 
Summary: ALGORITHMS FOR COMPUTING SATURATIONS OF IDEALS
IN FINITELY GENERATED COMMUTATIVE RINGS
MATTHIAS ASCHENBRENNER
Abstract. This short note constitutes an appendix to a paper by Poonen.
The paper Automorphisms mapping a point to a subvariety by Bjorn Poonen con
tains an appeal to the fact that there exists an algorithm which accomplishes the
following basic task:
Given a finitely generated commutative Zalgebra A (specified by
generators and relations) and a finite list of generators for an ideal
I of A, compute a finite list of generators for the inverse image of
the ideal I Q under the natural morphism A A Q.
The existence of such an algorithm is wellknown, and the purpose of this note
is to briefly describe two different procedures for the task at hand, and to make
some additional related remarks. Before this, we observe that representing the Z
algebra A as a quotient A = Z[X]/J where J is an ideal of the polynomial ring
Z[X] = Z[X1, . . . , XN ] (where N and generators for J are part of the input data)
one sees that it suffices to consider the case where A is a polynomial ring over Z.
In this case, the pullback ideal in question may also simply be described as the
saturation IQ[X] Z[X] of I with respect to the multiplicative subset Z \ {0} of
Z[X]. That is, we need to give an algorithm which does the following:
