Macaulay bases of modules

We define Macaulay bases of modules, which are a common generalization of Groebner bases and Macaulay $H$-bases to suitably graded modules over a commutative graded $\mathbf{k}$-algebra, where the index sets of the two gradings may differ. This includes Groebner bases of modules as a special case, in contrast to previous work on Macaulay bases of modules. We show that the standard results on Groebner bases and Macaulay $H$-bases generalize in fields of arbitrary characteristic to Macaulay bases, including the reduction algorithm and Buchberger's criterion and algorithm. A key result is that Macaulay bases, in contrast to Groebner bases, respect symmetries when there is a group $G$ acting homogeneously on a graded module, in which case the reduction algorithm is $G$-equivariant and the $\mathbf{k}$-span of a Macaulay basis is $G$-invariant. We also show that some of the standard applications of Groebner bases can be generalized to Macaulay bases, including elimination and computation of syzygy modules, which require the generalization to modules that was not present in previous work.