# What is an “algèbre augmentée sur un corps?” (EGA I)

In EGA I, Chapter 0, (1.1.10), Grothendieck is giving examples of terminal objects in different categories. He says "dans la catégorie des algèbres augmentées sur un corps $K$ (où les morphismes sont les homomorphismes d'algèbres compatibles avec les augmentations), $K$ est un objet final."

What is an augmented algebra? I thought he just might mean the category of $K$-algebras, but this doesn't make sense because $K$ is an initial object in this category, not a final one.

If $A$ is a $K$-algebra, an augmentation is a morphism of $K$-algebras from $A$ to $K$. An augmented $K$-algebra is a $K$-algebra $A$ together with an augmentation $\varepsilon \colon A\rightarrow K$. A morphism of augmented $K$-algebras from $\varepsilon \colon A\rightarrow K$ to $\delta\colon B\rightarrow K$ is a morphism $f\colon A\rightarrow B$ of $K$-algebras such that $\delta\circ f=\varepsilon$.
It is then clear that the augmented $K$-algebra $\mathrm{id}\colon K\rightarrow K$ is a final object. Indeed, the augmentation map $\varepsilon \colon A\rightarrow K$ is the unique morphism of augmented $K$-algebras from $\varepsilon \colon A\rightarrow K$ to $\mathrm{id}\colon K\rightarrow K$.
Geometrically, an augmentation $\varepsilon\colon A\rightarrow K$ corresponds to a $K$-valued point of the scheme $\mathrm{Spec}(A)$. Therefore, one thinks of augmented $K$-algebras as $K$-schemes with a base point. The morphisms of augmented $K$-algebras are morphism of $K$-schemes that preserve base points.
An augmented or augmentation $k$-algebra $A$ is just a $k$-algebra with a preferred nontrivial map $A \to k$. One of the canonical examples is a group algebra $k[X]$, where the augmentation map is given by $[g] \to 1$.