Derived quot schemes

TL;DR

This paper introduces a derived version of the classical quot scheme, proves its representability, and explores its geometric and symplectic structures, connecting it to recent developments in derived algebraic geometry.

Contribution

It defines and constructs a derived quot scheme, demonstrating its representability and geometric properties, and relates it to shifted symplectic structures in derived geometry.

Findings

Derived quot scheme is representable and has the expected tangent complex.

The scheme can be covered by affine charts as spectra of cdgas.

An explicit shifted symplectic form is constructed and conjectured to match known structures.

Abstract

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).