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Let $\mathbb{G}_m= Spec(k[z,z^{-1}])$ be the usual multiplicative group over a field $k$ viewed as a discrete commutative dg-algebra, and let $A$ be some arbitrary commutative dg-algebra concentrated in degrees $\leq 0$. How do I compute the $A$-values of $\mathbb{G}_m$ in the $\infty$-category of derived affine schemes?

On the level of spaces, I'm pretty sure $Maps(Spec(A), \mathbb{G}_m)$ can be described as the union of invertible path components of $Maps(Spec(A), \mathbb{A}^1)=Hom_k(k, A)$, where the space inherits a multiplication from the coalgebra structure on $k$. However, I'd like to have a more concrete description of this. What does an invertible map $k \to A$ look like? I'd also be satisfied with a concrete example, e.g. when $A$ is the Koszul complex or something like that.

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    $\begingroup$ Mapping spaces in the dg setting tend to be messy, but if you're willing to assume $A_0 \to H_0A$ complete (noting that such completion is harmless for Noetherian DGAs), then a model for $\mathbf{R} \mathbb{G}_m(A)$ is as follows. Let $I= \ker(A \to H_0A)$, and $N^{-1}$ be Dold-Kan denormalisation, then take the pushout of the diagram $N^{-1}I \leftarrow I_0 \xrightarrow{\exp} A_0^{\times}$ of simplicial abelian groups. $\endgroup$ – Jon Pridham Oct 26 '19 at 20:29
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Since the polynomial algebra $k[z]$ is free in the infinity category of cdgas, the space of maps $Spec(A) \to \mathbb{A}^1$ is equivalent to the "underlying space" of the cdga $A$. Here "space" means $\infty$-groupoid, or object of the $\infty$-category of spaces.

The underlying space of $A$ is what you get when you forget the addition and multiplication. First, if you forget the multiplication of a cdga, you end up with just a (connective) chain complex. By Dold-Kan that's the same thing as a simplicial abelian group. Forget the addition and you're left with just a simplicial set. The homotopy type of that simplicial set is the "underlying space" of the cdga. As a simple exercise you can work this out for some example, e.g. for $A$ the cdga of functions on the derived self-intersection of the origin in $\mathbb{A}^1$.

Then by the universal property of the localization $k[z,z^{-1}]$, the space of maps $Spec(A) \to \mathbb{G}_m$ is the union of connected components of $Maps(Spec(A), \mathbb{A}^1)$, namely those connected components $f : Spec(A) \to \mathbb{A}^1$ which correspond to a unit of $H^0(A)$. So you can model this by taking the simplicial set $X$ above, which models the underlying space of $A$, and then taking the appropriate simplicial subset. That is, take the pullback $X \times_{N\pi_0(X)} NU$, where $U \subset \pi_0(X)$ is the subset of units and $N$ stands for nerve.

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