  Based on Schreyer's algorithm (Schreyer, 1980, 1991; Berkesch and Schreyer,
2014), we present two refined algorithms for the computation of syzygies. The
two main ideas of the first algorithm, called LiftHybrid, are the following:
First, we may leave out certain terms of module elements during the computation
which do not contribute to the result. These terms are called "lower order
terms", see Definition 4.2. Second, we do not need to order the remaining terms
of these module elements during the computation. This significantly reduces the
number of monomial comparisons for the arithmetic operations. For the second
algorithm, called LiftTree, we additionally cache some partial results and
reuse them at the remaining steps.
