Graded Algebra

A graded algebra is a graded module equipped with a degree-preserving linear map .

Cohomology is a graded algebra. In addition, the grading set is a monoid having a compatibility relation such that if is in the grading of the algebra , and is in the grading of the algebra , then is in the grading of the algebra (where and are multiplied in , and and are multiplied in the index monoid). For example, cohomology of a space is a graded algebra over the integers (i.e., a graded ring), since if is an -dimensional cohomology class and is an -dimensional cohomology class, then the cup product is an dimensional cohomology class.

The group ring of a group over a commutative ring is a graded -algebra with grading .