On the symmetric algebra of certain first syzygy modules

Abstract

Let (R, $\mathfrak{m}$) be a standard graded K-algebra over a field K. Then R can be written as S/I, where I ⊆ (x₁,…, x_n)² is a graded ideal of a polynomial ring S = K[x₁,…, x_n]. Assume that n ⩽ 3 and I is a strongly stable monomial ideal. We study the symmetric algebra Sym_R(Syz₁($\mathfrak{m}$)) of the first syzygy module Syz₁($\mathfrak{m}$) of $\mathfrak{m}$. When the minimal generators of I are all of degree 2, the dimension of Sym_R (Syz₁($\mathfrak{m}$)) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.