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 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.

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 grIR.


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