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I'm using cohomological gradings. For $C\in k\text-\mathsf{cdga}$ (where $k$ can be taken of characteristic 0), a morphism $C\to A$ to a connective dg-algebra $A\in k\text-\mathsf{cdga}_{\leq0}$ induces a functor $\mathsf{Ho}(C\text-\mathsf{dg\text-mod})$ to $\mathsf{Ho}(A\text-\mathsf{dg\text-mod})$, where the modules are unbounded.

Is there a caracterisation of those $C$ such that the family of functors $$\{-\otimes^{\mathbb L}_CA\ |\ A\in k\text-\mathsf{cdga}_{\leq0},f:C\to A\}$$ is conservative, i.e. such that $M\in C\text-\mathsf{dg\text-mod}$ is uniquely determined (in the derived category) by the data of all those dg-modules $M\otimes^{\mathbb L}_C A$ for all connective commutative $k$-dg-algebra $A$ ?

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  • $\begingroup$ Do you have any non-connective examples? I suggest you look at extensions of $C$ by $C[n]$, which will become trivial after base change for $n \ge 0$. $\endgroup$
    – Jon Pridham
    Commented Jul 12, 2019 at 9:02
  • $\begingroup$ I think I asked the wrong question. I have many difficulties writing down the explicit problem, but as I see that your domain of research includes derived stacks, here is maybe a better question : given an unbounded cdga $B$, there is an associated derived stack $\mathbb R\mathrm{Spec}(C)$, that is of course not a derived affine scheme, but we can hope that for example, its quasi-coherent category may be exactly $C$-$\mathsf{dg}$-$\mathsf{mod}$, under some conditions on $C$. That's the thing I wanted to ask : when does that happen ? $\endgroup$
    – elidiot
    Commented Jul 14, 2019 at 16:03
  • $\begingroup$ What's the relation between $B$ and $C$? $\mathbf{R}Spec$ tends only to be defined for connective things. I think you're aiming for something like Toen's affine stacks. $\endgroup$
    – Jon Pridham
    Commented Jul 14, 2019 at 19:25
  • $\begingroup$ Sorry I wanted to write $C$ everywhere. Yes indeed, but derived affine stacks in a way, because my site is $(\mathsf{cdga}_{\leq0})^{op}$ and the derived affine stack associated to $C\in\mathsf{cdga}$ is $\mathbb R\mathsf{Map}_{\mathsf{cdga}}(C,-)_{|\mathsf{cdga}_{\leq0}}$ (Toens's site is juste $(k-\mathsf{Alg})^{op}$) $\endgroup$
    – elidiot
    Commented Jul 15, 2019 at 8:16
  • $\begingroup$ So for $x$ of degree $2$, you're setting $\mathbf{R}\mathrm{Spec} k[x,x^{-1}]=\emptyset$. I expect you need $C$ to be maps from that presheaf to $\mathbb{A}^1$, which would imply it's a (sifted?) homotopy limit of connective cdgas. $\endgroup$
    – Jon Pridham
    Commented Jul 16, 2019 at 8:05

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