Derived quot schemes and the derived linearity locus

Question

I am studying derived quot schemes in the paper Ciocan-Fontanine and Kapranov, “Derived Quot schemes” .

On page 36 ~ 37, the derived linearity locus is defined.

Let $S$ be a $\mathbb{Z}_-$-graded dg-scheme, $A$ a $\mathbb{Z}_-$-graded and $M, N$ two quasi-coherent dg-sheaves on $S$. Suppose $M, N$ are locally free and dg-modules over $A \otimes_\mathbb{K} \mathcal{O}_S$ and we have a morphism $f:M \rightarrow N $ of $\mathcal{O}_S$-dg-modules. Then, the derived linearity locus $RLin_A(f)$ is defined to be the fiber product $|R\mathcal{Hom}_{A \otimes \mathcal{O}_S}(M,N)| \times_{|\mathcal{Hom}_{\mathcal{O}_S}(M, N)|} S$.

And, according to Proposition 3.8.4, $RLin_A(f)$ is obtained as $ Cone\{\mathcal{O}_S \overset{\delta f}{\rightarrow} \mathcal{Hom}_{\mathcal{O}_S} (Bar_A(M), N) \}[1] $

, where $Bar_A(M)$ is the Bar resolution of $A$-module $M$ and $\delta f \in \mathcal{Hom}_{\mathcal{O}_S}(A \otimes M , N)$ takes $a \otimes m \mapsto f(a \otimes m) - a f(m)$.

However, I can't understand what this is as a dg-scheme and why this is correct.

I'd be grateful if you could tell me these. Thank you !

Answer 1

Your tensor product in the definition of the fibre product should just be ×, as these are dg schemes, not cdgas. The expression in Prop 3.8.4 is probably intended to be read as taking Spec Symm of the dual.

So I think they've just put the wrong shift in: take -1 instead.