On the Stability of Gröbner Bases Under Specializations

where lm(I) denotes the ideal generated by the leading monomials of the elements of I. This condition has already been studied in Bayer et al. (1991) and it has been shown that (1.1) holds for any ideal and any term order if and only if π is flat. In this paper we study condition (1.1) under the additional assumption that R′ is not a general Noetherian commutative ring with identity but a field. First we prove the following necessary and sufficient condition for (1.1). Let {g1, . . . , gs} be a Gröbner basis of an ideal I ⊆ R[x1, . . . , xn] with respect to an order ≺ and assume that the gis are ordered in such a way that the leading coefficients of precisely the first r polynomials are not in the kernel ker(π). Then (1.1) holds for I and ≺ if and only if the polynomials π(gr+1), . . . , π(gs) can be reduced to 0 modulo {π(g1), . . . , π(gr)}. Sufficient but not necessary conditions that (1.1) holds for an ideal and an order can be found in Bayer et al. (1991), Pauer (1992), Gräbe (1993) and Assi (1994). If R′ is a field ker(π) is a prime ideal. Let J be a subideal of ker(π). We show that the following two conditions are equivalent.