Reduction Mod P of Standard Bases

We investigate the behavior of standard bases (in the sense of Hironaka and Grauert) for ideals in rings of formal power series over commutative rings with respect to specializations of the coefficients. For instance, we show that any ideal I of the ring of formal power series A[[X]] = A[[X1, . . . , XN ]] with coefficients in a Noetherian ring A admits a standard basis whose image under every specialization of A onto a field is a standard basis of the image of I. Applications include a modular criterion for ideal membership in Z[[X]] and a constructibility result for ideal membership in K[[X]], where K is a field. Partially supported by National Science Foundation grant DMS 03-03618.