Augmentation (algebra)

{{one source |date=May 2024}}

In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism $A\; \backslash to\; k$, 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

:$A\; \backslash to\; k,\backslash ,\; \backslash sum\; a\_i\; x\_i\; \backslash mapsto\; \backslash sum\; a\_i$

is an augmentation.

If A is a graded algebra which is connected, i.e. $A\_0=k$, then the homomorphism $A\backslash to\; k$ which maps an element to its homogeneous component of degree 0 is an augmentation. For example,

:$k[x]\backslash to\; k,\; \backslash sum\; a\_ix^i\; \backslash mapsto\; a\_0$

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