commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

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

$R t \bigoplus_^ I^n t^n\subseteq R$

The extended Rees algebra of ''I'' (which some authors refer to as the Rees algebra of ''I'') is defined as

$R t,t^\bigoplus_^I^nt^n\subseteq R,t^$

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 This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

defined by the ideal.Eisenbud-Harris, ''The geometry of schemes''. Springer-Verlag, 197, 2000

Properties

* Assume ''R'' is Noetherian; then ''R t' is also Noetherian. The

Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...

of the Rees algebra is

\dim R+1

if ''I'' is not contained in any prime ideal ''P'' with

\dim(R/P)=\dim R

; otherwise

\dim R

. The Krull dimension of the extended Rees algebra is

\dim R

t, t^ T-comma (majuscule: Ț, minuscule: ț) is a letter which is part of the Romanian alphabet, used to represent the Romanian language sound , the voiceless alveolar affricate (like the letter C in Slavic languages that use the Latin alphabet). It is ...

\dim R+1. * If

J\subseteq I

are ideals in a Noetherian ring ''R'', then the ring extension

R t subseteq R t /math> is

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

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

$\operatorname_I(R)=R t IR t$

If ''R'' is a Noetherian

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...

with maximal ideal

\mathfrak

, then the special fiber ring of ''I'' is given by

$\mathcal_I(R)=R t \mathfrak{m}R t$

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

References

External links

''What Is the Rees Algebra of a Module?''Geometry behind Rees algebra (deformation to the normal cone)
Commutative algebra Algebraic geometry