Augmentation (algebra)

From HandWiki

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

[math]\displaystyle{ A \to k,\, \sum a_i x_i \mapsto \sum a_i }[/math]

is an augmentation.

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

[math]\displaystyle{ k[x]\to k, \sum a_ix^i \mapsto a_0 }[/math]

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