Derived quot schemes

We define a derived enhancement of the classical quot functor of quotients associated to a coherent sheaf on a nonsingular quasiprojective variety. We prove its representability and show that it has the expected tangent complex. The derived quot scheme of points can be covered by affine charts obtained as spectra of commutative graded differential algebras (cdgas) and we compute an example. As a demonstration of the usefulness of this presentation, we write down an explicit shifted form on this example and conjecture that it agrees with the shifted symplectic structure on the derived stack of perfect complexes constructed by Pantev-To\"en-Vaqui\'e-Vezzosi (arXiv:1111.3209) and Brav-Dyckerhoff (arXiv:1812.11913).