About augmented algebra

Question

An augmented algebra is equipped with a morphism of algebras $ \epsilon: A \to \mathbb{K}$. In this case $ A \equiv \mathbb{K} \oplus \ker(\epsilon)$. May you please clarify what the meaning of $ A \equiv \mathbb{K} \oplus \ker(\epsilon)$? What is the role of augmented algebras in the case of relations between non-unital algebra and unital ones?

Answer 1

An augmentation for a unital algebra $A$ is a morphism of unital algebras $\varepsilon \colon A \rightarrow \mathbb{K}$. In particular, it satisfies $\varepsilon(1_A) = 1_{\mathbb{K}}$ and so it is non-zero and onto. Let's denote the kernel by $\overline{A} = \ker \varepsilon$. The kernel $\overline{A}$ is a two-sided ideal of $A$ calle the augmentation ideal and can be thought of as a not-necessarily unital algebra. We have a vector space (internal) direct-sum decomposition

$$ A = \mathbb{K}\cdot 1_A \oplus \overline{A}. $$

This means that an element $x \in A$ can be written uniquely as $x = c \cdot 1_A + y$ where $c\in \mathbb{K}$ and $y \in \overline{A}$. Explicitly, we have

$$ x = \varepsilon(x) \cdot 1_A + (x - \varepsilon(x) \cdot 1_A). $$

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The relation between not-necessarily unital and augmented algebra is as follows. Given a not-necessarily unital algebra $I$, one can adjoin to $I$ and define $$ I_{+} = \mathbb{K} \oplus I$$ (here, on the right side we have an external direct sum of vector spaces). The multiplication on $I_{+}$ is defined by

$$ (\lambda + a) \cdot (\mu + b) = \lambda \mu + (\lambda b + \mu a + ab) $$ where $\lambda,\mu \in \mathbb{K}$ and $a,b\in I$. This gives $I_{+}$ the structure of a unital algebra with unit $1_{\mathbb{K}} + 0_{I}$ and in addition, it has a "canonical" copy of $I$ sitting inside it. Stated differently, $I_{+}$ is not only a unital algebra but it has a natural augmentation $\varepsilon \colon I_{+} \rightarrow \mathbb{K}$ given by the projection onto the first factor and we have $\overline{I_{+}} = I$.

Hence, we have two operations:

1. Given an augmented unital algebra $(A,\varepsilon)$ over $\mathbb{K}$, we can get a not-necessarily unital algebra $\overline{A}$.
2. Given a not-necessarily unital algebra $I$, we get an augmented unital algebra $\overline{I}_{+}$.

One readily sees that the two operations are functorial and are inverses of each other (up to natural identifications) so we get an equivalence of categories between the category of unital augmented algebras and the category of not-necessarily unital algebras.