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

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

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 $\dim R[It,t^{-1}]=\dim R+1$ .[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

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.

External links

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[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

[:0-3] Swanson, Irena; Huneke, Craig (2006). Integral Closure of Ideals, Rings, and Modules. Cambridge University Press. ISBN 9780521688604.