Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

Abstract

We use the Thom–Whitney construction to show that infinitesimal deformations of a coherent sheaf $\mathcal{F}$ are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf $\mathcal{E}n{d}^{\ast }({\mathcal{E}}^{\cdot })$, where ${\mathcal{E}}^{\cdot }$ is any locally free resolution of $\mathcal{F}$. In particular, one recovers the well known fact that the tangent space to ${\mathrm{Def}}_{\mathcal{F}}$ is ${\mathrm{Ext}}^1(\mathcal{F},\mathcal{F})$, and obstructions are contained in ${\mathrm{Ext}}^2(\mathcal{F},\mathcal{F})$. The main tool is the identification of the deformation functor associated with the Thom–Whitney DGLA of a semicosimplicial DGLA ${\mathfrak{g}}^{\Delta }$, whose cohomology is concentrated in nonnegative degrees, with a noncommutative \v{C}ech cohomology-type functor $H_{\mathrm{sc}}^1(\exp {\mathfrak{g}}^{\Delta })$.