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Let $g$ be a nilpotent $L_{\infty}$-algebra. For every commutative differential graded algebra $A$, one can form the extension $g\otimes A$ and endow it with a nilpotent $L_{\infty}$-algebra structure. Let us denote by $MC(g)$ the set of Maurer-Cartan elements of $g$. My question is the following: is there in the litterature any well known result expressing $MC(g\otimes A)$ in function of $MC(g)$ ? (or maybe it is something really simple that I did not see, in this case I apologize)

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up vote 3 down vote accepted

I don't think such a result exists.

Consider the following $L_\infty$-algebra $\mathfrak g$, which is strict and abelian: $\mathfrak g=\mathfrak g^0=k$ is $1$ dimensional and concentrated in degree $0$.

Let $A=k[\epsilon]$, where $\epsilon$ has degree $1$, with zero differential.

We have that $MC(\mathfrak g)=\{0\}$ and $MC(\mathfrak g\otimes A)=\{0,\epsilon\}$.

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