The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jm^k = m^{k+1}. Vasconcelos conjectured that r(R/I)=r(R/in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture.
Reduction numbers and initial ideals
CONCA, ALDO
2003-01-01
Abstract
The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jm^k = m^{k+1}. Vasconcelos conjectured that r(R/I)=r(R/in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture.
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