gradedcommutativity

Graded commutativity is a generalization of commutativity for graded rings or algebras. A graded ring A decomposes as a direct sum A = ⊕_{g∈G} A_g over an abelian group G, with multiplication that respects the grading. A is called graded-commutative if for homogeneous elements a ∈ A_g and b ∈ A_h, their product satisfies ab = χ(g,h) ba, where χ: G × G → R^× is a symmetric bicharacter. In the common Z-graded setting, this reduces to ab = (-1)^{deg(a) deg(b)} ba, so elements of odd degree anticommute with each other while even-degree elements commute with everything.

A prominent special case is when G = Z/2Z, yielding supercommutativity: even elements commute with all elements, while odd elements anticommute with each other. The sign rule in this setting is commonly referred to as the Koszul sign rule.

Examples and contexts

The exterior algebra Λ(V) on a vector space V is graded-commutative: generators of degree 1 anticommute, and products of even-degree elements commute with all elements. In contrast, a polynomial algebra on even-degree generators is strictly commutative, while mixing degrees is handled by the same graded sign rule when swapping homogeneous elements.

Differential graded objects

Differential graded algebras (DG-algebras) are graded-commutative rings equipped with a differential of degree 1 that satisfies the Leibniz rule with signs. Graded commutativity interacts with tensor products and homological constructions, leading to the Koszul sign conventions in chain complexes and derived categories.

Generalization

More broadly, a G-graded-commutative algebra can be defined with a bicharacter χ, so ab = χ(|a|,|b|) ba for homogeneous a, b. Graded commutativity is a foundational principle in algebraic topology, algebraic geometry, and deformation theory, ensuring coherent sign behavior across constructions.