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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.

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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.

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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 $[g] \to 1$.

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    $\begingroup$ Thanks for explaining ! Even in French, I do not understand a single word of the sentence ! Cheers $\endgroup$ – Claude Leibovici May 31 '16 at 15:25

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