In commutative algebra, the Rees algebra or Rees ring of an ideal I in a commutative ring R is defined to be R [ I t ] = ⨁ n = 0 ∞ I n t n ⊆ R [ t ] . {\displaystyle 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 [ I t , t − 1 ] = ⨁ n = − ∞ ∞ I n t n ⊆ R [ t , t − 1 ] . {\displaystyle 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 (see Ideal sheaf § Algebraic geometry).
Properties The Rees algebra is an algebra over Z [ t − 1 ] {\displaystyle \mathbb {Z} [t^{-1}]} , and it is defined so that, quotienting by t − 1 = 0 {\displaystyle t^{-1}=0} or t=λ for λ any invertible element in R, we get gr I R ← R [ I t ] → R . {\displaystyle {\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 [ I t ] = dim R + 1 {\displaystyle \dim R[It]=\dim R+1} if I is not contained in any prime ideal P with dim ( R / P ) = dim R {\displaystyle \dim(R/P)=\dim R} ; otherwise dim R [ I t ] = dim R {\displaystyle \dim R[It]=\dim R} . The Krull dimension of the extended Rees algebra is dim R [ I t , t − 1 ] = dim R + 1 {\displaystyle \dim R[It,t^{-1}]=\dim R+1} . If J ⊆ I {\displaystyle J\subseteq I} are ideals in a Noetherian ring R, then the ring extension R [ J t ] ⊆ R [ I t ] {\displaystyle 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 gr I ( R ) = R [ I t ] / I R [ I t ] . {\displaystyle \operatorname {gr} _{I}(R)=R[It]/IR[It].} If R is a Noetherian local ring with maximal ideal m {\displaystyle {\mathfrak {m}}} , then the special fiber ring of I is given by F I ( R ) = R [ I t ] / m R [ I t ] . {\displaystyle {\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 Weisstein, Eric W. "Rees Ring". MathWorld. Retrieved 2024-08-31. What Is the Rees Algebra of a Module? Geometry behind Rees algebra (deformation to the normal cone)
