Augmentation (algebra)
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].
References
- Loday, Jean-Louis; Vallette, Bruno (2012). Algebraic operads. Grundlehren der Mathematischen Wissenschaften. 346. Berlin: Springer-Verlag. p. 2. ISBN 978-3-642-30361-6.
 |  |
