On the symmetric algebra of certain first syzygy modules

Let (R, m) be a standard graded K-algebra over a field K. Then R can be written as S/I, whereI ⊆(x1,...,xn) isagradedidealofapolynomialringS =K[x1,...,xn]. Assume that n > 3 and I is a strongly stable monomial ideal. We study the symmetric algebra SymR(Syz1(m)) of the first syzygy module Syz1(m) of m. When the minimal gen- erators of I are all of degree 2, the dimension of SymR(Syz1(m)) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.