Let $(\mathfrak{g},[-,-])$ be a pronilpotent Lie algebra (considered of degree zero). We can consider $(\mathfrak{g},[-,-])$ as a differential graded Lie algebra endowed with the $0$-differential. Let $A$ be a commutative non-negatively graded differential algebra. Then $A\widehat{\otimes }\mathfrak{g}$ is a differential graded Lie algebra whit bracket


$[a\otimes V,b\otimes W]:=ab\otimes[V,W] $


and where the differential is the obvious differential in the tensor product. We denote such a differential with $d$. Let $MC(A\widehat{\otimes}\mathfrak{g})$ be the set of maurer-Cartan elements, i.e solutions of the equations


$d\alpha +\frac{1}{2}[\alpha,\alpha]=0$.


There are two type of equivalences on this differential graded algebra.


1) **gauge equivalences**


The set $(A\widehat{\otimes }\mathfrak{g})^{0}$ acts on the set of maurer-Cartan elements via the action


$d+\alpha\mapsto e^{u}(d+\alpha)e^{-u}=d+e^{ad\_{u}}\alpha+\frac{1-e^{ad\_{u}}}{ad\_{u}}du$.


Two maurer-Cartan elements $\alpha\_{0},\alpha\_{0}$ are gauge equivalent if for there exists an $u$ as above connecting them.


2)**homotopy equivalence**
Let $\Omega(1)$ be the differential graded algebra of polynomial forms on $[0,1]$. Then $(\Omega(1)\otimes A)\widehat{\otimes }\mathfrak{g}$ is again a differential graded algebra. Two maurer-Cartan elements $\alpha\_{0},\alpha\_{0}$ are homotopy equivalent if there exists a Maurer-Cartan element $\alpha$ in $(\Omega(1)\otimes A)\widehat{\otimes }\mathfrak{g}$ such that $\alpha(0)=\alpha\_{0}$ and $\alpha(1)=\alpha\_{1}$.


Here my question: Under which condition of $A$ is 1 equivalent to 2? Do I need some special condition on $A$?


Edit: here some observations: Notice that if $A$ is finite dimensional then $A\widehat{\otimes }\mathfrak{g}$ is again pronilpotent and the two definitions are equivalent. On the other hand if $A$ is not finite dimensional, then $A\widehat{\otimes }\mathfrak{g}$ can be written as the projective limit of non necessarlily finite dimensional Lie algebras. So my question can be writtens as: what happen if $A$ is not finite-dimensional?



Homotopy equivalence vs gauge equivalence
The two conditions are always equivalent, since $A\widehat{\otimes}\mathfrak{g}$ is complete. This is present in the literature, I think a possible reference is [A tale of three homotopies](https://arxiv.org/abs/1208.4695) by Dotsenko and Poncin, as well as my own article [Representing the Deligne-Hinich-Getzler ∞-groupoid](https://arxiv.org/abs/1702.02529), where this fact is a direct consequence of Corollary 5.3.


