A decomposition of a module into a direct sum of submodules. The index set for the collection of submodules is then called the grading set. Graded modules arise naturally in homology. In particular, for every integer i, there exists an ith homology group of a space H_i(X), and usually the "total homology" of the space is considered to be the direct sum of all the H_i(X)s. This makes the "total" homology of X a module graded over the integers.

graded algebra | graded free resolution