Exact pairs of homogeneous zero divisors

Abstract

Abstract. Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that if S is a compressed level algebra, then S does not contain any homogeneous zero divisors.In [17], Henriques and Şega defined the pair of elements (a, b) in a commutative ring S to be an exact pair of zero divisors if (0 : S a) = (b) and (0 : S b) = (a). We take S be a standard graded Artinian algebra over a field and we identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair (θ 1 , θ 2 ) of homogeneous zero divisors. In Theorem 2.9 we prove that the main numerical constraint depends on the sum deg θ 1 + deg θ 2 , but not on the individual numbers deg θ 1 or deg θ 2 . In other words, the numerical constraint imposed on S by having an exact pair of homogeneous zero divisors of degrees d 1 and d 2 is the same as the constraint imposed by having an exact pair of homogeneous zero divisors of degrees 1 and d 1 + d 2 − 1. This result is especially curious because it is possible for S to have an exact pair of homogeneous zero divisors of degrees 2 and 2 without having any homogeneous exact zero divisors of degree 1; see Example 3.1. Our main result is Theorem 2.9.Theorem 2.9. Let S be a standard graded Artinian k-algebra. Suppose that (θ 1 , θ 2 ) is an exact pair of homogeneous zero divisors in S. If D = deg θ 1 + deg θ 2 , then the Hilbert series of S is divisible by2010 Mathematics Subject Classification. 13D02, 13A02.