Rees algebra

In commutative algebra, the Rees algebra or Rees ring of an ideal I in a commutative ring R is defined to be <blockquote></blockquote> The extended Rees algebra of I (which some authors refer to as the Rees algebra of I) is defined as<blockquote></blockquote>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 ).

Properties

The Rees algebra is an algebra over , and it is defined so that, quotienting by or t=λ for λ any invertible element in R, we get <blockquote></blockquote> 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 if I is not contained in any prime ideal P with ; otherwise . The Krull dimension of the extended Rees algebra is .
• If are ideals in a Noetherian ring R, then the ring extension 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<blockquote></blockquote>If R is a Noetherian local ring with maximal ideal , then the special fiber ring of I is given by<blockquote></blockquote>The Krull dimension of the special fiber ring is called the analytic spread of I.

References

External links

• What Is the Rees Algebra of a Module?
• Geometry behind Rees algebra (deformation to the normal cone)