# nLab augmented algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 round,

$\epsilon \colon A \to R$

then it is called an augmented 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 $\mathcal{O}_{X,x}$.