Rees algebra

In commutative algebra, the Rees algebra or Rees ring 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{{Cite book|title = Commutative Algebra with a View Toward Algebraic Geometry|last = Eisenbud|first = David|publisher = Springer-Verlag|year = 1995|isbn = 978-3-540-78122-6}} 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 (see {{section link|Ideal sheaf|Algebraic geometry}}).Eisenbud-Harris, The geometry of schemes. Springer-Verlag, 197, 2000

Properties

The Rees algebra is an algebra over $\mathbb{Z}[t^{-1}]$ , and it is defined so that, quotienting by $t^{-1}=0$ or t=λ for λ any invertible element in R, we get

$\text{gr}_I R \ \leftarrow\ R[It]\ \to\ R.$

Thus it interpolates between R and its associated graded ring gr_IR.

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 $\dim R[It, t^{-1}]=\dim R+1$ .{{Cite book|title = Integral Closure of Ideals, Rings, and Modules|last = Swanson|first = Irena|author1-link= Irena Swanson |publisher = Cambridge University Press|year = 2006|isbn = 9780521688604|last2 = Huneke|first2 = Craig}}
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.
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

External links

{{MathWorld |id=ReesRing |title=Rees Ring |access-date=2024-08-31 }}
[https://www.ams.org/journals/proc/2003-131-03/S0002-9939-02-06575-9/S0002-9939-02-06575-9.pdf What Is the Rees Algebra of a Module?]
[https://mathoverflow.net/q/143746 Geometry behind Rees algebra (deformation to the normal cone)]

Category:Commutative algebra

Category:Algebraic geometry