nLab complete augmented algebra

Definition

A complete augmented algebra over a field $k$ is an augmented algebra $R$ over $k$ equipped with a decreasing filtration of two-sided ideals

$$R=F_0 R\supset F_1 R\supset F_2 R \supset \cdots,$$

where $1\in F_0 R$ and $F_p R \cdot F_q R \subset F_{p+q} R$, and, furthermore, the map $F_1 R\to F_0 R$ is the augmentation ideal, i.e., the kernel of the augmentation map $R\to k$, the associated graded algebra of $R$ is generated by its degree $1$ component, and the filtration is complete, i.e., the canonical map

$$R\to lim_n F_n R$$

is an isomorphism.

A morphism of complete augmented algebras is a morphism of augmented algebras that is also a morphism of filtered algebras.

Examples

The forgetful functor from complete augmented algebras to augmented algebras admits a left adjoint functor, which sends an augmented algebra $B$ to the complete augmented algebra $\hat B=lim_n B/I^n$, where $I$ is the augmentation ideal of $B$.

In particular, for any set $I$, the algebra $k\langle\langle x_i\rangle\rangle_{i\in I}$ of noncommutative formal power series? in variables $x_i$ ($i\in I$) is a complete augmented algebra, since it is the completion of the algebra of noncommutative polynomials in the same variables. Such algebras are precisely the projective objects in the category of complete augmented algebras. (Corollary A.1.9 in Quillen.)

The quotient $R/J$ of a complete augmentation algebra $R$ by a closed ideal $J$ such that the augmentation map vanishes on $J$ is again a complete augmented algebra.

Properties

Any complete augmented algebra is a quotient of the algebra of noncommutative formal power series? by a closed ideal. The quotient map can be chosen to induce an isomorphism of associated graded algebras in degree 1. (Corollary A.1.7 in Quillen.)

A morphism of complete augmented algebras is an effective epimorphism if and only if it is surjective, equivalently, the degree 1 component of its associated graded is surjective. (Proposition A.1.8 in Quillen.)

The category of complete augmented algebras admits small limits and has a projective generator?, namely, $k\langle\langle x\rangle\rangle$. (Proposition A.1.10 in Quillen.)

Constructions

The forgetful functor from the category of complete augmented algebras to the category of groups that sends a complete augmented algebra $R$ to the group $1+\bar R$, where $\bar R$ denotes the augmentation ideal of $R$, admits a left adjoint functor, which sends a group $G$ to the completion of its group algebra.

The forgetful functor from the category of complete augmented algebras to the category of Lie algebras that sends a complete augmented algebra $R$ to the Lie algebra $\bar R$ with Lie bracket $[x,y]=xy-yx$, where $\bar R$ denotes the augmentation ideal of $R$, admits a left adjoint functor, which sends a Lie algebra $\mathfrak{g}$ to the completion of its universal enveloping algebra.

See (1.12) in Quillen.

Monoidal structure

The category of complete augmented algebras admits a symmetric monoidal structure, given by the completion of the tensor product of the underlying filtered vector spaces.

The associated graded functor is a strong monoidal functor from the category of complete augmented algebras to the category of graded algebras.

The monoidal product has a universal property: morphisms $R\otimes R'\to S$ are in a natural bijection with pairs of morphisms $R\to S$ and $R'\to S$ whose images in $S$ commute.

Related concepts

• augmented algebra

• complete Hopf algebra

• Malcev group

• Malcev completion

References

• Daniel Quillen, Rational homotopy theory, Annals of Mathematics 90:2 (1969), 205. doi.