Rees algebra

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

]

.