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complete augmented algebra

Definition

A complete augmented algebra over a field kk is an augmented algebra RR over kk equipped with a decreasing filtration of two-sided ideals

R=F 0RF 1RF 2R,R=F_0 R\supset F_1 R\supset F_2 R \supset \cdots,

where 1F 0R1\in F_0 R and F pRF qRF p+qRF_p R \cdot F_q R \subset F_{p+q} R, and, furthermore, the map F 1RF 0RF_1 R\to F_0 R is the augmentation ideal, i.e., the kernel of the augmentation map RkR\to k, the associated graded algebra of RR is generated by its degree 11 component, and the filtration is complete, i.e., the canonical map

Rlim nF nRR\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 BB to the complete augmented algebra B^=lim nB/I n\hat B=lim_n B/I^n, where II is the augmentation ideal of BB.

In particular, for any set II, the algebra kx i iIk\langle\langle x_i\rangle\rangle_{i\in I} of noncommutative formal power series? in variables x ix_i (iIi\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 Quillen.)

The quotient R/JR/J of a complete augmentation algebra RR by a closed ideal JJ such that the augmentation map vanishes on JJ 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 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 Quillen.)

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

Constructions

The forgetful functor from the category of complete augmented algebras to the category of groups that sends a complete augmented algebra RR to the group 1+R¯1+\bar R, where R¯\bar R denotes the augmentation ideal of RR, admits a left adjoint functor, which sends a group GG 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 RR to the Lie algebra R¯\bar R with Lie bracket [x,y]=xyyx[x,y]=xy-yx, where R¯\bar R denotes the augmentation ideal of RR, admits a left adjoint functor, which sends a Lie algebra 𝔤\mathfrak{g} to the completion of its universal enveloping algebra.

See (1.12) in Quillen 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 RRSR\otimes R'\to S are in a natural bijection with pairs of morphisms RSR\to S and RSR'\to S whose images in SS commute.

References

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

Last revised on June 15, 2021 at 14:30:12. See the history of this page for a list of all contributions to it.