On Lucky Primes

Winkler (1988) and Pauer (1992) present algorithms for a Hensel lifting of a modular Gröbner basis over lucky primes to a rational one. They have to solve a linear system with modular polynomial entries that requires another (modular) Gröbner basis computation. After an extension of luckiness to arbitrary (commutative noetherian) base rings we show in this paper that for a homogeneous polynomial ideal I one can lift not only its Gröbner basis but also a homogeneous basis of its syzygy module. The same result holds for arbitrary ideals and liftings from Z/p to Q. Moreover the same lifting can be obtained from a true Gröbner trace by linear algebra over Q only. Parallel modular techniques allow to find such a true Gröbner trace and a lucky prime with high probability. All these results generalize in an obvious way to homogeneous modules generated by the rows of matrices with polynomial entries. Since luckiness can be weakened to a condition that transfers from I to higher syzygy modules the lifting theorem generalizes to a lifting theorem for the resolution of I.