Initial Ideals, Veronese Subrings, and Rates of Algebras

Abstract

⌉Let S be a polynomial ring over an infinite field and let I be a homogeneous ideal of S. Let T d be a polynomial ring whose variables correspond to the monomials of degree d in S. We study the initial ideals of the ideals V d ( I) ⊂ T d that define the Veronese subrings of S/ I. In suitable orders, they are easily deduced from the initial ideal of I. We show that in V d ( I) is generated in degree ≤ max (⌉ reg( I)/ d ⌈, 2), where reg( I) is the regularity of the ideal I. (In other words, the dth Veronese subrings of any commutative graded ring S/ I has a Gröbner basis of degree ≤ max (⌉( I)/ d ⌈, 2).) We also give bounds on the regularity of I in terms of the degrees of the generators of in( I) and some combinatorial data. This implies a version of Backelin′s theorem that high Veronese subrings of any ring are homogeneous Koszul algebras in the sense of Priddy [ Trans. Amer. Math. Soc, 152 (1970), 39-60]. We also give a general obstruction for a homogeneous ideal I ⊂ S to have an initial ideal in( I) that is generated by quadrics, beyond the obvious requirement that I itself should be generated by quadrics, and the stronger statement that S/ I is Koszul. We use the obstruction to show that in certain dimensions, a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics. For the application to Backelin′s theorem, we require a result of Backelin whose proof has never appeared. We give a simple proof of a sharpened version, bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials, following a method of Bruns and Herzog.