Misplaced Pages

Rees algebra

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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=\bigoplus _{n=0}^{\infty }I^{n}t^{n}\subseteq R.}

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=\bigoplus _{n=-\infty }^{\infty }I^{n}t^{n}\subseteq R.}

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 Z [ t − 1 ] {\displaystyle \mathbb {Z} } , 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\ \to \ R.}

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

  • Assume R is Noetherian; then R is also Noetherian. The Krull dimension of the Rees algebra is dim ⁡ R [ I t ] = dim ⁡ R + 1 {\displaystyle \dim R=\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=\dim R} . The Krull dimension of the extended Rees algebra is dim ⁡ R [ I t , t − 1 ] = dim ⁡ R + 1 {\displaystyle \dim R=\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\subseteq R} 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/IR.}

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/{\mathfrak {m}}R.}

The Krull dimension of the special fiber ring is called the analytic spread of I.

References

  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
  3. ^ Swanson, Irena; Huneke, Craig (2006). Integral Closure of Ideals, Rings, and Modules. Cambridge University Press. ISBN 9780521688604.

External links

Categories:
Rees algebra Add topic