# Derived quot schemes and the derived linearity locus

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 !

• Your tensor product in the definition of the fibre product should just be $\times$, 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. Feb 27, 2021 at 10:30
• Thank you for pointing it out. \otimes shoud be \times. It is exactly as you said about the expression in Prop 3.8.4. However, this complex seems to have a non-zero term in negative degree. Feb 27, 2021 at 13:41

## 1 Answer

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.