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The Maurer–Cartan series converges in a complete filtered L∞-algebra

Statement

Let $W$ be the shifted graded space of a complete filtered L∞-algebra over a characteristic-zero field, and let $w \in F^{1} W^{0}$ . Then the series

$\sum_{n \geq 1} \frac{1}{n!} q_{n} (w, \dots, w)$

converges uniquely in the filtration topology to an element of $W^{1}$ .

Facts & Assumptions

Given: A complete filtered L∞-algebra over a characteristic-zero field and an element $w \in F^{1} W^{0}$ .

[F1]

Complete filtered L∞-algebra: the filtration is complete and $q_{n} (F^{r_{1}}, \dots, F^{r_{n}}) \subseteq F^{r_{1} + \dots + r_{n}}$ .

[F2]

Field, The characteristic of a ring: the least $n \geq 1$ with $n \cdot 1_{R} = 0$ when one exists, and $0$ otherwise, and Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring: in characteristic zero, each $n!$ is nonzero in the field and therefore invertible.

Proof

1.1givenF1F2

For every $n \geq 1$ , [F1] with each $r_{i} = 1$ gives $q_{n} (w, \dots, w) \in F^{n} W$ . Since $q_{n}$ has degree $+ 1$ and $w$ has degree zero, each term lies in $F^{n} W^{1}$ . By [F2], division by $n!$ is defined and does not change the filtration level.

2.1F1step 1.1algebra

Fix $r \geq 1$ . If $n \geq r$ , then $F^{n} W \subseteq F^{r} W$ , so every term of index $n \geq r$ vanishes in $W / F^{r} W$ . Therefore the partial sums stabilize modulo $F^{r} W$ once the upper index is at least $r - 1$ .

3.1F1step 2.1construct∎

These stabilized residues are compatible under the quotient maps $W / F^{r + 1} W \to W / F^{r} W$ , because they come from the same nested partial sums. All residues have degree one, so degreewise completeness in [F1] gives a unique element of $W^{1}$ having all these residues. The partial sums converge to it in the filtration topology.

Depends on

Used by

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Sources

Jean-Louis Loday and Bruno Vallette, Algebraic Operads, §13.2 (standard reference, not scraped)