I study the construction of derived Quot schemes in paper “ Shifted symplectic structures on derived Quot-stacks ”(arXiv:1908.03021).

Derived quot stacks are constructed from sheaves of non-positively graded dg algebras in section 3 of the paper.

In particular, I have some question about differentials of the dg algebras.

**Question**

1) The last line of page 14, a differential is constructed by the morphism

$mathcal{V}_j otimes left (bigotimes_{1 leq l leq m+1} mathcal{A}_{i_l} right) otimes (W_i)^vee rightarrow mathcal{V}_j otimes left (bigotimes_{1 leq l leq m} mathcal{A}_{i_l} right) otimes (W_{i+i_{m+1}})^vee$ .

However it seems to me this morphism does not degree 0 morphism and this should be the morphism

$mathcal{V}_j otimes left (bigotimes_{1 leq l leq m+1} mathcal{A}_{i_l} right) otimes (W_{i+i_{m+1}})^vee rightarrow mathcal{V}_j otimes left (bigotimes_{1 leq l leq m} mathcal{A}_{i_l} right) otimes (W_{i})^vee$.

Is this correct ?

2) Does the differential $delta_W$ constructed from the above morphism really become a differential ?

It seems to me that ${delta_W}^2 neq 0$

3) Do we need the differential on $C^{bullet}$ constructed from the multiplication on $oplus mathcal{A_i}$ like that on $B^{bullet}$ on line 10 of page 13.

Thank you !