INITIAL IDEALS, VERONESE SUBRINGS, AND RATES OF ALGEBRAS

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)subset of 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 less than or equal to max (inverted right perpendicular reg(I)/d inverted left perpendicular, 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 Grobner basis of degree less than or equal to max (inverted right perpendicular(I)/d inverted left perpendicular, 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. Sec, 152 (1970), 39-60]. We also give a general obstruction for a homogeneous ideal I subset of 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 module an ideal generated by monomials, following a method of Bruns and Herzog. (C) 1994 Academic Press, Inc.