nLab augmented algebra

Definition

For $R$ a ring, an associative algebra over $R$ is a ring $A$ equipped with a ring inclusion $R \hookrightarrow A$.

Definition

If the $R$-algebra $A$ is equipped with an $R$-algebra homomorphism the other way around,

$$\epsilon \colon A \to R \,,$$

then it is called an augmented $R$-algebra.

Remark

In Cartan-Eilenberg this is called a supplemented algebra.

Definition

The kernel of $\epsilon$ is called the corresponding augmentation ideal in $A$.

Examples

Example

An augmentation of a bare ring itself, being an associative algebra over the ring of integers $\mathbb{Z}$, is a ring homomorphism to the integers

$$\epsilon \colon R \to \mathbb{Z}$$

Example

Every group algebra $R[G]$ is canonically augmented, the augmentation map being the operation that forms the sum of coefficients of the canonical basis elements.

Example

If $X$ is a variety over an algebraically closed field $k$ and $x\in X(k)$ is a closed point, then the local ring $\mathcal{O}_{X,x}$ naturally has the structure of an augmented $k$-algebra. The augmentation map $\mathcal{O}_{X,x}\rightarrow k$ is the evaluation map, and the augmentation ideal is the maximal ideal of $\mathcal{O}_{X,x}$.

Related concepts

• augmentation

• augmented A-∞ algebra

References

• Henri Cartan, Samuel Eilenberg, Homological algebra