Augmentation (algebra)

In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism ${\displaystyle A\to k}$, 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 ${\displaystyle A=k[G]}$ is the group algebra of a finite group G, then

${\displaystyle A\to k,\,\sum a_{i}x_{i}\mapsto \sum a_{i}}$

is an augmentation.

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

${\displaystyle k[x]\to k,\sum a_{i}x^{i}\mapsto a_{0}}$

is an augmentation on the polynomial ring ${\displaystyle k[x]}$.