Английская Википедия:Augmentation (algebra)

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In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism <math>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>A =k[G]</math> is the group algebra of a finite group G, then

<math>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>A_0=k</math>, then the homomorphism <math>A\to k</math> which maps an element to its homogeneous component of degree 0 is an augmentation. For example,

<math>k[x]\to k, \sum a_ix^i \mapsto a_0</math>

is an augmentation on the polynomial ring <math>k[x]</math>.

References

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