Abstract: Syzygies interpolate between torsion-freeness and freeness. In this talk, we will introduce the concept of syzygies and review criteria for a module to attain a certain syzygy order. Then we will discuss a result of Franz which relates the syzygy order of equivariant cohomology of a toric variety to the combinatorics of the underlying fan. Finally, by restricting the torus action on toric varieties to subtori, we will investigate the resulting changes in the syzygy order of their equivariant cohomology.