The reduction theory goes back to the influential 1954 paper by Northcott and Rees, the paper that introduced the basic notions. In algebraic geometry, the theory is among the essential tools to extract detailed information about the behaviors of blow-ups.
Given ideals J ⊂ I in a ring R, the ideal J is said to be a reduction of I if there is some integer m > 0 such that
JIm=Im+1
JkIm=Jk-1Im+1= … =Im+k
If R is a Noetherian ring, then J is a reduction of I if and only if the Rees algebra R[''It''] is finite over R[''Jt'']. (This is the reason for the relation to a blow up.)
A closely related notion is that of analytic spread. By definition, the fiber cone ring of a Noetherian local ring (R,
ak{m}
l{F}I(R)=R[It] ⊗ R\kappa(ak{m})\simeq
infty | |
oplus | |
n=0 |
In/ak{m}In
l{F}I(R)
J\subsetI
R/akm
\ell
\ell