Reduction numbers and initial ideals

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 $\mathbf m$ of $A$ such that $J\mathbf m^k=\mathbf m^{k+1}$. Vasconcelos conjectured that $r(R/I)\leq r(R/\mathrm {in}(I))$ where $\mathrm {in}(I)$ 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.