The derived moduli space of stable sheaves

TL;DR

This paper constructs a derived moduli space of stable sheaves on a smooth projective variety using derived algebraic geometry and graded modules, linking GIT stability with sheaf stability.

Contribution

It introduces a new derived scheme of stable sheaves via graded modules and demonstrates the equivalence of GIT stability for modules with sheaf stability.

Findings

Constructed the derived scheme of stable sheaves.

Established the correspondence between GIT stability and sheaf stability.

Provided a new framework for studying moduli of sheaves using derived algebraic geometry.

Abstract

We construct the derived scheme of stable sheaves on a smooth projective variety via derived moduli of finite graded modules over a graded ring. We do this by dividing the derived scheme of actions of Ciocan-Fontanine and Kapranov by a suitable algebraic gauge group. We show that the natural notion of GIT-stability for graded modules reproduces stability for sheaves.