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The notation originates from the paper virtual Cartier divisor and derived blow up I read recently, in its proof of proposition 3.2.6, there is a derived stack: $\\$$\underline{Pic} ^{\simeq}$: $(DSch^{aff})^{op}\rightarrow Spc$, $\ $ $S\rightarrow (\underline{Pic}(S))^{\simeq}$ where $\underline{Pic}(S)$ is the infinite category of locally free sheaf of rank one and $(-)^{\simeq}$ denotes the operation of discarding the non-invertible morphisms. And it claims that this derived stack is $BG_{m}$.

Here are my questions:

About 'the infinite category of locally free sheaf of rank one'

$\textbf{Question 1}$: how do we define the 'sheaf' on the derived schemes (or derived stacks), and how come the collection of sheaves on the derived schemes become an infinite category?

And in such case, how should we define the quasi-coherence of sheaf on derived scheme? Let's start with the affine derived scheme: it's natural to consider the sheaf on an affine derived scheme $Spec(A)$ with $A\in Fun(Poly^{op},Spc)$ as simplicial ring as an $A$ module.

So here is our $\textbf{Question 2}$: how do we define the infinite category of module over a simplicial ring $A$?

For our affine case, the infinite category of locally free sheaf on Spec(A) with A a simplicial ring ($A\in Fun(Poly^{op},Spc)$) is equivalent to the infinite category of locally free module over $A$.

Thus I want to know how we define the 'modules over a simplicial ring' and how they form an infinite category.

$\textbf{Question 3:}$Is the derived stack $BG_{m}$ defined here is the same thing as the underived version stack $BG_{m}$?

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    $\begingroup$ You might have more luck with that one at MO. $\endgroup$
    – Aphelli
    Commented Jul 10 at 8:13

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