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 $k[x]$.

References

• Loday, Jean-Louis; Vallette, Bruno (2012). Algebraic operads. Grundlehren der Mathematischen Wissenschaften. Vol. 346. Berlin: Springer-Verlag. p. 2. ISBN 978-3-642-30361-6. Zbl 1260.18001.