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[ 1] 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.[ 2]
Properties
The Rees algebra is an algebra over
Z
[
t
− − -->
1
]
{\displaystyle \mathbb {Z} [t^{-1}]}
t
− − -->
1
=
0
{\displaystyle t^{-1}=0}
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 grI R .
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}
I is not contained in any prime ideal P with
dim
-->
(
R
/
P
)
=
dim
-->
R
{\displaystyle \dim(R/P)=\dim R}
dim
-->
R
[
I
t
]
=
dim
-->
R
{\displaystyle \dim R[It]=\dim R}
dim
-->
R
[
I
t
,
t
− − -->
1
]
=
dim
-->
R
+
1
{\displaystyle \dim R[It,t^{-1}]=\dim R+1}
[ 3]
If
J
⊆ ⊆ -->
I
{\displaystyle J\subseteq I}
R , then the ring extension
R
[
J
t
]
⊆ ⊆ -->
R
[
I
t
]
{\displaystyle R[Jt]\subseteq R[It]}
integral if and only if J is a reduction of I .[ 3]
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}}}
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
^ Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry . Springer-Verlag. ISBN 978-3-540-78122-6 ^ Eisenbud-Harris, The geometry of schemes . Springer-Verlag, 197, 2000
^ a b Swanson, Irena ; Huneke, Craig (2006). Integral Closure of Ideals, Rings, and Modules . Cambridge University Press. ISBN 9780521688604
External links
![{\displaystyle R[It]=\bigoplus _{n=0}^{\infty }I^{n}t^{n}\subseteq R[t].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33d6fdd6b4c416324e50c4b07416c79698435844)
![{\displaystyle R[It,t^{-1}]=\bigoplus _{n=-\infty }^{\infty }I^{n}t^{n}\subseteq R[t,t^{-1}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/484221506c060c9fdfbc23488fecd0f0dbcd26f0)
![{\displaystyle \mathbb {Z} [t^{-1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/825c17f7e95d6a9059e9e82c7da2599d8df57c41)
![{\displaystyle {\text{gr}}_{I}R\ \leftarrow \ R[It]\ \to \ R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6b3dd1fa5d1c616e325f2900ee9f0d3b3cacb5)
![{\displaystyle \dim R[It]=\dim R+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c056a97e4c9942c3f3ae8feb887347d749c314cb)
![{\displaystyle \dim R[It]=\dim R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8d9033df9263328142e90985b71eceaf25915c)
![{\displaystyle \dim R[It,t^{-1}]=\dim R+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248ba396d3f75aaee03e9edff719bcc53ea3f8d6)
![{\displaystyle R[Jt]\subseteq R[It]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7312421e01e132443dcbcd2919ac1ea39fc94c4)
![{\displaystyle \operatorname {gr} _{I}(R)=R[It]/IR[It].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef3d45f4328de060c08ac1b4c650f72a5b94906)
![{\displaystyle {\mathcal {F}}_{I}(R)=R[It]/{\mathfrak {m}}R[It].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df2691dfe4a9e49a8c254eac63f38e8c2762e226)