Augmentation (algebra)

In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism $A\to k$, 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 finite group G, then

$A\to k,\sum a_i{x}_i\mapsto \sum a_i$

is an augmentation.

If A is a graded algebra which is connected, i.e. $A_0=k$, then the homomorphism $A\to k$ which maps an element to its homogeneous component of degree 0 is an augmentation. For example,

$k[x]\to k,\sum a_i{x}^i\mapsto a_0$

is an augmentation on the polynomial ring $k[x]$.