defined as

$R[It,t^{-1}]=\bigoplus_{n=-\infty}^{\infty}I^nt^n\subseteq R[t,t^{-1}].$

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 $\dim R[It]=\dim R+1$ if I is not contained in any prime ideal P with $\dim(R/P)=\dim R$; otherwise $\dim R[It]=\dim R$. The Krull dimension of the extended Rees algebra is $\dim R[It]=\dim R+1$.^[3]
• If $J\subseteq I$ are ideals in a Noetherian ring R, then the ring extension $R[Jt]\subseteq R[It]$ 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

$\operatorname{gr}_I(R)=R[It]/IR[It].$

If R is a Noetherian local ring with maximal ideal $\mathfrak{m}$, then the special fiber ring of I is given by

$\mathcal{F}_I(R)=R[It]/\mathfrak{m}R[It].$

The Krull dimension of the special fiber ring is called the analytic spread of I.

References

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.

External links

• What Is the Rees Algebra of a Module?
• http://mathoverflow.net/questions/143746/geometry-behind-rees-algebra-deformation-to-the-normal-cone?rq=1