In mathematics, an alternating algebra is a -graded algebra for which for all nonzero homogeneous elements and (i.e. it is an anticommutative algebra) and has the further property that for every homogeneous element of odd degree. Examples The differential forms on a differentiable manifold form an alternating algebra. The exterior algebra is an alternating algebra. The cohomology ring of a topological space is an alternating algebra. Properties The algebra formed as the direct sum of the homogeneous subspaces of even degree of an anticommutative algebra is a subalgebra contained in the centre of , and is thus commutative. An anticommutative algebra over a (commutative) base ring in which 2 is not a zero divisor is alternating. See also Alternating multilinear map Exterior algebra Graded-symmetric algebra References Algebraic geometry