Rees algebra

In commutative algebra, the Rees algebra 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=igoplus _{n=0}^{infty }I^{n}t^{n}subseteq R.} R [ I t , t − 1 ] = ⨁ n = − ∞ ∞ I n t n ⊆ R [ t , t − 1 ] . {displaystyle R=igoplus _{n=-infty }^{infty }I^{n}t^{n}subseteq R.} gr I ⁡ ( R ) = R [ I t ] / I R [ I t ] . {displaystyle operatorname {gr} _{I}(R)=R/IR.} F I ( R ) = R [ I t ] / m R [ I t ] . {displaystyle {mathcal {F}}_{I}(R)=R/{mathfrak {m}}R.} In commutative algebra, the Rees algebra of an ideal I in a commutative ring R is defined to be The extended Rees algebra of I (which some authors 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. The associated graded ring of I may be defined as If R is a Noetherian local ring with maximal ideal m {displaystyle {mathfrak {m}}} , 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.