Rees algebra について

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Rees algebra ：ウィキペディア英語版

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

$R()=\bigoplus_^ I^n t^n\subseteq R().$

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

$R()=\bigoplus_^I^nt^n\subseteq R().$

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.〔Eisenbud-Harris, ''The geometry of schemes''. Springer-Verlag, 197, 2000〕
== Properties ==

* Assume ''R'' is Noetherian. The Krull dimension 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+1

.
* If

J\subseteq I

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

R()\subseteq R()

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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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