ALDO CONCA Lemma

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 J m = m. 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. 1. Reduction numbers and initial ideals Let K be an infinite field and let A = ⊕i∈NAi be a homogeneous K-algebra, that is, an algebra of the form R/I where R = K[x1, . . . , xn] is a polynomial ring and I is a homogeneous ideal. The reduction number r(A) of A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that J m = m. It is not difficult to see that r(A) is the largest integer k such that the Hilbert function of A/J at k does not vanishes; here J is the ideal of A generated by d = dimA generic linear forms. Vasconcelos conjectured [10, Conjecture 7.2] that r(R/I) ≤ r(R/ inτ (I)) where inτ (I) is the initial ideal of I with respect to a term order τ . The conjecture has been proved by Bresinsky and Hoa [2] for the generic initial ideal Ginτ (I), or, more generally, when inτ (I) is Borel-fixed. Trung [8] showed that r(R/I) = r(R/GinRL(I)) where the GinRL(I) is the generic initial ideal of I with respect to the degree reverse lexicographic order RL (revlex for short). The goal of this note is to prove the conjecture in general. After this paper was written we were informed that Trung [9] has independently solved the conjecture in general by a completely different method. What we prove is the following generalization of Vasconcelos’ conjecture: Theorem 1.1. Let p be an integer, 0 ≤ p ≤ n, and let inτ (I) be the initial ideal of I with respect to a term order τ . Let J be an ideal generated by p generic linear forms. Then the Hilbert function of R/I + J is ≤ that of R/ inτ (I) + J , that is dimK [R/I + J ]j ≤ dimK [R/ inτ (I) + J ]j