In commutative algebra, the Rees algebra or Rees ring of an ideal I in a commutative ring R is defined to be
The extended Rees algebra of I (which some authors[1] refer to as the Rees algebra of I) is defined as
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).[2]
The Rees algebra is an algebra over , and it is defined so that, quotienting by or t=λ for λ any invertible element in R, we get
Thus it interpolates between R and its associated graded ring grIR.
The associated graded ring of I may be defined as
If R is a Noetherian local ring with maximal ideal , then the special fiber ring of I is given by
The Krull dimension of the special fiber ring is called the analytic spread of I.
![{\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)