gradedcommutative

Gradedcommutative, typically written as graded-commutative or graded-commutative algebra, refers to a graded algebra A = ⊕_{n∈Z} A_n over a commutative ring R whose multiplication satisfies ab = (-1)^{|a||b|} ba for all homogeneous elements a ∈ A_{|a|}, b ∈ A_{|b|}. Here |a| and |b| denote the degrees of a and b. This sign rule, known as the Koszul sign rule, generalizes ordinary commutativity to the graded setting. When the grading is taken modulo 2, so that elements are even or odd, one obtains the notion of a supercommutative (or graded-commutative) algebra.

Examples include the exterior algebra Λ(V) on a vector space V, which is graded-commutative: elements of odd degree anticommute and even-degree elements commute. A polynomial algebra in generators all of even degree is strictly commutative in the graded sense. More generally, any Z-graded algebra in which all homogeneous elements have even degree is strictly commutative, while the presence of odd-degree elements introduces sign changes according to the product of degrees.

Properties of graded-commutative algebras include the compatibility of grading with multiplication, so A_i A_j ⊆ A_{i+j}, and the requirement that morphisms preserve both the grading and the product. They arise naturally in various areas of mathematics, including algebraic topology (differential forms form a graded-commutative algebra under the wedge product), algebraic geometry, and the theory of differential graded algebras (DGAs). In many contexts, the grading interacts with differentials via the relation d(ab) = d(a)b + (-1)^{|a|} a d(b).