Testing flatness and computing rank of a module using syzygies
Using syzygies computed via Gröbner bases techniques, we present algorithms for testing some homological properties for submodules of the free module $A^m$, where $A=R[x_1,\dots,x_n]$ and $R$ is a Noetherian commutative ring. We will test if a given submodule $M$ of $A^m$ is flat. We will also check if $M$ is locally free of constant dimension. Moreover, we present an algorithm that computes the rank of a flat submodule $M$ of $A^m$ and also an algorithm that computes the projective dimension of an arbitrary submodule of $A^m$. All algorithms are illustrated with examples.