In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism A → k , typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the augmentation is a two-sided ideal called the augmentation ideal of A.
For example, if A = k [ G ] is the group algebra of a group G, then
A → k , ∑ a i x i ↦ ∑ a i is an augmentation.
If A is a graded algebra which is connected, i.e. A 0 = k , then the homomorphism A → k which maps an element to its homogeneous component of degree 0 is an augmentation. For example,
k [ x ] → k , ∑ a i x i ↦ a 0 is an augmentation.
