Harman Patil (Editor)

Rees algebra

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In commutative algebra, the Rees algebra of an ideal I in a commutative ring R is defined to be

Contents

R [ I t ] = ⨁ n = 0 ∞ I n t n ⊆ 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 ] .

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

  • 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 if I is not contained in any prime ideal P with dim ⁡ ( R / P ) = dim ⁡ R ; otherwise dim ⁡ R [ I t ] = dim ⁡ R . The Krull dimension of the extended Rees algebra is dim ⁡ R [ I t ] = dim ⁡ R + 1 .
  • If J ⊆ I are ideals in a Noetherian ring R, then the ring extension R [ J t ] ⊆ R [ I t ] 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 ] .

    If R is a Noetherian local ring with maximal ideal m , then the special fiber ring of I is given by

    F I ( R ) = R [ I t ] / m R [ I t ] .

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

    References

    Rees algebra Wikipedia
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