IDEALS GENERATED BY QUADRATIC POLYNOMIALS

Let R be a polynomial ring in N variables over an arbitrary field K and let I be an ideal of R generated by n polynomials of degree at most 2. We show that there is a bound on the projective dimension of R/I that depends only on n, and not on N. The proof depends on showing that if K is infinite and n is a positive integer, there exists a positive integer C(n), independent of N, such that any n forms of degree at most 2 in R are contained in a subring of R generated over K by at most t <= C(n) forms G(1), ... ,G(t) of degree 1 or 2 such that G(1), ... ,G(t) is a regular sequence in R. C(n) is asymptotic to 2n(2n).