Fe b 19 92 Gröbner Bases and Extension of Scalars

Let A be a Noetherian commutative ring with identity, let A[x] = A[x1, . . . , xn] be a polynomial ring over A, and let I ⊂ A[x] be an ideal. Geometrically, I defines a family of schemes over the base scheme SpecA; the fiber over each point p ∈ SpecA is a subscheme of the affine space Ank(p) = Speck(p)[x], where k(p) = Ap/pp is the residue field of p. Let > be a total order on the monomials of A[x] satisfying x > x ⇒ xx > xx , and satisfying xi > 1 for each i. For f ∈ A[x], define in(f) to be the initial (greatest) term cx of f with respect to the order >, where c ∈ A is nonzero. For I ⊂ A[x], define the initial ideal in(I) to be the ideal (in(f) | f ∈ I) generated by all initial terms of elements of I. in(I) ⊂ A[x] is generated by single terms of the form cx ; we call such an ideal a monomial ideal. {f1, . . . , fr} ⊂ I is a Gröbner basis for I if and only if {in(f1), . . . , in(fr)} generates in(I). For an ideal I in a ring of formal power series A[[x]], Hironaka defined the corresponding notion (the standard basis of I) and proved a generalized Weierstrass division theorem.The relation between flatness and division was considered in [HLT 73] and later in [Gal 79] in order to obtain a presentation