supercommutative

Supercommutative refers to a property of certain algebras that are graded by the two-element group Z/2 and obey a sign rule that generalizes ordinary commutativity. A supercommutative algebra A over a commutative ring R has a decomposition A = A0 ⊕ A1 into even and odd parts, and its multiplication respects the grading: for homogeneous elements a and b with degrees |a|, |b| in {0,1}, one has ab = (-1)^{|a||b|} ba. In particular, even elements (degree 0) commute with all elements, while odd elements (degree 1) anticommute with each other.

Consequences follow from the sign rule. The even part A0 is a commutative R-algebra, and A1 is an A0-bimodule with the product A1 × A1 mapping into A0. If the base ring has characteristic not equal to 2, then the square of any odd element is zero (since a a = - a a implies 2 a a = 0). In characteristic 2 the sign becomes trivial and the grading loses its antisymmetry.

Key examples include the exterior (Grassmann) algebra Λ(V) on a vector space V, which is the free supercommutative algebra generated by V placed in odd degree. More generally, supercommutative algebras arise as coordinate rings in the theory of supermanifolds and superschemes, providing a framework for handling both commuting (even) and anticommuting (odd) coordinates.

Relation to broader theory: supercommutativity is a special case of graded-commutativity, where the sign rules come from a Z/2 grading. The concept is central in both algebraic geometry and mathematical physics, particularly in the study of supersymmetry and fermionic variables.